crypto/ed25519: replace internal/edwards25519 with filippo.io/edwards25519

This change replaces the crypto/ed25519/internal/edwards25519 package
with code from filippo.io/edwards25519, a significantly faster, safer,
well tested (over 1600 lines of new tests, 99% test coverage), and
better documented (600 lines of new comments) implementation.

Some highlights:

* an unsaturated 51-bit limb field implementation optimized for 64-bit
  architectures and math/bits.Mul64 intrinsics

* more efficient variable time scalar multiplication using multi-width
  non-adjacent form with a larger lookup table for fixed-base

* a safe math/big.Int-like API for the Scalar, Point, and field.Element
  types with fully abstracted reduction invariants

* a test suite including a testing/quick fuzzer that explores edge case
  values that would be impossible to hit randomly, and systematic tests
  for arguments and receiver aliasing

* point decoding rules that strictly match the original logic of
  crypto/ed25519/internal/edwards25519, to avoid consensus issues

* AssemblyPolicy-compliant assembly cores for arm64 and amd64, the
  former under 20 lines, and the latter generated by a program based on
  github.com/mmcloughlin/avo that can be reviewed line-by-line against
  the generic implementation

Intel(R) Core(TM) i5-7400 CPU @ 3.00GHz

name              old time/op    new time/op    delta
KeyGeneration-4     59.5µs ± 1%    26.1µs ± 1%  -56.20%  (p=0.000 n=10+10)
NewKeyFromSeed-4    59.3µs ± 1%    25.8µs ± 1%  -56.48%  (p=0.000 n=9+10)
Signing-4           60.4µs ± 1%    31.4µs ± 1%  -48.05%  (p=0.000 n=10+10)
Verification-4       169µs ± 1%      73µs ± 2%  -56.55%  (p=0.000 n=10+10)

Apple M1

name              old time/op    new time/op    delta
KeyGeneration-8     35.1µs ± 0%    20.2µs ± 2%  -42.46%  (p=0.000 n=8+10)
NewKeyFromSeed-8    35.1µs ± 0%    20.0µs ± 1%  -42.93%  (p=0.000 n=8+9)
Signing-8           36.2µs ± 0%    25.6µs ± 1%  -29.25%  (p=0.000 n=8+9)
Verification-8      96.1µs ± 0%    57.6µs ± 1%  -40.14%  (p=0.000 n=10+10)

The code in this CL is a copy of the filippo.io/edwards25519 module at
version v1.0.0-beta.3.0.20210405211453-c6be47d67779 with only the
following functions removed as irrelevant to crypto/ed25519:

- (*Point).BytesMontgomery()
- (*Point).MultByCofactor()
- (*Scalar).Invert()
- (*Point).MultiScalarMult()
- (*Point).VarTimeMultiScalarMult()

This codebase took a long journey outside the standard library before
making its way back here. Its oldest parts started as a faster field
implementation rewrite by George Tankersley almost four years ago,
eventually submitted as CL 71950 but never merged. That code was then
merged into github.com/gtank/ristretto255, which also started as an
internal/edwards25519 fork. There it was worked on by me, George, and
Henry de Valence as a backend for our Go ristretto255 implementation.
Finally, I extracted the edwards25519 code into a reusable package as
filippo.io/edwards25519.

Now, we're ready for the standard library to become the source of truth
for this code again, while filippo.io/edwards25519 will become a
re-packaged and extended version for external use, since we don't want
to expose unsafe curve operations in x/crypto or the standard library.

Submitted under the Google CLA on behalf of:

- Henry de Valence
  https://github.com/gtank/ristretto255/issues/34
- George Tankersley
  https://golang.org/cl/71950
  https://github.com/gtank/ristretto255-private/issues/28
- Luke Champine
  https://github.com/FiloSottile/edwards25519/pull/7
- Adrian Hamelink
  https://github.com/FiloSottile/edwards25519/pull/12

Changes 32506b5 and 18c803c are trivial and don't require a CLA.

The full history of this code since diverging from internal/edwards25519
is available at https://github.com/FiloSottile/edwards25519, and
summarized below.

+ c6be47d - edwards25519: update TestScalarSetBytesWithClamping <Filippo Valsorda>
+ c882e8e - edwards25519: rewrite amd64 assembly with avo <Filippo Valsorda>
+ 8eb02eb - edwards25519: refactor feMulGeneric and feSquareGeneric <Filippo Valsorda>
+ 8afd860 - edwards25519: remove Go 1.12 compatibility hack <Filippo Valsorda>
+ 1765c13 - edwards25519: don't clobber BP in amd64 assembly <Filippo Valsorda>
+ b73a7c8 - edwards25519: fix ScalarMult when receiver is not the identity (FiloSottile/edwards25519#12) <Adrian Hamelink>
+ 32a46d7 - edwards25519: document why this can't implement X25519 <Filippo Valsorda>
+ c547797 - edwards25519: make SqrtRatio slightly more efficient <Filippo Valsorda>
+ 700f4f4 - edwards25519: panic if an uninitialized Point is used <Filippo Valsorda>
+ d791cf8 - edwards25519: use testing.AllocsPerRun for TestAllocations <Filippo Valsorda>
+ 8cc8037 - edwards25519: smooth a couple test coverage rough edges <Filippo Valsorda>
+ 9063a14 - edwards25519: test that operations cause zero heap allocations <Filippo Valsorda>
+ 6944ac7 - edwards25519: relax the limb schedule slightly <Filippo Valsorda>
+ 21ebdac - edwards25519: rewrite carryPropagate in arm64 assembly <Filippo Valsorda>
+ a260082 - edwards25519: merge carryPropagate[12] <Filippo Valsorda>
+ dbe1792 - edwards25519: add TestScalarSetBytesWithClamping <Filippo Valsorda>
+ c1fe95a - edwards25519: add MultByCofactor <Filippo Valsorda>
+ 132d95c - edwards25519: sprinkle on-curve checks around tests <Filippo Valsorda>
+ ffb3e31 - edwards25519: specify the behavior of Invert(0) and I.BytesMontgomery() <Filippo Valsorda>
+ 9e6a931 - edwards25519: add (*Scalar).MultiplyAdd <lukechampine>
+ 3b045f3 - edwards25519: outline (*Point).Bytes (FiloSottile/edwards25519#6) <Luke Champine>
+ ec6f8a6 - edwards25519: make (*Scalar).SetCanonicalBytes return the receiver <Filippo Valsorda>
+ 77d7b31 - edwards25519: add (*Point).BytesMontgomery <Filippo Valsorda>
+ 6e8d645 - edwards25519: implement (*Point).Bytes and (*Point).SetBytes <Filippo Valsorda>
+ 1c833da - edwards25519: clarify ScalarBaseMult docs <Filippo Valsorda>
+ 3a13cf1 - edwards25519: apply gc build tag <Filippo Valsorda>
+ 90c35a7 - edwards25519: hide FieldElement and (*Point).ExtendedCoords <Filippo Valsorda>
+ 498fb1e - edwards25519: replace FillBytes with Bytes, again <Filippo Valsorda>
+ 9c7303a - edwards25519: remove (*Point).Identity and (*Point).Generator <Filippo Valsorda>
+ 2e52ce2 - edwards25519: drop unused (*Scalar).Zero <Filippo Valsorda>
+ 7c14a36 - edwards25519: rename FromBytes to SetBytes <Filippo Valsorda>
+ e3d0e45 - edwards25519: ensure only test files import math/big <Filippo Valsorda>
+ daa2507 - edwards25519: minor doc and string touch-ups <Filippo Valsorda>
+ e8698cd - edwards25519: implement (*Scalar).FromBytesWithClamping <Filippo Valsorda>
+ f28d75a - edwards25519: change constructors <Filippo Valsorda>
+ 36d8598 - edwards25519: test the invariant that Scalars are always reduced <Filippo Valsorda>
+ feed48c - edwards25519: cleanup the FieldElement API <Filippo Valsorda>
+ f6ee187 - edwards25519: make Point opaque <Filippo Valsorda>
+ 176388b - edwards25519: cleanup Scalar API to match ristretto255 <Filippo Valsorda>
+ c5c2e9e - edwards25519: rename ProjP3 to Point and unexport other point types <Filippo Valsorda>
+ 8542076 - edwards25519: add Scalar aliasing test <Filippo Valsorda>
+ 1a86a9c - edwards25519: make Scalar opaque <Filippo Valsorda>
+ 07a7683 - edwards25519: hide some more exposed symbols <Filippo Valsorda>
+ d3569cb - all: flatten the package and make FieldElement opaque <Filippo Valsorda>
+ 6f5f582 - all: expose edwards25519, base, and scalar packages <Filippo Valsorda>
+ 7ab4a68 - all: ensure compatibility with older Go versions <Filippo Valsorda>
+ e9b8baa - internal/radix51: implement (*FieldElement).Mul32 <Filippo Valsorda>
+ eac4de5 - internal/radix51: restructure according to golang.org/wiki/TargetSpecific <Filippo Valsorda>
+ 32506b5 - internal/radix51: fix !amd64 build (lightReduce -> carryPropagate) (gtank/ristretto255#29) <Sunny Aggarwal>
+ d64d989 - internal/scalar: fix FromUniformBytes <Filippo Valsorda>
+ 044bb44 - internal/scalar: address review comments <Filippo Valsorda>
+ 7dba54f - all: apply suggestions from code review <Filippo Valsorda>
+ 94bd1d9 - ristretto255: expose scalar multiplication APIs <Filippo Valsorda>
+ 5bd5476 - internal/edwards25519: fix shadowing of B in TestAddSubNegOnBasePoint <Filippo Valsorda>
+ 66bf647 - internal/scalar: replace FromBytes/IsCanonical with FromUniformBytes/FromCanonicalBytes <Filippo Valsorda>
+ 024f3f7 - internal/edwards25519,internal/scalar: apply some Go style touches <Filippo Valsorda>
+ 5e0c5c6 - internal/scalar: add scalar inversion <Henry de Valence>
+ 74fd625 - internal/ed25519: rearrange VartimeDoubleBaseMul args <Henry de Valence>
+ 81ae7ea - internal/ed25519: add benchmarks for scalar mul <Henry de Valence>
+ 9f1f939 - internal/ed25519: add variable-time multiscalar mul <Henry de Valence>
+ 7a96974 - internal/ed25519: add vartime double-base scmul <Henry de Valence>
+ 2bc256c - internal/ed25519: add precomputed NAF table for basepoint <Henry de Valence>
+ a0f0b96 - internal/ed25519: lower quickcheck size for point ops <Henry de Valence>
+ 2f385a1 - internal/ed25519: implement MultiscalarMul <Henry de Valence>
+ 8ae211b - internal/ed25519: implement BasepointMul <Henry de Valence>
+ 7b4858d - internal/ed25519: extract common test variables <Henry de Valence>
+ 16e7c48 - internal/ed25519: add a basepoint multiple table. <Henry de Valence>
+ 988e521 - internal/ed25519: add constant-time variable-base scmul. <Henry de Valence>
+ b695f6b - internal/ed25519: move basepoint constant & correct it <Henry de Valence>
+ ddd014e - internal/scalar: fix high bit check <Henry de Valence>
+ c88ea89 - internal/scalar: make casts clearer <Henry de Valence>
+ b75f989 - internal/scalar: add invariant checks on Scalar digits <Henry de Valence>
+ 36216ca - internal/scalar: use one scMulAdd for Sub <Henry de Valence>
+ 8bf40f3 - internal/scalar: fix constant-time signed radix 16 implementation <Henry de Valence>
+ e6d9ef6 - Update internal/radix51/fe_test.go <Filippo Valsorda>
+ 3aa63de - Update internal/radix51/fe_test.go <Filippo Valsorda>
+ 3e66ff0 - Update internal/radix51/fe_test.go <Filippo Valsorda>
+ 94e6c15 - internal/ed25519: add TODO note and doc ref <Henry de Valence>
+ 3647548 - internal/ed25519: rename twoD to D2 <Henry de Valence>
+ 1cf853c - internal/ed25519: add lookup tables for scalar mul. <Henry de Valence>
+ 3af304a - internal/radix51: add a conditional swap <Henry de Valence>
+ 4673217 - ristretto255: use multi-model arithmetic <Henry de Valence>
+ cca757a - internal/ed25519: remove single-model code <Henry de Valence>
+ d26e77b - internal/ed25519: add addition for Edwards points <Henry de Valence>
+ e0fbb35 - internal/ed25519: use twoD <Henry de Valence>
+ fd9b37b - internal/ed25519: add tests for multi-model point types. <Henry de Valence>
+ dacabb0 - internal/ed25519: add multi-model point types. <Henry de Valence>
+ dddc72e - internal/scalar: add constant-time signed radix 16 <Henry de Valence>
+ 92cdb35 - internal/scalar: add non-adjacent form <Henry de Valence>
+ d147963 - internal/scalar: don't zero memory that is about to be copied over <George Tankersley>
+ 8da186c - internal/scalar: add scalar field implementation <George Tankersley>
+ f38e583 - internal/radix51: add a "weird" testing/quick generation strategy <Filippo Valsorda>
+ 6454f61 - Move comment inside function <Henry de Valence>
+ 1983365 - implement Add, Sub, Neg for ed25519 and ristretto255 points. <Henry de Valence>
+ 9f25562 - internal/group: rename to internal/edwards25519 <Filippo Valsorda>
+ 48e66d3 - internal/group: restore ScalarMult code <Filippo Valsorda>
+ 0078d66 - internal/radix51: rename lightReduce to carryPropagate and touch up docs <Filippo Valsorda>
+ 05f4107 - internal/radix51: add benchmarks <Filippo Valsorda>
+ fd36334 - internal/radix51: test that operations don't exceed bounds <Filippo Valsorda>
+ 703421d - internal/radix51: make Generate produce random light-reduced elements <Filippo Valsorda>
+ f8d8297 - internal/radix51: simplify lightReduce <Filippo Valsorda>
+ 413120f - internal/radix51: minor tests cleanup <Filippo Valsorda>
+ abc8c5a - internal/radix51: make reduction an invariant and unexport Reduce <Filippo Valsorda>
+ 4fd198d - internal/radix51: actually apply go:noescape <Filippo Valsorda>
+ 18c803c - all: fix typos <Dimitris Apostolou>
+ bbfe059 - internal/radix51: test field encoding roundtrip with fixed vectors <George Tankersley>
+ c428b18 - internal/radix51: rename AppendBytes to Bytes <Filippo Valsorda>
+ c59bc1a - internal/radix51: rewrite FromBytes and AppendBytes with encoding/binary <Filippo Valsorda>
+ 57c0cd5 - internal/radix51: add docs and some light readability refactors <Filippo Valsorda>
+ cb1b734 - internal/radix51: remove unused (and a bit broken) SetInt <Filippo Valsorda>
+ beb8abd - internal/radix51: refactor ToBig and FromBig <Filippo Valsorda>
+ 87c0a53 - internal/radix51: replace ToBytes with AppendBytes <Filippo Valsorda>
+ b7e1e45 - internal/radix51: fix aliasing bug in CondNeg (gtank/ristretto255#21) <George Tankersley>
+ ed3748d - internal/radix51: actually, uhm, check the result of TestAliasing <Filippo Valsorda>
+ ec0e293 - radix51: change API of FromBytes and ToBytes to use slices <George Tankersley>
+ 29f6815 - internal/radix51: test all combinations of argument and receiver aliasing <Filippo Valsorda>
+ cd53d90 - internal/radix51: add property-based tests that multiplication distributes over addition <Henry de Valence>
+ c3bc45f - radix51: use go1.12 intrinsics for 128-bit multiplications <George Tankersley>
+ 7e7043e - internal/radix51: define a mask64Bits constant <Filippo Valsorda>
+ 4fdd06d - internal/group: set Z to 1, not 0 in FromAffine <Filippo Valsorda>
+ ffa7be7 - internal/group: fix typo <Filippo Valsorda>
+ 1f452ac - internal/group: derive twoD from D <Filippo Valsorda>
+ 2424c78 - internal/radix51: add MinusOne <Filippo Valsorda>
+ 76978fc - internal/group: make conversion APIs caller-allocated <Filippo Valsorda>
+ d17d202 - internal/group: rewrite DoubleZ1 because stack is cheaper than mental state <Filippo Valsorda>
+ 72b97c1 - internal: make all APIs chainable <Filippo Valsorda>
+ 993d979 - internal/radix51: make all APIs not consider the receiver an input <Filippo Valsorda>
+ b2a1d7d - all: refactor field API to be methods based <Filippo Valsorda>
+ cdf9b90 - internal/radix51: add constant time field operations <Filippo Valsorda>
+ e490a48 - internal/radix51: remove FeEqual <Filippo Valsorda>
+ 2de114c - internal/radix51: remove FeCSwap <Filippo Valsorda>
+ 08b80c1 - make things more generally presentable <George Tankersley>
+ 2178536 - Cache the field representation of d <George Tankersley>
+ 4135059 - Remove 32-bit code and update license. <George Tankersley>
+ 5d95cb3 - Use Bits() for FeToBig. <George Tankersley>
+ 146e33c - Implement ScalarMult using Montgomery pattern and dedicated extended-coordinates doubling. This will be slow. <George Tankersley>
+ 12a673a - use faster FeFromBig & a horrible assortment of other random changes <George Tankersley>
+ 901f40c - group logic WIP <George Tankersley>
+ a9c89cd - add equality for field elements <George Tankersley>
+ 214873b - Add radix51 FieldElement implementation <George Tankersley>
+ 8fd5cae - Implement an elliptic.Curve for ed25519 <George Tankersley>

Change-Id: Ifbcdd13e8b6304f9906c0ef2b73f1fdc493a7dfa
Co-authored-by: George Tankersley <george.tankersley@gmail.com>
Co-authored-by: Henry de Valence <hdevalence@hdevalence.ca>
Reviewed-on: https://go-review.googlesource.com/c/go/+/276272
Run-TryBot: Filippo Valsorda <filippo@golang.org>
TryBot-Result: Go Bot <gobot@golang.org>
Trust: Filippo Valsorda <filippo@golang.org>
Trust: Katie Hockman <katie@golang.org>
Reviewed-by: Katie Hockman <katie@golang.org>

diff --git a/src/cmd/compile/internal/types2/stdlib_test.go b/src/cmd/compile/internal/types2/stdlib_test.go
index c04f5e1c46d6b8ec124d9738e0a4fed93297a6d3..7d6753e5c25b168c71ff80d5402bedc4a8279b20 100644 (file)
--- a/src/cmd/compile/internal/types2/stdlib_test.go
+++ b/src/cmd/compile/internal/types2/stdlib_test.go
@@ -301,6 +301,13 @@ func (w *walker) walk(dir string) {
                return
        }
 
+       // skip submodules, which might not be vendored
+       for _, f := range files {
+               if f.Name() == "go.mod" {
+                       return
+               }
+       }
+
        // apply pkgh to the files in directory dir
        // but ignore files directly under $GOROOT/src (might be temporary test files).
        if dir != filepath.Join(runtime.GOROOT(), "src") {

diff --git a/src/cmd/internal/moddeps/moddeps_test.go b/src/cmd/internal/moddeps/moddeps_test.go
index 62687403ecce84c31adb420fd0147226eec2764b..ba574f400495694a6d43bb68129c86d1ab8c4388 100644 (file)
--- a/src/cmd/internal/moddeps/moddeps_test.go
+++ b/src/cmd/internal/moddeps/moddeps_test.go
@@ -422,6 +422,12 @@ func findGorootModules(t *testing.T) []gorootModule {
                                // running time of this test anyway.)
                                return filepath.SkipDir
                        }
+                       if strings.HasPrefix(info.Name(), "_") || strings.HasPrefix(info.Name(), ".") {
+                               // _ and . prefixed directories can be used for internal modules
+                               // without a vendor directory that don't contribute to the build
+                               // but might be used for example as code generators.
+                               return filepath.SkipDir
+                       }
                        if info.IsDir() || info.Name() != "go.mod" {
                                return nil
                        }

diff --git a/src/crypto/ed25519/ed25519.go b/src/crypto/ed25519/ed25519.go
index 6f59bb5cffbe9d2632cb96549cfe7903789a10fb..09c5269d0c6a033047011c82d073a1fb65d145f9 100644 (file)
--- a/src/crypto/ed25519/ed25519.go
+++ b/src/crypto/ed25519/ed25519.go
@@ -12,9 +12,6 @@
 // 8032 private key as the “seed”.
 package ed25519
 
-// This code is a port of the public domain, “ref10” implementation of ed25519
-// from SUPERCOP.
-
 import (
        "bytes"
        "crypto"
@@ -128,20 +125,14 @@ func newKeyFromSeed(privateKey, seed []byte) {
                panic("ed25519: bad seed length: " + strconv.Itoa(l))
        }
 
-       digest := sha512.Sum512(seed)
-       digest[0] &= 248
-       digest[31] &= 127
-       digest[31] |= 64
+       h := sha512.Sum512(seed)
+       s := edwards25519.NewScalar().SetBytesWithClamping(h[:32])
+       A := (&edwards25519.Point{}).ScalarBaseMult(s)
 
-       var A edwards25519.ExtendedGroupElement
-       var hBytes [32]byte
-       copy(hBytes[:], digest[:])
-       edwards25519.GeScalarMultBase(&A, &hBytes)
-       var publicKeyBytes [32]byte
-       A.ToBytes(&publicKeyBytes)
+       publicKey := A.Bytes()
 
        copy(privateKey, seed)
-       copy(privateKey[32:], publicKeyBytes[:])
+       copy(privateKey[32:], publicKey)
 }
 
 // Sign signs the message with privateKey and returns a signature. It will
@@ -158,44 +149,33 @@ func sign(signature, privateKey, message []byte) {
        if l := len(privateKey); l != PrivateKeySize {
                panic("ed25519: bad private key length: " + strconv.Itoa(l))
        }
+       seed, publicKey := privateKey[:SeedSize], privateKey[SeedSize:]
+
+       h := sha512.Sum512(seed)
+       s := edwards25519.NewScalar().SetBytesWithClamping(h[:32])
+       prefix := h[32:]
+
+       mh := sha512.New()
+       mh.Write(prefix)
+       mh.Write(message)
+       messageDigest := make([]byte, 0, sha512.Size)
+       messageDigest = mh.Sum(messageDigest)
+       r := edwards25519.NewScalar().SetUniformBytes(messageDigest)
+
+       R := (&edwards25519.Point{}).ScalarBaseMult(r)
 
-       h := sha512.New()
-       h.Write(privateKey[:32])
-
-       var digest1, messageDigest, hramDigest [64]byte
-       var expandedSecretKey [32]byte
-       h.Sum(digest1[:0])
-       copy(expandedSecretKey[:], digest1[:])
-       expandedSecretKey[0] &= 248
-       expandedSecretKey[31] &= 63
-       expandedSecretKey[31] |= 64
-
-       h.Reset()
-       h.Write(digest1[32:])
-       h.Write(message)
-       h.Sum(messageDigest[:0])
-
-       var messageDigestReduced [32]byte
-       edwards25519.ScReduce(&messageDigestReduced, &messageDigest)
-       var R edwards25519.ExtendedGroupElement
-       edwards25519.GeScalarMultBase(&R, &messageDigestReduced)
-
-       var encodedR [32]byte
-       R.ToBytes(&encodedR)
-
-       h.Reset()
-       h.Write(encodedR[:])
-       h.Write(privateKey[32:])
-       h.Write(message)
-       h.Sum(hramDigest[:0])
-       var hramDigestReduced [32]byte
-       edwards25519.ScReduce(&hramDigestReduced, &hramDigest)
-
-       var s [32]byte
-       edwards25519.ScMulAdd(&s, &hramDigestReduced, &expandedSecretKey, &messageDigestReduced)
-
-       copy(signature[:], encodedR[:])
-       copy(signature[32:], s[:])
+       kh := sha512.New()
+       kh.Write(R.Bytes())
+       kh.Write(publicKey)
+       kh.Write(message)
+       hramDigest := make([]byte, 0, sha512.Size)
+       hramDigest = kh.Sum(hramDigest)
+       k := edwards25519.NewScalar().SetUniformBytes(hramDigest)
+
+       S := edwards25519.NewScalar().MultiplyAdd(k, s, r)
+
+       copy(signature[:32], R.Bytes())
+       copy(signature[32:], S.Bytes())
 }
 
 // Verify reports whether sig is a valid signature of message by publicKey. It
@@ -209,38 +189,27 @@ func Verify(publicKey PublicKey, message, sig []byte) bool {
                return false
        }
 
-       var A edwards25519.ExtendedGroupElement
-       var publicKeyBytes [32]byte
-       copy(publicKeyBytes[:], publicKey)
-       if !A.FromBytes(&publicKeyBytes) {
+       A, err := (&edwards25519.Point{}).SetBytes(publicKey)
+       if err != nil {
                return false
        }
-       edwards25519.FeNeg(&A.X, &A.X)
-       edwards25519.FeNeg(&A.T, &A.T)
-
-       h := sha512.New()
-       h.Write(sig[:32])
-       h.Write(publicKey[:])
-       h.Write(message)
-       var digest [64]byte
-       h.Sum(digest[:0])
-
-       var hReduced [32]byte
-       edwards25519.ScReduce(&hReduced, &digest)
-
-       var R edwards25519.ProjectiveGroupElement
-       var s [32]byte
-       copy(s[:], sig[32:])
-
-       // https://tools.ietf.org/html/rfc8032#section-5.1.7 requires that s be in
-       // the range [0, order) in order to prevent signature malleability.
-       if !edwards25519.ScMinimal(&s) {
+
+       kh := sha512.New()
+       kh.Write(sig[:32])
+       kh.Write(publicKey)
+       kh.Write(message)
+       hramDigest := make([]byte, 0, sha512.Size)
+       hramDigest = kh.Sum(hramDigest)
+       k := edwards25519.NewScalar().SetUniformBytes(hramDigest)
+
+       S, err := edwards25519.NewScalar().SetCanonicalBytes(sig[32:])
+       if err != nil {
                return false
        }
 
-       edwards25519.GeDoubleScalarMultVartime(&R, &hReduced, &A, &s)
+       // [S]B = R + [k]A --> [k](-A) + [S]B = R
+       minusA := (&edwards25519.Point{}).Negate(A)
+       R := (&edwards25519.Point{}).VarTimeDoubleScalarBaseMult(k, minusA, S)
 
-       var checkR [32]byte
-       R.ToBytes(&checkR)
-       return bytes.Equal(sig[:32], checkR[:])
+       return bytes.Equal(sig[:32], R.Bytes())
 }

diff --git a/src/crypto/ed25519/ed25519_test.go b/src/crypto/ed25519/ed25519_test.go
index adb09e409a502c58a4f7910c4308a18708e5f9cc..ba9970e23e5dae1489515e8c2beae8008a125172 100644 (file)
--- a/src/crypto/ed25519/ed25519_test.go
+++ b/src/crypto/ed25519/ed25519_test.go
@@ -9,7 +9,6 @@ import (
        "bytes"
        "compress/gzip"
        "crypto"
-       "crypto/ed25519/internal/edwards25519"
        "crypto/rand"
        "encoding/hex"
        "os"
@@ -26,24 +25,6 @@ func (zeroReader) Read(buf []byte) (int, error) {
        return len(buf), nil
 }
 
-func TestUnmarshalMarshal(t *testing.T) {
-       pub, _, _ := GenerateKey(rand.Reader)
-
-       var A edwards25519.ExtendedGroupElement
-       var pubBytes [32]byte
-       copy(pubBytes[:], pub)
-       if !A.FromBytes(&pubBytes) {
-               t.Fatalf("ExtendedGroupElement.FromBytes failed")
-       }
-
-       var pub2 [32]byte
-       A.ToBytes(&pub2)
-
-       if pubBytes != pub2 {
-               t.Errorf("FromBytes(%v)->ToBytes does not round-trip, got %x\n", pubBytes, pub2)
-       }
-}
-
 func TestSignVerify(t *testing.T) {
        var zero zeroReader
        public, private, _ := GenerateKey(zero)
@@ -204,6 +185,21 @@ func TestMalleability(t *testing.T) {
        }
 }
 
+func TestAllocations(t *testing.T) {
+       if allocs := testing.AllocsPerRun(100, func() {
+               seed := make([]byte, SeedSize)
+               message := []byte("Hello, world!")
+               priv := NewKeyFromSeed(seed)
+               pub := priv.Public().(PublicKey)
+               signature := Sign(priv, message)
+               if !Verify(pub, message, signature) {
+                       t.Fatal("signature didn't verify")
+               }
+       }); allocs > 0 {
+               t.Errorf("expected zero allocations, got %0.1v", allocs)
+       }
+}
+
 func BenchmarkKeyGeneration(b *testing.B) {
        var zero zeroReader
        for i := 0; i < b.N; i++ {
@@ -215,7 +211,6 @@ func BenchmarkKeyGeneration(b *testing.B) {
 
 func BenchmarkNewKeyFromSeed(b *testing.B) {
        seed := make([]byte, SeedSize)
-       b.ReportAllocs()
        for i := 0; i < b.N; i++ {
                _ = NewKeyFromSeed(seed)
        }
@@ -228,7 +223,6 @@ func BenchmarkSigning(b *testing.B) {
                b.Fatal(err)
        }
        message := []byte("Hello, world!")
-       b.ReportAllocs()
        b.ResetTimer()
        for i := 0; i < b.N; i++ {
                Sign(priv, message)

diff --git a/src/crypto/ed25519/internal/edwards25519/const.go b/src/crypto/ed25519/internal/edwards25519/const.go
deleted file mode 100644 (file)
index e39f086..0000000
--- a/src/crypto/ed25519/internal/edwards25519/const.go
+++ /dev/null
@@ -1,1422 +0,0 @@
-// Copyright 2016 The Go Authors. All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-package edwards25519
-
-// These values are from the public domain, “ref10” implementation of ed25519
-// from SUPERCOP.
-
-// d is a constant in the Edwards curve equation.
-var d = FieldElement{
-       -10913610, 13857413, -15372611, 6949391, 114729, -8787816, -6275908, -3247719, -18696448, -12055116,
-}
-
-// d2 is 2*d.
-var d2 = FieldElement{
-       -21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199,
-}
-
-// SqrtM1 is the square-root of -1 in the field.
-var SqrtM1 = FieldElement{
-       -32595792, -7943725, 9377950, 3500415, 12389472, -272473, -25146209, -2005654, 326686, 11406482,
-}
-
-// A is a constant in the Montgomery-form of curve25519.
-var A = FieldElement{
-       486662, 0, 0, 0, 0, 0, 0, 0, 0, 0,
-}
-
-// bi contains precomputed multiples of the base-point. See the Ed25519 paper
-// for a discussion about how these values are used.
-var bi = [8]PreComputedGroupElement{
-       {
-               FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605},
-               FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378},
-               FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546},
-       },
-       {
-               FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024},
-               FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574},
-               FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357},
-       },
-       {
-               FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380},
-               FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306},
-               FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942},
-       },
-       {
-               FieldElement{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766},
-               FieldElement{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701},
-               FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300},
-       },
-       {
-               FieldElement{-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211, -1361450, -13062696, 13821877},
-               FieldElement{-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028, -7212327, 18853322, -14220951},
-               FieldElement{4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358, -10431137, 2207753, -3209784},
-       },
-       {
-               FieldElement{-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364, -663000, -31111463, -16132436},
-               FieldElement{25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789, 15725684, 171356, 6466918},
-               FieldElement{23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339, -14088058, -30714912, 16193877},
-       },
-       {
-               FieldElement{-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398, 4729455, -18074513, 9256800},
-               FieldElement{-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405, 9761698, -19827198, 630305},
-               FieldElement{-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551, -15960994, -2449256, -14291300},
-       },
-       {
-               FieldElement{-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575, 15033784, 25105118, -7894876},
-               FieldElement{-24326370, 15950226, -31801215, -14592823, -11662737, -5090925, 1573892, -2625887, 2198790, -15804619},
-               FieldElement{-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022, -16236442, -32461234, -12290683},
-       },
-}
-
-// base contains precomputed multiples of the base-point. See the Ed25519 paper
-// for a discussion about how these values are used.
-var base = [32][8]PreComputedGroupElement{
-       {
-               {
-                       FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605},
-                       FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378},
-                       FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546},
-               },
-               {
-                       FieldElement{-12815894, -12976347, -21581243, 11784320, -25355658, -2750717, -11717903, -3814571, -358445, -10211303},
-                       FieldElement{-21703237, 6903825, 27185491, 6451973, -29577724, -9554005, -15616551, 11189268, -26829678, -5319081},
-                       FieldElement{26966642, 11152617, 32442495, 15396054, 14353839, -12752335, -3128826, -9541118, -15472047, -4166697},
-               },
-               {
-                       FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024},
-                       FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574},
-                       FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357},
-               },
-               {
-                       FieldElement{-17036878, 13921892, 10945806, -6033431, 27105052, -16084379, -28926210, 15006023, 3284568, -6276540},
-                       FieldElement{23599295, -8306047, -11193664, -7687416, 13236774, 10506355, 7464579, 9656445, 13059162, 10374397},
-                       FieldElement{7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664, -3839045, -641708, -101325},
-               },
-               {
-                       FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380},
-                       FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306},
-                       FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942},
-               },
-               {
-                       FieldElement{-15371964, -12862754, 32573250, 4720197, -26436522, 5875511, -19188627, -15224819, -9818940, -12085777},
-                       FieldElement{-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240, -15689887, 1762328, 14866737},
-                       FieldElement{-18199695, -15951423, -10473290, 1707278, -17185920, 3916101, -28236412, 3959421, 27914454, 4383652},
-               },
-               {
-                       FieldElement{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766},
-                       FieldElement{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701},
-                       FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300},
-               },
-               {
-                       FieldElement{14499471, -2729599, -33191113, -4254652, 28494862, 14271267, 30290735, 10876454, -33154098, 2381726},
-                       FieldElement{-7195431, -2655363, -14730155, 462251, -27724326, 3941372, -6236617, 3696005, -32300832, 15351955},
-                       FieldElement{27431194, 8222322, 16448760, -3907995, -18707002, 11938355, -32961401, -2970515, 29551813, 10109425},
-               },
-       },
-       {
-               {
-                       FieldElement{-13657040, -13155431, -31283750, 11777098, 21447386, 6519384, -2378284, -1627556, 10092783, -4764171},
-                       FieldElement{27939166, 14210322, 4677035, 16277044, -22964462, -12398139, -32508754, 12005538, -17810127, 12803510},
-                       FieldElement{17228999, -15661624, -1233527, 300140, -1224870, -11714777, 30364213, -9038194, 18016357, 4397660},
-               },
-               {
-                       FieldElement{-10958843, -7690207, 4776341, -14954238, 27850028, -15602212, -26619106, 14544525, -17477504, 982639},
-                       FieldElement{29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899, -4120128, -21047696, 9934963},
-                       FieldElement{5793303, 16271923, -24131614, -10116404, 29188560, 1206517, -14747930, 4559895, -30123922, -10897950},
-               },
-               {
-                       FieldElement{-27643952, -11493006, 16282657, -11036493, 28414021, -15012264, 24191034, 4541697, -13338309, 5500568},
-                       FieldElement{12650548, -1497113, 9052871, 11355358, -17680037, -8400164, -17430592, 12264343, 10874051, 13524335},
-                       FieldElement{25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038, 5080568, -22528059, 5376628},
-               },
-               {
-                       FieldElement{-26088264, -4011052, -17013699, -3537628, -6726793, 1920897, -22321305, -9447443, 4535768, 1569007},
-                       FieldElement{-2255422, 14606630, -21692440, -8039818, 28430649, 8775819, -30494562, 3044290, 31848280, 12543772},
-                       FieldElement{-22028579, 2943893, -31857513, 6777306, 13784462, -4292203, -27377195, -2062731, 7718482, 14474653},
-               },
-               {
-                       FieldElement{2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965, -7236665, 24316168, -5253567},
-                       FieldElement{13741529, 10911568, -33233417, -8603737, -20177830, -1033297, 33040651, -13424532, -20729456, 8321686},
-                       FieldElement{21060490, -2212744, 15712757, -4336099, 1639040, 10656336, 23845965, -11874838, -9984458, 608372},
-               },
-               {
-                       FieldElement{-13672732, -15087586, -10889693, -7557059, -6036909, 11305547, 1123968, -6780577, 27229399, 23887},
-                       FieldElement{-23244140, -294205, -11744728, 14712571, -29465699, -2029617, 12797024, -6440308, -1633405, 16678954},
-                       FieldElement{-29500620, 4770662, -16054387, 14001338, 7830047, 9564805, -1508144, -4795045, -17169265, 4904953},
-               },
-               {
-                       FieldElement{24059557, 14617003, 19037157, -15039908, 19766093, -14906429, 5169211, 16191880, 2128236, -4326833},
-                       FieldElement{-16981152, 4124966, -8540610, -10653797, 30336522, -14105247, -29806336, 916033, -6882542, -2986532},
-                       FieldElement{-22630907, 12419372, -7134229, -7473371, -16478904, 16739175, 285431, 2763829, 15736322, 4143876},
-               },
-               {
-                       FieldElement{2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801, -14594663, 23527084, -16458268},
-                       FieldElement{33431127, -11130478, -17838966, -15626900, 8909499, 8376530, -32625340, 4087881, -15188911, -14416214},
-                       FieldElement{1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055, 4357868, -4774191, -16323038},
-               },
-       },
-       {
-               {
-                       FieldElement{6721966, 13833823, -23523388, -1551314, 26354293, -11863321, 23365147, -3949732, 7390890, 2759800},
-                       FieldElement{4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353, -4264057, 1244380, -12919645},
-                       FieldElement{-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413, 9208236, 15886429, 16489664},
-               },
-               {
-                       FieldElement{1996075, 10375649, 14346367, 13311202, -6874135, -16438411, -13693198, 398369, -30606455, -712933},
-                       FieldElement{-25307465, 9795880, -2777414, 14878809, -33531835, 14780363, 13348553, 12076947, -30836462, 5113182},
-                       FieldElement{-17770784, 11797796, 31950843, 13929123, -25888302, 12288344, -30341101, -7336386, 13847711, 5387222},
-               },
-               {
-                       FieldElement{-18582163, -3416217, 17824843, -2340966, 22744343, -10442611, 8763061, 3617786, -19600662, 10370991},
-                       FieldElement{20246567, -14369378, 22358229, -543712, 18507283, -10413996, 14554437, -8746092, 32232924, 16763880},
-                       FieldElement{9648505, 10094563, 26416693, 14745928, -30374318, -6472621, 11094161, 15689506, 3140038, -16510092},
-               },
-               {
-                       FieldElement{-16160072, 5472695, 31895588, 4744994, 8823515, 10365685, -27224800, 9448613, -28774454, 366295},
-                       FieldElement{19153450, 11523972, -11096490, -6503142, -24647631, 5420647, 28344573, 8041113, 719605, 11671788},
-                       FieldElement{8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916, -15266516, 27000813, -10195553},
-               },
-               {
-                       FieldElement{-15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065, 5336097, 6750977, -14521026},
-                       FieldElement{11836410, -3979488, 26297894, 16080799, 23455045, 15735944, 1695823, -8819122, 8169720, 16220347},
-                       FieldElement{-18115838, 8653647, 17578566, -6092619, -8025777, -16012763, -11144307, -2627664, -5990708, -14166033},
-               },
-               {
-                       FieldElement{-23308498, -10968312, 15213228, -10081214, -30853605, -11050004, 27884329, 2847284, 2655861, 1738395},
-                       FieldElement{-27537433, -14253021, -25336301, -8002780, -9370762, 8129821, 21651608, -3239336, -19087449, -11005278},
-                       FieldElement{1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092, 5821408, 10478196, 8544890},
-               },
-               {
-                       FieldElement{32173121, -16129311, 24896207, 3921497, 22579056, -3410854, 19270449, 12217473, 17789017, -3395995},
-                       FieldElement{-30552961, -2228401, -15578829, -10147201, 13243889, 517024, 15479401, -3853233, 30460520, 1052596},
-                       FieldElement{-11614875, 13323618, 32618793, 8175907, -15230173, 12596687, 27491595, -4612359, 3179268, -9478891},
-               },
-               {
-                       FieldElement{31947069, -14366651, -4640583, -15339921, -15125977, -6039709, -14756777, -16411740, 19072640, -9511060},
-                       FieldElement{11685058, 11822410, 3158003, -13952594, 33402194, -4165066, 5977896, -5215017, 473099, 5040608},
-                       FieldElement{-20290863, 8198642, -27410132, 11602123, 1290375, -2799760, 28326862, 1721092, -19558642, -3131606},
-               },
-       },
-       {
-               {
-                       FieldElement{7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786, 8076149, -27868496, 11538389},
-                       FieldElement{-19935666, 3899861, 18283497, -6801568, -15728660, -11249211, 8754525, 7446702, -5676054, 5797016},
-                       FieldElement{-11295600, -3793569, -15782110, -7964573, 12708869, -8456199, 2014099, -9050574, -2369172, -5877341},
-               },
-               {
-                       FieldElement{-22472376, -11568741, -27682020, 1146375, 18956691, 16640559, 1192730, -3714199, 15123619, 10811505},
-                       FieldElement{14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363, 15776356, -28886779, -11974553},
-                       FieldElement{-28241164, -8072475, -4978962, -5315317, 29416931, 1847569, -20654173, -16484855, 4714547, -9600655},
-               },
-               {
-                       FieldElement{15200332, 8368572, 19679101, 15970074, -31872674, 1959451, 24611599, -4543832, -11745876, 12340220},
-                       FieldElement{12876937, -10480056, 33134381, 6590940, -6307776, 14872440, 9613953, 8241152, 15370987, 9608631},
-                       FieldElement{-4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868, 15866074, -28210621, -8814099},
-               },
-               {
-                       FieldElement{26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233, 858697, 20571223, 8420556},
-                       FieldElement{14620715, 13067227, -15447274, 8264467, 14106269, 15080814, 33531827, 12516406, -21574435, -12476749},
-                       FieldElement{236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519, 7256740, 8791136, 15069930},
-               },
-               {
-                       FieldElement{1276410, -9371918, 22949635, -16322807, -23493039, -5702186, 14711875, 4874229, -30663140, -2331391},
-                       FieldElement{5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175, -7912378, -33069337, 9234253},
-                       FieldElement{20590503, -9018988, 31529744, -7352666, -2706834, 10650548, 31559055, -11609587, 18979186, 13396066},
-               },
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-                       FieldElement{22648901, 1402143, -22799984, 13746059, 7936347, 365344, -8668633, -1674433, -3758243, -2304625},
-                       FieldElement{-15491917, 8012313, -2514730, -12702462, -23965846, -10254029, -1612713, -1535569, -16664475, 8194478},
-               },
-               {
-                       FieldElement{27338066, -7507420, -7414224, 10140405, -19026427, -6589889, 27277191, 8855376, 28572286, 3005164},
-                       FieldElement{26287124, 4821776, 25476601, -4145903, -3764513, -15788984, -18008582, 1182479, -26094821, -13079595},
-                       FieldElement{-7171154, 3178080, 23970071, 6201893, -17195577, -4489192, -21876275, -13982627, 32208683, -1198248},
-               },
-               {
-                       FieldElement{-16657702, 2817643, -10286362, 14811298, 6024667, 13349505, -27315504, -10497842, -27672585, -11539858},
-                       FieldElement{15941029, -9405932, -21367050, 8062055, 31876073, -238629, -15278393, -1444429, 15397331, -4130193},
-                       FieldElement{8934485, -13485467, -23286397, -13423241, -32446090, 14047986, 31170398, -1441021, -27505566, 15087184},
-               },
-               {
-                       FieldElement{-18357243, -2156491, 24524913, -16677868, 15520427, -6360776, -15502406, 11461896, 16788528, -5868942},
-                       FieldElement{-1947386, 16013773, 21750665, 3714552, -17401782, -16055433, -3770287, -10323320, 31322514, -11615635},
-                       FieldElement{21426655, -5650218, -13648287, -5347537, -28812189, -4920970, -18275391, -14621414, 13040862, -12112948},
-               },
-               {
-                       FieldElement{11293895, 12478086, -27136401, 15083750, -29307421, 14748872, 14555558, -13417103, 1613711, 4896935},
-                       FieldElement{-25894883, 15323294, -8489791, -8057900, 25967126, -13425460, 2825960, -4897045, -23971776, -11267415},
-                       FieldElement{-15924766, -5229880, -17443532, 6410664, 3622847, 10243618, 20615400, 12405433, -23753030, -8436416},
-               },
-               {
-                       FieldElement{-7091295, 12556208, -20191352, 9025187, -17072479, 4333801, 4378436, 2432030, 23097949, -566018},
-                       FieldElement{4565804, -16025654, 20084412, -7842817, 1724999, 189254, 24767264, 10103221, -18512313, 2424778},
-                       FieldElement{366633, -11976806, 8173090, -6890119, 30788634, 5745705, -7168678, 1344109, -3642553, 12412659},
-               },
-               {
-                       FieldElement{-24001791, 7690286, 14929416, -168257, -32210835, -13412986, 24162697, -15326504, -3141501, 11179385},
-                       FieldElement{18289522, -14724954, 8056945, 16430056, -21729724, 7842514, -6001441, -1486897, -18684645, -11443503},
-                       FieldElement{476239, 6601091, -6152790, -9723375, 17503545, -4863900, 27672959, 13403813, 11052904, 5219329},
-               },
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-       {
-               {
-                       FieldElement{20678546, -8375738, -32671898, 8849123, -5009758, 14574752, 31186971, -3973730, 9014762, -8579056},
-                       FieldElement{-13644050, -10350239, -15962508, 5075808, -1514661, -11534600, -33102500, 9160280, 8473550, -3256838},
-                       FieldElement{24900749, 14435722, 17209120, -15292541, -22592275, 9878983, -7689309, -16335821, -24568481, 11788948},
-               },
-               {
-                       FieldElement{-3118155, -11395194, -13802089, 14797441, 9652448, -6845904, -20037437, 10410733, -24568470, -1458691},
-                       FieldElement{-15659161, 16736706, -22467150, 10215878, -9097177, 7563911, 11871841, -12505194, -18513325, 8464118},
-                       FieldElement{-23400612, 8348507, -14585951, -861714, -3950205, -6373419, 14325289, 8628612, 33313881, -8370517},
-               },
-               {
-                       FieldElement{-20186973, -4967935, 22367356, 5271547, -1097117, -4788838, -24805667, -10236854, -8940735, -5818269},
-                       FieldElement{-6948785, -1795212, -32625683, -16021179, 32635414, -7374245, 15989197, -12838188, 28358192, -4253904},
-                       FieldElement{-23561781, -2799059, -32351682, -1661963, -9147719, 10429267, -16637684, 4072016, -5351664, 5596589},
-               },
-               {
-                       FieldElement{-28236598, -3390048, 12312896, 6213178, 3117142, 16078565, 29266239, 2557221, 1768301, 15373193},
-                       FieldElement{-7243358, -3246960, -4593467, -7553353, -127927, -912245, -1090902, -4504991, -24660491, 3442910},
-                       FieldElement{-30210571, 5124043, 14181784, 8197961, 18964734, -11939093, 22597931, 7176455, -18585478, 13365930},
-               },
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-                       FieldElement{-9660625, 12374379, -27006999, -7026148, -7724114, -12314514, 11879682, 5400171, 519526, -1235876},
-               },
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-                       FieldElement{-8081308, -8464597, -8223311, 9719710, 19259459, -15348212, 23994942, -5281555, -9468848, 4763278},
-                       FieldElement{-21699244, 9220969, -15730624, 1084137, -25476107, -2852390, 31088447, -7764523, -11356529, 728112},
-               },
-               {
-                       FieldElement{26047220, -11751471, -6900323, -16521798, 24092068, 9158119, -4273545, -12555558, -29365436, -5498272},
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-                       FieldElement{-10312768, 3936952, 9156313, -8897683, 16498692, -994647, -27481051, -666732, 3424691, 7540221},
-               },
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-                       FieldElement{30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422, -16317219, -9244265, 15258046},
-                       FieldElement{13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406, 2711395, 1062915, -5136345},
-                       FieldElement{-19240248, -11254599, -29509029, -7499965, -5835763, 13005411, -6066489, 12194497, 32960380, 1459310},
-               },
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-       {
-               {
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-                       FieldElement{-9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222, 9677543, -32294889, -6456008},
-               },
-               {
-                       FieldElement{-2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579, -7839692, -7852844, -8138429},
-                       FieldElement{-15236356, -15433509, 7766470, 746860, 26346930, -10221762, -27333451, 10754588, -9431476, 5203576},
-                       FieldElement{31834314, 14135496, -770007, 5159118, 20917671, -16768096, -7467973, -7337524, 31809243, 7347066},
-               },
-               {
-                       FieldElement{-9606723, -11874240, 20414459, 13033986, 13716524, -11691881, 19797970, -12211255, 15192876, -2087490},
-                       FieldElement{-12663563, -2181719, 1168162, -3804809, 26747877, -14138091, 10609330, 12694420, 33473243, -13382104},
-                       FieldElement{33184999, 11180355, 15832085, -11385430, -1633671, 225884, 15089336, -11023903, -6135662, 14480053},
-               },
-               {
-                       FieldElement{31308717, -5619998, 31030840, -1897099, 15674547, -6582883, 5496208, 13685227, 27595050, 8737275},
-                       FieldElement{-20318852, -15150239, 10933843, -16178022, 8335352, -7546022, -31008351, -12610604, 26498114, 66511},
-                       FieldElement{22644454, -8761729, -16671776, 4884562, -3105614, -13559366, 30540766, -4286747, -13327787, -7515095},
-               },
-               {
-                       FieldElement{-28017847, 9834845, 18617207, -2681312, -3401956, -13307506, 8205540, 13585437, -17127465, 15115439},
-                       FieldElement{23711543, -672915, 31206561, -8362711, 6164647, -9709987, -33535882, -1426096, 8236921, 16492939},
-                       FieldElement{-23910559, -13515526, -26299483, -4503841, 25005590, -7687270, 19574902, 10071562, 6708380, -6222424},
-               },
-               {
-                       FieldElement{2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017, 9328700, 29955601, -11678310},
-                       FieldElement{3096359, 9271816, -21620864, -15521844, -14847996, -7592937, -25892142, -12635595, -9917575, 6216608},
-                       FieldElement{-32615849, 338663, -25195611, 2510422, -29213566, -13820213, 24822830, -6146567, -26767480, 7525079},
-               },
-               {
-                       FieldElement{-23066649, -13985623, 16133487, -7896178, -3389565, 778788, -910336, -2782495, -19386633, 11994101},
-                       FieldElement{21691500, -13624626, -641331, -14367021, 3285881, -3483596, -25064666, 9718258, -7477437, 13381418},
-                       FieldElement{18445390, -4202236, 14979846, 11622458, -1727110, -3582980, 23111648, -6375247, 28535282, 15779576},
-               },
-               {
-                       FieldElement{30098053, 3089662, -9234387, 16662135, -21306940, 11308411, -14068454, 12021730, 9955285, -16303356},
-                       FieldElement{9734894, -14576830, -7473633, -9138735, 2060392, 11313496, -18426029, 9924399, 20194861, 13380996},
-                       FieldElement{-26378102, -7965207, -22167821, 15789297, -18055342, -6168792, -1984914, 15707771, 26342023, 10146099},
-               },
-       },
-       {
-               {
-                       FieldElement{-26016874, -219943, 21339191, -41388, 19745256, -2878700, -29637280, 2227040, 21612326, -545728},
-                       FieldElement{-13077387, 1184228, 23562814, -5970442, -20351244, -6348714, 25764461, 12243797, -20856566, 11649658},
-                       FieldElement{-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944, 6114064, 33514190, 2333242},
-               },
-               {
-                       FieldElement{-21433588, -12421821, 8119782, 7219913, -21830522, -9016134, -6679750, -12670638, 24350578, -13450001},
-                       FieldElement{-4116307, -11271533, -23886186, 4843615, -30088339, 690623, -31536088, -10406836, 8317860, 12352766},
-                       FieldElement{18200138, -14475911, -33087759, -2696619, -23702521, -9102511, -23552096, -2287550, 20712163, 6719373},
-               },
-               {
-                       FieldElement{26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530, -3763210, 26224235, -3297458},
-                       FieldElement{-17168938, -14854097, -3395676, -16369877, -19954045, 14050420, 21728352, 9493610, 18620611, -16428628},
-                       FieldElement{-13323321, 13325349, 11432106, 5964811, 18609221, 6062965, -5269471, -9725556, -30701573, -16479657},
-               },
-               {
-                       FieldElement{-23860538, -11233159, 26961357, 1640861, -32413112, -16737940, 12248509, -5240639, 13735342, 1934062},
-                       FieldElement{25089769, 6742589, 17081145, -13406266, 21909293, -16067981, -15136294, -3765346, -21277997, 5473616},
-                       FieldElement{31883677, -7961101, 1083432, -11572403, 22828471, 13290673, -7125085, 12469656, 29111212, -5451014},
-               },
-               {
-                       FieldElement{24244947, -15050407, -26262976, 2791540, -14997599, 16666678, 24367466, 6388839, -10295587, 452383},
-                       FieldElement{-25640782, -3417841, 5217916, 16224624, 19987036, -4082269, -24236251, -5915248, 15766062, 8407814},
-                       FieldElement{-20406999, 13990231, 15495425, 16395525, 5377168, 15166495, -8917023, -4388953, -8067909, 2276718},
-               },
-               {
-                       FieldElement{30157918, 12924066, -17712050, 9245753, 19895028, 3368142, -23827587, 5096219, 22740376, -7303417},
-                       FieldElement{2041139, -14256350, 7783687, 13876377, -25946985, -13352459, 24051124, 13742383, -15637599, 13295222},
-                       FieldElement{33338237, -8505733, 12532113, 7977527, 9106186, -1715251, -17720195, -4612972, -4451357, -14669444},
-               },
-               {
-                       FieldElement{-20045281, 5454097, -14346548, 6447146, 28862071, 1883651, -2469266, -4141880, 7770569, 9620597},
-                       FieldElement{23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528, -1694323, -33502340, -14767970},
-                       FieldElement{1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801, 1220118, 30494170, -11440799},
-               },
-               {
-                       FieldElement{-5037580, -13028295, -2970559, -3061767, 15640974, -6701666, -26739026, 926050, -1684339, -13333647},
-                       FieldElement{13908495, -3549272, 30919928, -6273825, -21521863, 7989039, 9021034, 9078865, 3353509, 4033511},
-                       FieldElement{-29663431, -15113610, 32259991, -344482, 24295849, -12912123, 23161163, 8839127, 27485041, 7356032},
-               },
-       },
-       {
-               {
-                       FieldElement{9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142, 2625015, 28431036, -16771834},
-                       FieldElement{-23839233, -8311415, -25945511, 7480958, -17681669, -8354183, -22545972, 14150565, 15970762, 4099461},
-                       FieldElement{29262576, 16756590, 26350592, -8793563, 8529671, -11208050, 13617293, -9937143, 11465739, 8317062},
-               },
-               {
-                       FieldElement{-25493081, -6962928, 32500200, -9419051, -23038724, -2302222, 14898637, 3848455, 20969334, -5157516},
-                       FieldElement{-20384450, -14347713, -18336405, 13884722, -33039454, 2842114, -21610826, -3649888, 11177095, 14989547},
-                       FieldElement{-24496721, -11716016, 16959896, 2278463, 12066309, 10137771, 13515641, 2581286, -28487508, 9930240},
-               },
-               {
-                       FieldElement{-17751622, -2097826, 16544300, -13009300, -15914807, -14949081, 18345767, -13403753, 16291481, -5314038},
-                       FieldElement{-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774, 6957617, 4368891, 9788741},
-                       FieldElement{16660756, 7281060, -10830758, 12911820, 20108584, -8101676, -21722536, -8613148, 16250552, -11111103},
-               },
-               {
-                       FieldElement{-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584, 10604807, -30190403, 4782747},
-                       FieldElement{-1354539, 14736941, -7367442, -13292886, 7710542, -14155590, -9981571, 4383045, 22546403, 437323},
-                       FieldElement{31665577, -12180464, -16186830, 1491339, -18368625, 3294682, 27343084, 2786261, -30633590, -14097016},
-               },
-               {
-                       FieldElement{-14467279, -683715, -33374107, 7448552, 19294360, 14334329, -19690631, 2355319, -19284671, -6114373},
-                       FieldElement{15121312, -15796162, 6377020, -6031361, -10798111, -12957845, 18952177, 15496498, -29380133, 11754228},
-                       FieldElement{-2637277, -13483075, 8488727, -14303896, 12728761, -1622493, 7141596, 11724556, 22761615, -10134141},
-               },
-               {
-                       FieldElement{16918416, 11729663, -18083579, 3022987, -31015732, -13339659, -28741185, -12227393, 32851222, 11717399},
-                       FieldElement{11166634, 7338049, -6722523, 4531520, -29468672, -7302055, 31474879, 3483633, -1193175, -4030831},
-                       FieldElement{-185635, 9921305, 31456609, -13536438, -12013818, 13348923, 33142652, 6546660, -19985279, -3948376},
-               },
-               {
-                       FieldElement{-32460596, 11266712, -11197107, -7899103, 31703694, 3855903, -8537131, -12833048, -30772034, -15486313},
-                       FieldElement{-18006477, 12709068, 3991746, -6479188, -21491523, -10550425, -31135347, -16049879, 10928917, 3011958},
-                       FieldElement{-6957757, -15594337, 31696059, 334240, 29576716, 14796075, -30831056, -12805180, 18008031, 10258577},
-               },
-               {
-                       FieldElement{-22448644, 15655569, 7018479, -4410003, -30314266, -1201591, -1853465, 1367120, 25127874, 6671743},
-                       FieldElement{29701166, -14373934, -10878120, 9279288, -17568, 13127210, 21382910, 11042292, 25838796, 4642684},
-                       FieldElement{-20430234, 14955537, -24126347, 8124619, -5369288, -5990470, 30468147, -13900640, 18423289, 4177476},
-               },
-       },
-}

diff --git a/src/crypto/ed25519/internal/edwards25519/doc.go b/src/crypto/ed25519/internal/edwards25519/doc.go
new file mode 100644 (file)
index 0000000..ff31cd2
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/doc.go
@@ -0,0 +1,22 @@
+// Copyright (c) 2021 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package edwards25519 implements group logic for the twisted Edwards curve
+//
+//     -x^2 + y^2 = 1 + -(121665/121666)*x^2*y^2
+//
+// This is better known as the Edwards curve equivalent to Curve25519, and is
+// the curve used by the Ed25519 signature scheme.
+//
+// Most users don't need this package, and should instead use crypto/ed25519 for
+// signatures, golang.org/x/crypto/curve25519 for Diffie-Hellman, or
+// github.com/gtank/ristretto255 for prime order group logic.
+//
+// However, developers who do need to interact with low-level edwards25519
+// operations can use filippo.io/edwards25519, an extended version of this
+// package repackaged as an importable module.
+//
+// (Note that filippo.io/edwards25519 and github.com/gtank/ristretto255 are not
+// maintained by the Go team and are not covered by the Go 1 Compatibility Promise.)
+package edwards25519

diff --git a/src/crypto/ed25519/internal/edwards25519/edwards25519.go b/src/crypto/ed25519/internal/edwards25519/edwards25519.go
index b091481b7734d4f6b5916caa3c00a2acee241733..313e6c281cb74dacc732e49b73b2dbdef180eb0f 100644 (file)
--- a/src/crypto/ed25519/internal/edwards25519/edwards25519.go
+++ b/src/crypto/ed25519/internal/edwards25519/edwards25519.go
@@ -1,1793 +1,427 @@
-// Copyright 2016 The Go Authors. All rights reserved.
+// Copyright (c) 2017 The Go Authors. All rights reserved.
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
 package edwards25519
 
-import "encoding/binary"
+import (
+       "crypto/ed25519/internal/edwards25519/field"
+       "errors"
+)
 
-// This code is a port of the public domain, “ref10” implementation of ed25519
-// from SUPERCOP.
+// Point types.
 
-// FieldElement represents an element of the field GF(2^255 - 19).  An element
-// t, entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
-// t[3]+2^102 t[4]+...+2^230 t[9].  Bounds on each t[i] vary depending on
-// context.
-type FieldElement [10]int32
-
-var zero FieldElement
-
-func FeZero(fe *FieldElement) {
-       copy(fe[:], zero[:])
+type projP1xP1 struct {
+       X, Y, Z, T field.Element
 }
 
-func FeOne(fe *FieldElement) {
-       FeZero(fe)
-       fe[0] = 1
+type projP2 struct {
+       X, Y, Z field.Element
 }
 
-func FeAdd(dst, a, b *FieldElement) {
-       dst[0] = a[0] + b[0]
-       dst[1] = a[1] + b[1]
-       dst[2] = a[2] + b[2]
-       dst[3] = a[3] + b[3]
-       dst[4] = a[4] + b[4]
-       dst[5] = a[5] + b[5]
-       dst[6] = a[6] + b[6]
-       dst[7] = a[7] + b[7]
-       dst[8] = a[8] + b[8]
-       dst[9] = a[9] + b[9]
-}
+// Point represents a point on the edwards25519 curve.
+//
+// This type works similarly to math/big.Int, and all arguments and receivers
+// are allowed to alias.
+//
+// The zero value is NOT valid, and it may be used only as a receiver.
+type Point struct {
+       // The point is internally represented in extended coordinates (X, Y, Z, T)
+       // where x = X/Z, y = Y/Z, and xy = T/Z per https://eprint.iacr.org/2008/522.
+       x, y, z, t field.Element
 
-func FeSub(dst, a, b *FieldElement) {
-       dst[0] = a[0] - b[0]
-       dst[1] = a[1] - b[1]
-       dst[2] = a[2] - b[2]
-       dst[3] = a[3] - b[3]
-       dst[4] = a[4] - b[4]
-       dst[5] = a[5] - b[5]
-       dst[6] = a[6] - b[6]
-       dst[7] = a[7] - b[7]
-       dst[8] = a[8] - b[8]
-       dst[9] = a[9] - b[9]
+       // Make the type not comparable (i.e. used with == or as a map key), as
+       // equivalent points can be represented by different Go values.
+       _ incomparable
 }
 
-func FeCopy(dst, src *FieldElement) {
-       copy(dst[:], src[:])
-}
+type incomparable [0]func()
 
-// Replace (f,g) with (g,g) if b == 1;
-// replace (f,g) with (f,g) if b == 0.
-//
-// Preconditions: b in {0,1}.
-func FeCMove(f, g *FieldElement, b int32) {
-       b = -b
-       f[0] ^= b & (f[0] ^ g[0])
-       f[1] ^= b & (f[1] ^ g[1])
-       f[2] ^= b & (f[2] ^ g[2])
-       f[3] ^= b & (f[3] ^ g[3])
-       f[4] ^= b & (f[4] ^ g[4])
-       f[5] ^= b & (f[5] ^ g[5])
-       f[6] ^= b & (f[6] ^ g[6])
-       f[7] ^= b & (f[7] ^ g[7])
-       f[8] ^= b & (f[8] ^ g[8])
-       f[9] ^= b & (f[9] ^ g[9])
+func checkInitialized(points ...*Point) {
+       for _, p := range points {
+               if p.x == (field.Element{}) && p.y == (field.Element{}) {
+                       panic("edwards25519: use of uninitialized Point")
+               }
+       }
 }
 
-func load3(in []byte) int64 {
-       var r int64
-       r = int64(in[0])
-       r |= int64(in[1]) << 8
-       r |= int64(in[2]) << 16
-       return r
+type projCached struct {
+       YplusX, YminusX, Z, T2d field.Element
 }
 
-func load4(in []byte) int64 {
-       var r int64
-       r = int64(in[0])
-       r |= int64(in[1]) << 8
-       r |= int64(in[2]) << 16
-       r |= int64(in[3]) << 24
-       return r
+type affineCached struct {
+       YplusX, YminusX, T2d field.Element
 }
 
-func FeFromBytes(dst *FieldElement, src *[32]byte) {
-       h0 := load4(src[:])
-       h1 := load3(src[4:]) << 6
-       h2 := load3(src[7:]) << 5
-       h3 := load3(src[10:]) << 3
-       h4 := load3(src[13:]) << 2
-       h5 := load4(src[16:])
-       h6 := load3(src[20:]) << 7
-       h7 := load3(src[23:]) << 5
-       h8 := load3(src[26:]) << 4
-       h9 := (load3(src[29:]) & 8388607) << 2
-
-       FeCombine(dst, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9)
-}
+// Constructors.
 
-// FeToBytes marshals h to s.
-// Preconditions:
-//   |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
-//
-// Write p=2^255-19; q=floor(h/p).
-// Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
-//
-// Proof:
-//   Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
-//   Also have |h-2^230 h9|<2^230 so |19 2^(-255)(h-2^230 h9)|<1/4.
-//
-//   Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
-//   Then 0<y<1.
-//
-//   Write r=h-pq.
-//   Have 0<=r<=p-1=2^255-20.
-//   Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
-//
-//   Write x=r+19(2^-255)r+y.
-//   Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
-//
-//   Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
-//   so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
-func FeToBytes(s *[32]byte, h *FieldElement) {
-       var carry [10]int32
-
-       q := (19*h[9] + (1 << 24)) >> 25
-       q = (h[0] + q) >> 26
-       q = (h[1] + q) >> 25
-       q = (h[2] + q) >> 26
-       q = (h[3] + q) >> 25
-       q = (h[4] + q) >> 26
-       q = (h[5] + q) >> 25
-       q = (h[6] + q) >> 26
-       q = (h[7] + q) >> 25
-       q = (h[8] + q) >> 26
-       q = (h[9] + q) >> 25
-
-       // Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20.
-       h[0] += 19 * q
-       // Goal: Output h-2^255 q, which is between 0 and 2^255-20.
-
-       carry[0] = h[0] >> 26
-       h[1] += carry[0]
-       h[0] -= carry[0] << 26
-       carry[1] = h[1] >> 25
-       h[2] += carry[1]
-       h[1] -= carry[1] << 25
-       carry[2] = h[2] >> 26
-       h[3] += carry[2]
-       h[2] -= carry[2] << 26
-       carry[3] = h[3] >> 25
-       h[4] += carry[3]
-       h[3] -= carry[3] << 25
-       carry[4] = h[4] >> 26
-       h[5] += carry[4]
-       h[4] -= carry[4] << 26
-       carry[5] = h[5] >> 25
-       h[6] += carry[5]
-       h[5] -= carry[5] << 25
-       carry[6] = h[6] >> 26
-       h[7] += carry[6]
-       h[6] -= carry[6] << 26
-       carry[7] = h[7] >> 25
-       h[8] += carry[7]
-       h[7] -= carry[7] << 25
-       carry[8] = h[8] >> 26
-       h[9] += carry[8]
-       h[8] -= carry[8] << 26
-       carry[9] = h[9] >> 25
-       h[9] -= carry[9] << 25
-       // h10 = carry9
-
-       // Goal: Output h[0]+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
-       // Have h[0]+...+2^230 h[9] between 0 and 2^255-1;
-       // evidently 2^255 h10-2^255 q = 0.
-       // Goal: Output h[0]+...+2^230 h[9].
-
-       s[0] = byte(h[0] >> 0)
-       s[1] = byte(h[0] >> 8)
-       s[2] = byte(h[0] >> 16)
-       s[3] = byte((h[0] >> 24) | (h[1] << 2))
-       s[4] = byte(h[1] >> 6)
-       s[5] = byte(h[1] >> 14)
-       s[6] = byte((h[1] >> 22) | (h[2] << 3))
-       s[7] = byte(h[2] >> 5)
-       s[8] = byte(h[2] >> 13)
-       s[9] = byte((h[2] >> 21) | (h[3] << 5))
-       s[10] = byte(h[3] >> 3)
-       s[11] = byte(h[3] >> 11)
-       s[12] = byte((h[3] >> 19) | (h[4] << 6))
-       s[13] = byte(h[4] >> 2)
-       s[14] = byte(h[4] >> 10)
-       s[15] = byte(h[4] >> 18)
-       s[16] = byte(h[5] >> 0)
-       s[17] = byte(h[5] >> 8)
-       s[18] = byte(h[5] >> 16)
-       s[19] = byte((h[5] >> 24) | (h[6] << 1))
-       s[20] = byte(h[6] >> 7)
-       s[21] = byte(h[6] >> 15)
-       s[22] = byte((h[6] >> 23) | (h[7] << 3))
-       s[23] = byte(h[7] >> 5)
-       s[24] = byte(h[7] >> 13)
-       s[25] = byte((h[7] >> 21) | (h[8] << 4))
-       s[26] = byte(h[8] >> 4)
-       s[27] = byte(h[8] >> 12)
-       s[28] = byte((h[8] >> 20) | (h[9] << 6))
-       s[29] = byte(h[9] >> 2)
-       s[30] = byte(h[9] >> 10)
-       s[31] = byte(h[9] >> 18)
+func (v *projP2) Zero() *projP2 {
+       v.X.Zero()
+       v.Y.One()
+       v.Z.One()
+       return v
 }
 
-func FeIsNegative(f *FieldElement) byte {
-       var s [32]byte
-       FeToBytes(&s, f)
-       return s[0] & 1
-}
+// identity is the point at infinity.
+var identity, _ = new(Point).SetBytes([]byte{
+       1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0})
 
-func FeIsNonZero(f *FieldElement) int32 {
-       var s [32]byte
-       FeToBytes(&s, f)
-       var x uint8
-       for _, b := range s {
-               x |= b
-       }
-       x |= x >> 4
-       x |= x >> 2
-       x |= x >> 1
-       return int32(x & 1)
+// NewIdentityPoint returns a new Point set to the identity.
+func NewIdentityPoint() *Point {
+       return new(Point).Set(identity)
 }
 
-// FeNeg sets h = -f
-//
-// Preconditions:
-//    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
-//
-// Postconditions:
-//    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
-func FeNeg(h, f *FieldElement) {
-       h[0] = -f[0]
-       h[1] = -f[1]
-       h[2] = -f[2]
-       h[3] = -f[3]
-       h[4] = -f[4]
-       h[5] = -f[5]
-       h[6] = -f[6]
-       h[7] = -f[7]
-       h[8] = -f[8]
-       h[9] = -f[9]
-}
+// generator is the canonical curve basepoint. See TestGenerator for the
+// correspondence of this encoding with the values in RFC 8032.
+var generator, _ = new(Point).SetBytes([]byte{
+       0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
+       0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
+       0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
+       0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66})
 
-func FeCombine(h *FieldElement, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 int64) {
-       var c0, c1, c2, c3, c4, c5, c6, c7, c8, c9 int64
-
-       /*
-         |h0| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38))
-           i.e. |h0| <= 1.2*2^59; narrower ranges for h2, h4, h6, h8
-         |h1| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19))
-           i.e. |h1| <= 1.5*2^58; narrower ranges for h3, h5, h7, h9
-       */
-
-       c0 = (h0 + (1 << 25)) >> 26
-       h1 += c0
-       h0 -= c0 << 26
-       c4 = (h4 + (1 << 25)) >> 26
-       h5 += c4
-       h4 -= c4 << 26
-       /* |h0| <= 2^25 */
-       /* |h4| <= 2^25 */
-       /* |h1| <= 1.51*2^58 */
-       /* |h5| <= 1.51*2^58 */
-
-       c1 = (h1 + (1 << 24)) >> 25
-       h2 += c1
-       h1 -= c1 << 25
-       c5 = (h5 + (1 << 24)) >> 25
-       h6 += c5
-       h5 -= c5 << 25
-       /* |h1| <= 2^24; from now on fits into int32 */
-       /* |h5| <= 2^24; from now on fits into int32 */
-       /* |h2| <= 1.21*2^59 */
-       /* |h6| <= 1.21*2^59 */
-
-       c2 = (h2 + (1 << 25)) >> 26
-       h3 += c2
-       h2 -= c2 << 26
-       c6 = (h6 + (1 << 25)) >> 26
-       h7 += c6
-       h6 -= c6 << 26
-       /* |h2| <= 2^25; from now on fits into int32 unchanged */
-       /* |h6| <= 2^25; from now on fits into int32 unchanged */
-       /* |h3| <= 1.51*2^58 */
-       /* |h7| <= 1.51*2^58 */
-
-       c3 = (h3 + (1 << 24)) >> 25
-       h4 += c3
-       h3 -= c3 << 25
-       c7 = (h7 + (1 << 24)) >> 25
-       h8 += c7
-       h7 -= c7 << 25
-       /* |h3| <= 2^24; from now on fits into int32 unchanged */
-       /* |h7| <= 2^24; from now on fits into int32 unchanged */
-       /* |h4| <= 1.52*2^33 */
-       /* |h8| <= 1.52*2^33 */
-
-       c4 = (h4 + (1 << 25)) >> 26
-       h5 += c4
-       h4 -= c4 << 26
-       c8 = (h8 + (1 << 25)) >> 26
-       h9 += c8
-       h8 -= c8 << 26
-       /* |h4| <= 2^25; from now on fits into int32 unchanged */
-       /* |h8| <= 2^25; from now on fits into int32 unchanged */
-       /* |h5| <= 1.01*2^24 */
-       /* |h9| <= 1.51*2^58 */
-
-       c9 = (h9 + (1 << 24)) >> 25
-       h0 += c9 * 19
-       h9 -= c9 << 25
-       /* |h9| <= 2^24; from now on fits into int32 unchanged */
-       /* |h0| <= 1.8*2^37 */
-
-       c0 = (h0 + (1 << 25)) >> 26
-       h1 += c0
-       h0 -= c0 << 26
-       /* |h0| <= 2^25; from now on fits into int32 unchanged */
-       /* |h1| <= 1.01*2^24 */
-
-       h[0] = int32(h0)
-       h[1] = int32(h1)
-       h[2] = int32(h2)
-       h[3] = int32(h3)
-       h[4] = int32(h4)
-       h[5] = int32(h5)
-       h[6] = int32(h6)
-       h[7] = int32(h7)
-       h[8] = int32(h8)
-       h[9] = int32(h9)
+// NewGeneratorPoint returns a new Point set to the canonical generator.
+func NewGeneratorPoint() *Point {
+       return new(Point).Set(generator)
 }
 
-// FeMul calculates h = f * g
-// Can overlap h with f or g.
-//
-// Preconditions:
-//    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
-//    |g| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
-//
-// Postconditions:
-//    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
-//
-// Notes on implementation strategy:
-//
-// Using schoolbook multiplication.
-// Karatsuba would save a little in some cost models.
-//
-// Most multiplications by 2 and 19 are 32-bit precomputations;
-// cheaper than 64-bit postcomputations.
-//
-// There is one remaining multiplication by 19 in the carry chain;
-// one *19 precomputation can be merged into this,
-// but the resulting data flow is considerably less clean.
-//
-// There are 12 carries below.
-// 10 of them are 2-way parallelizable and vectorizable.
-// Can get away with 11 carries, but then data flow is much deeper.
-//
-// With tighter constraints on inputs, can squeeze carries into int32.
-func FeMul(h, f, g *FieldElement) {
-       f0 := int64(f[0])
-       f1 := int64(f[1])
-       f2 := int64(f[2])
-       f3 := int64(f[3])
-       f4 := int64(f[4])
-       f5 := int64(f[5])
-       f6 := int64(f[6])
-       f7 := int64(f[7])
-       f8 := int64(f[8])
-       f9 := int64(f[9])
-
-       f1_2 := int64(2 * f[1])
-       f3_2 := int64(2 * f[3])
-       f5_2 := int64(2 * f[5])
-       f7_2 := int64(2 * f[7])
-       f9_2 := int64(2 * f[9])
-
-       g0 := int64(g[0])
-       g1 := int64(g[1])
-       g2 := int64(g[2])
-       g3 := int64(g[3])
-       g4 := int64(g[4])
-       g5 := int64(g[5])
-       g6 := int64(g[6])
-       g7 := int64(g[7])
-       g8 := int64(g[8])
-       g9 := int64(g[9])
-
-       g1_19 := int64(19 * g[1]) /* 1.4*2^29 */
-       g2_19 := int64(19 * g[2]) /* 1.4*2^30; still ok */
-       g3_19 := int64(19 * g[3])
-       g4_19 := int64(19 * g[4])
-       g5_19 := int64(19 * g[5])
-       g6_19 := int64(19 * g[6])
-       g7_19 := int64(19 * g[7])
-       g8_19 := int64(19 * g[8])
-       g9_19 := int64(19 * g[9])
-
-       h0 := f0*g0 + f1_2*g9_19 + f2*g8_19 + f3_2*g7_19 + f4*g6_19 + f5_2*g5_19 + f6*g4_19 + f7_2*g3_19 + f8*g2_19 + f9_2*g1_19
-       h1 := f0*g1 + f1*g0 + f2*g9_19 + f3*g8_19 + f4*g7_19 + f5*g6_19 + f6*g5_19 + f7*g4_19 + f8*g3_19 + f9*g2_19
-       h2 := f0*g2 + f1_2*g1 + f2*g0 + f3_2*g9_19 + f4*g8_19 + f5_2*g7_19 + f6*g6_19 + f7_2*g5_19 + f8*g4_19 + f9_2*g3_19
-       h3 := f0*g3 + f1*g2 + f2*g1 + f3*g0 + f4*g9_19 + f5*g8_19 + f6*g7_19 + f7*g6_19 + f8*g5_19 + f9*g4_19
-       h4 := f0*g4 + f1_2*g3 + f2*g2 + f3_2*g1 + f4*g0 + f5_2*g9_19 + f6*g8_19 + f7_2*g7_19 + f8*g6_19 + f9_2*g5_19
-       h5 := f0*g5 + f1*g4 + f2*g3 + f3*g2 + f4*g1 + f5*g0 + f6*g9_19 + f7*g8_19 + f8*g7_19 + f9*g6_19
-       h6 := f0*g6 + f1_2*g5 + f2*g4 + f3_2*g3 + f4*g2 + f5_2*g1 + f6*g0 + f7_2*g9_19 + f8*g8_19 + f9_2*g7_19
-       h7 := f0*g7 + f1*g6 + f2*g5 + f3*g4 + f4*g3 + f5*g2 + f6*g1 + f7*g0 + f8*g9_19 + f9*g8_19
-       h8 := f0*g8 + f1_2*g7 + f2*g6 + f3_2*g5 + f4*g4 + f5_2*g3 + f6*g2 + f7_2*g1 + f8*g0 + f9_2*g9_19
-       h9 := f0*g9 + f1*g8 + f2*g7 + f3*g6 + f4*g5 + f5*g4 + f6*g3 + f7*g2 + f8*g1 + f9*g0
-
-       FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9)
+func (v *projCached) Zero() *projCached {
+       v.YplusX.One()
+       v.YminusX.One()
+       v.Z.One()
+       v.T2d.Zero()
+       return v
 }
 
-func feSquare(f *FieldElement) (h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 int64) {
-       f0 := int64(f[0])
-       f1 := int64(f[1])
-       f2 := int64(f[2])
-       f3 := int64(f[3])
-       f4 := int64(f[4])
-       f5 := int64(f[5])
-       f6 := int64(f[6])
-       f7 := int64(f[7])
-       f8 := int64(f[8])
-       f9 := int64(f[9])
-       f0_2 := int64(2 * f[0])
-       f1_2 := int64(2 * f[1])
-       f2_2 := int64(2 * f[2])
-       f3_2 := int64(2 * f[3])
-       f4_2 := int64(2 * f[4])
-       f5_2 := int64(2 * f[5])
-       f6_2 := int64(2 * f[6])
-       f7_2 := int64(2 * f[7])
-       f5_38 := 38 * f5 // 1.31*2^30
-       f6_19 := 19 * f6 // 1.31*2^30
-       f7_38 := 38 * f7 // 1.31*2^30
-       f8_19 := 19 * f8 // 1.31*2^30
-       f9_38 := 38 * f9 // 1.31*2^30
-
-       h0 = f0*f0 + f1_2*f9_38 + f2_2*f8_19 + f3_2*f7_38 + f4_2*f6_19 + f5*f5_38
-       h1 = f0_2*f1 + f2*f9_38 + f3_2*f8_19 + f4*f7_38 + f5_2*f6_19
-       h2 = f0_2*f2 + f1_2*f1 + f3_2*f9_38 + f4_2*f8_19 + f5_2*f7_38 + f6*f6_19
-       h3 = f0_2*f3 + f1_2*f2 + f4*f9_38 + f5_2*f8_19 + f6*f7_38
-       h4 = f0_2*f4 + f1_2*f3_2 + f2*f2 + f5_2*f9_38 + f6_2*f8_19 + f7*f7_38
-       h5 = f0_2*f5 + f1_2*f4 + f2_2*f3 + f6*f9_38 + f7_2*f8_19
-       h6 = f0_2*f6 + f1_2*f5_2 + f2_2*f4 + f3_2*f3 + f7_2*f9_38 + f8*f8_19
-       h7 = f0_2*f7 + f1_2*f6 + f2_2*f5 + f3_2*f4 + f8*f9_38
-       h8 = f0_2*f8 + f1_2*f7_2 + f2_2*f6 + f3_2*f5_2 + f4*f4 + f9*f9_38
-       h9 = f0_2*f9 + f1_2*f8 + f2_2*f7 + f3_2*f6 + f4_2*f5
-
-       return
+func (v *affineCached) Zero() *affineCached {
+       v.YplusX.One()
+       v.YminusX.One()
+       v.T2d.Zero()
+       return v
 }
 
-// FeSquare calculates h = f*f. Can overlap h with f.
-//
-// Preconditions:
-//    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
-//
-// Postconditions:
-//    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
-func FeSquare(h, f *FieldElement) {
-       h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 := feSquare(f)
-       FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9)
+// Assignments.
+
+// Set sets v = u, and returns v.
+func (v *Point) Set(u *Point) *Point {
+       *v = *u
+       return v
 }
 
-// FeSquare2 sets h = 2 * f * f
-//
-// Can overlap h with f.
-//
-// Preconditions:
-//    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
-//
-// Postconditions:
-//    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
-// See fe_mul.c for discussion of implementation strategy.
-func FeSquare2(h, f *FieldElement) {
-       h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 := feSquare(f)
-
-       h0 += h0
-       h1 += h1
-       h2 += h2
-       h3 += h3
-       h4 += h4
-       h5 += h5
-       h6 += h6
-       h7 += h7
-       h8 += h8
-       h9 += h9
-
-       FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9)
+// Encoding.
+
+// Bytes returns the canonical 32-byte encoding of v, according to RFC 8032,
+// Section 5.1.2.
+func (v *Point) Bytes() []byte {
+       // This function is outlined to make the allocations inline in the caller
+       // rather than happen on the heap.
+       var buf [32]byte
+       return v.bytes(&buf)
 }
 
-func FeInvert(out, z *FieldElement) {
-       var t0, t1, t2, t3 FieldElement
-       var i int
+func (v *Point) bytes(buf *[32]byte) []byte {
+       checkInitialized(v)
 
-       FeSquare(&t0, z)        // 2^1
-       FeSquare(&t1, &t0)      // 2^2
-       for i = 1; i < 2; i++ { // 2^3
-               FeSquare(&t1, &t1)
-       }
-       FeMul(&t1, z, &t1)      // 2^3 + 2^0
-       FeMul(&t0, &t0, &t1)    // 2^3 + 2^1 + 2^0
-       FeSquare(&t2, &t0)      // 2^4 + 2^2 + 2^1
-       FeMul(&t1, &t1, &t2)    // 2^4 + 2^3 + 2^2 + 2^1 + 2^0
-       FeSquare(&t2, &t1)      // 5,4,3,2,1
-       for i = 1; i < 5; i++ { // 9,8,7,6,5
-               FeSquare(&t2, &t2)
-       }
-       FeMul(&t1, &t2, &t1)     // 9,8,7,6,5,4,3,2,1,0
-       FeSquare(&t2, &t1)       // 10..1
-       for i = 1; i < 10; i++ { // 19..10
-               FeSquare(&t2, &t2)
-       }
-       FeMul(&t2, &t2, &t1)     // 19..0
-       FeSquare(&t3, &t2)       // 20..1
-       for i = 1; i < 20; i++ { // 39..20
-               FeSquare(&t3, &t3)
-       }
-       FeMul(&t2, &t3, &t2)     // 39..0
-       FeSquare(&t2, &t2)       // 40..1
-       for i = 1; i < 10; i++ { // 49..10
-               FeSquare(&t2, &t2)
-       }
-       FeMul(&t1, &t2, &t1)     // 49..0
-       FeSquare(&t2, &t1)       // 50..1
-       for i = 1; i < 50; i++ { // 99..50
-               FeSquare(&t2, &t2)
-       }
-       FeMul(&t2, &t2, &t1)      // 99..0
-       FeSquare(&t3, &t2)        // 100..1
-       for i = 1; i < 100; i++ { // 199..100
-               FeSquare(&t3, &t3)
-       }
-       FeMul(&t2, &t3, &t2)     // 199..0
-       FeSquare(&t2, &t2)       // 200..1
-       for i = 1; i < 50; i++ { // 249..50
-               FeSquare(&t2, &t2)
-       }
-       FeMul(&t1, &t2, &t1)    // 249..0
-       FeSquare(&t1, &t1)      // 250..1
-       for i = 1; i < 5; i++ { // 254..5
-               FeSquare(&t1, &t1)
-       }
-       FeMul(out, &t1, &t0) // 254..5,3,1,0
+       var zInv, x, y field.Element
+       zInv.Invert(&v.z)       // zInv = 1 / Z
+       x.Multiply(&v.x, &zInv) // x = X / Z
+       y.Multiply(&v.y, &zInv) // y = Y / Z
+
+       out := copyFieldElement(buf, &y)
+       out[31] |= byte(x.IsNegative() << 7)
+       return out
 }
 
-func fePow22523(out, z *FieldElement) {
-       var t0, t1, t2 FieldElement
-       var i int
+var feOne = new(field.Element).One()
 
-       FeSquare(&t0, z)
-       for i = 1; i < 1; i++ {
-               FeSquare(&t0, &t0)
-       }
-       FeSquare(&t1, &t0)
-       for i = 1; i < 2; i++ {
-               FeSquare(&t1, &t1)
-       }
-       FeMul(&t1, z, &t1)
-       FeMul(&t0, &t0, &t1)
-       FeSquare(&t0, &t0)
-       for i = 1; i < 1; i++ {
-               FeSquare(&t0, &t0)
-       }
-       FeMul(&t0, &t1, &t0)
-       FeSquare(&t1, &t0)
-       for i = 1; i < 5; i++ {
-               FeSquare(&t1, &t1)
-       }
-       FeMul(&t0, &t1, &t0)
-       FeSquare(&t1, &t0)
-       for i = 1; i < 10; i++ {
-               FeSquare(&t1, &t1)
-       }
-       FeMul(&t1, &t1, &t0)
-       FeSquare(&t2, &t1)
-       for i = 1; i < 20; i++ {
-               FeSquare(&t2, &t2)
-       }
-       FeMul(&t1, &t2, &t1)
-       FeSquare(&t1, &t1)
-       for i = 1; i < 10; i++ {
-               FeSquare(&t1, &t1)
-       }
-       FeMul(&t0, &t1, &t0)
-       FeSquare(&t1, &t0)
-       for i = 1; i < 50; i++ {
-               FeSquare(&t1, &t1)
-       }
-       FeMul(&t1, &t1, &t0)
-       FeSquare(&t2, &t1)
-       for i = 1; i < 100; i++ {
-               FeSquare(&t2, &t2)
-       }
-       FeMul(&t1, &t2, &t1)
-       FeSquare(&t1, &t1)
-       for i = 1; i < 50; i++ {
-               FeSquare(&t1, &t1)
+// SetBytes sets v = x, where x is a 32-byte encoding of v. If x does not
+// represent a valid point on the curve, SetBytes returns nil and an error and
+// the receiver is unchanged. Otherwise, SetBytes returns v.
+//
+// Note that SetBytes accepts all non-canonical encodings of valid points.
+// That is, it follows decoding rules that match most implementations in
+// the ecosystem rather than RFC 8032.
+func (v *Point) SetBytes(x []byte) (*Point, error) {
+       // Specifically, the non-canonical encodings that are accepted are
+       //   1) the ones where the field element is not reduced (see the
+       //      (*field.Element).SetBytes docs) and
+       //   2) the ones where the x-coordinate is zero and the sign bit is set.
+       //
+       // This is consistent with crypto/ed25519/internal/edwards25519. Read more
+       // at https://hdevalence.ca/blog/2020-10-04-its-25519am, specifically the
+       // "Canonical A, R" section.
+
+       if len(x) != 32 {
+               return nil, errors.New("edwards25519: invalid point encoding length")
        }
-       FeMul(&t0, &t1, &t0)
-       FeSquare(&t0, &t0)
-       for i = 1; i < 2; i++ {
-               FeSquare(&t0, &t0)
+       y := new(field.Element).SetBytes(x)
+
+       // -x² + y² = 1 + dx²y²
+       // x² + dx²y² = x²(dy² + 1) = y² - 1
+       // x² = (y² - 1) / (dy² + 1)
+
+       // u = y² - 1
+       y2 := new(field.Element).Square(y)
+       u := new(field.Element).Subtract(y2, feOne)
+
+       // v = dy² + 1
+       vv := new(field.Element).Multiply(y2, d)
+       vv = vv.Add(vv, feOne)
+
+       // x = +√(u/v)
+       xx, wasSquare := new(field.Element).SqrtRatio(u, vv)
+       if wasSquare == 0 {
+               return nil, errors.New("edwards25519: invalid point encoding")
        }
-       FeMul(out, &t0, z)
-}
 
-// Group elements are members of the elliptic curve -x^2 + y^2 = 1 + d * x^2 *
-// y^2 where d = -121665/121666.
-//
-// Several representations are used:
-//   ProjectiveGroupElement: (X:Y:Z) satisfying x=X/Z, y=Y/Z
-//   ExtendedGroupElement: (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
-//   CompletedGroupElement: ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
-//   PreComputedGroupElement: (y+x,y-x,2dxy)
-
-type ProjectiveGroupElement struct {
-       X, Y, Z FieldElement
-}
+       // Select the negative square root if the sign bit is set.
+       xxNeg := new(field.Element).Negate(xx)
+       xx = xx.Select(xxNeg, xx, int(x[31]>>7))
 
-type ExtendedGroupElement struct {
-       X, Y, Z, T FieldElement
+       v.x.Set(xx)
+       v.y.Set(y)
+       v.z.One()
+       v.t.Multiply(xx, y) // xy = T / Z
+
+       return v, nil
 }
 
-type CompletedGroupElement struct {
-       X, Y, Z, T FieldElement
+func copyFieldElement(buf *[32]byte, v *field.Element) []byte {
+       copy(buf[:], v.Bytes())
+       return buf[:]
 }
 
-type PreComputedGroupElement struct {
-       yPlusX, yMinusX, xy2d FieldElement
+// Conversions.
+
+func (v *projP2) FromP1xP1(p *projP1xP1) *projP2 {
+       v.X.Multiply(&p.X, &p.T)
+       v.Y.Multiply(&p.Y, &p.Z)
+       v.Z.Multiply(&p.Z, &p.T)
+       return v
 }
 
-type CachedGroupElement struct {
-       yPlusX, yMinusX, Z, T2d FieldElement
+func (v *projP2) FromP3(p *Point) *projP2 {
+       v.X.Set(&p.x)
+       v.Y.Set(&p.y)
+       v.Z.Set(&p.z)
+       return v
 }
 
-func (p *ProjectiveGroupElement) Zero() {
-       FeZero(&p.X)
-       FeOne(&p.Y)
-       FeOne(&p.Z)
+func (v *Point) fromP1xP1(p *projP1xP1) *Point {
+       v.x.Multiply(&p.X, &p.T)
+       v.y.Multiply(&p.Y, &p.Z)
+       v.z.Multiply(&p.Z, &p.T)
+       v.t.Multiply(&p.X, &p.Y)
+       return v
 }
 
-func (p *ProjectiveGroupElement) Double(r *CompletedGroupElement) {
-       var t0 FieldElement
-
-       FeSquare(&r.X, &p.X)
-       FeSquare(&r.Z, &p.Y)
-       FeSquare2(&r.T, &p.Z)
-       FeAdd(&r.Y, &p.X, &p.Y)
-       FeSquare(&t0, &r.Y)
-       FeAdd(&r.Y, &r.Z, &r.X)
-       FeSub(&r.Z, &r.Z, &r.X)
-       FeSub(&r.X, &t0, &r.Y)
-       FeSub(&r.T, &r.T, &r.Z)
+func (v *Point) fromP2(p *projP2) *Point {
+       v.x.Multiply(&p.X, &p.Z)
+       v.y.Multiply(&p.Y, &p.Z)
+       v.z.Square(&p.Z)
+       v.t.Multiply(&p.X, &p.Y)
+       return v
 }
 
-func (p *ProjectiveGroupElement) ToBytes(s *[32]byte) {
-       var recip, x, y FieldElement
+// d is a constant in the curve equation.
+var d = new(field.Element).SetBytes([]byte{
+       0xa3, 0x78, 0x59, 0x13, 0xca, 0x4d, 0xeb, 0x75,
+       0xab, 0xd8, 0x41, 0x41, 0x4d, 0x0a, 0x70, 0x00,
+       0x98, 0xe8, 0x79, 0x77, 0x79, 0x40, 0xc7, 0x8c,
+       0x73, 0xfe, 0x6f, 0x2b, 0xee, 0x6c, 0x03, 0x52})
+var d2 = new(field.Element).Add(d, d)
 
-       FeInvert(&recip, &p.Z)
-       FeMul(&x, &p.X, &recip)
-       FeMul(&y, &p.Y, &recip)
-       FeToBytes(s, &y)
-       s[31] ^= FeIsNegative(&x) << 7
+func (v *projCached) FromP3(p *Point) *projCached {
+       v.YplusX.Add(&p.y, &p.x)
+       v.YminusX.Subtract(&p.y, &p.x)
+       v.Z.Set(&p.z)
+       v.T2d.Multiply(&p.t, d2)
+       return v
 }
 
-func (p *ExtendedGroupElement) Zero() {
-       FeZero(&p.X)
-       FeOne(&p.Y)
-       FeOne(&p.Z)
-       FeZero(&p.T)
-}
+func (v *affineCached) FromP3(p *Point) *affineCached {
+       v.YplusX.Add(&p.y, &p.x)
+       v.YminusX.Subtract(&p.y, &p.x)
+       v.T2d.Multiply(&p.t, d2)
 
-func (p *ExtendedGroupElement) Double(r *CompletedGroupElement) {
-       var q ProjectiveGroupElement
-       p.ToProjective(&q)
-       q.Double(r)
+       var invZ field.Element
+       invZ.Invert(&p.z)
+       v.YplusX.Multiply(&v.YplusX, &invZ)
+       v.YminusX.Multiply(&v.YminusX, &invZ)
+       v.T2d.Multiply(&v.T2d, &invZ)
+       return v
 }
 
-func (p *ExtendedGroupElement) ToCached(r *CachedGroupElement) {
-       FeAdd(&r.yPlusX, &p.Y, &p.X)
-       FeSub(&r.yMinusX, &p.Y, &p.X)
-       FeCopy(&r.Z, &p.Z)
-       FeMul(&r.T2d, &p.T, &d2)
-}
+// (Re)addition and subtraction.
 
-func (p *ExtendedGroupElement) ToProjective(r *ProjectiveGroupElement) {
-       FeCopy(&r.X, &p.X)
-       FeCopy(&r.Y, &p.Y)
-       FeCopy(&r.Z, &p.Z)
+// Add sets v = p + q, and returns v.
+func (v *Point) Add(p, q *Point) *Point {
+       checkInitialized(p, q)
+       qCached := new(projCached).FromP3(q)
+       result := new(projP1xP1).Add(p, qCached)
+       return v.fromP1xP1(result)
 }
 
-func (p *ExtendedGroupElement) ToBytes(s *[32]byte) {
-       var recip, x, y FieldElement
-
-       FeInvert(&recip, &p.Z)
-       FeMul(&x, &p.X, &recip)
-       FeMul(&y, &p.Y, &recip)
-       FeToBytes(s, &y)
-       s[31] ^= FeIsNegative(&x) << 7
+// Subtract sets v = p - q, and returns v.
+func (v *Point) Subtract(p, q *Point) *Point {
+       checkInitialized(p, q)
+       qCached := new(projCached).FromP3(q)
+       result := new(projP1xP1).Sub(p, qCached)
+       return v.fromP1xP1(result)
 }
 
-func (p *ExtendedGroupElement) FromBytes(s *[32]byte) bool {
-       var u, v, v3, vxx, check FieldElement
-
-       FeFromBytes(&p.Y, s)
-       FeOne(&p.Z)
-       FeSquare(&u, &p.Y)
-       FeMul(&v, &u, &d)
-       FeSub(&u, &u, &p.Z) // u = y^2-1
-       FeAdd(&v, &v, &p.Z) // v = dy^2+1
-
-       FeSquare(&v3, &v)
-       FeMul(&v3, &v3, &v) // v3 = v^3
-       FeSquare(&p.X, &v3)
-       FeMul(&p.X, &p.X, &v)
-       FeMul(&p.X, &p.X, &u) // x = uv^7
-
-       fePow22523(&p.X, &p.X) // x = (uv^7)^((q-5)/8)
-       FeMul(&p.X, &p.X, &v3)
-       FeMul(&p.X, &p.X, &u) // x = uv^3(uv^7)^((q-5)/8)
-
-       var tmpX, tmp2 [32]byte
-
-       FeSquare(&vxx, &p.X)
-       FeMul(&vxx, &vxx, &v)
-       FeSub(&check, &vxx, &u) // vx^2-u
-       if FeIsNonZero(&check) == 1 {
-               FeAdd(&check, &vxx, &u) // vx^2+u
-               if FeIsNonZero(&check) == 1 {
-                       return false
-               }
-               FeMul(&p.X, &p.X, &SqrtM1)
+func (v *projP1xP1) Add(p *Point, q *projCached) *projP1xP1 {
+       var YplusX, YminusX, PP, MM, TT2d, ZZ2 field.Element
 
-               FeToBytes(&tmpX, &p.X)
-               for i, v := range tmpX {
-                       tmp2[31-i] = v
-               }
-       }
+       YplusX.Add(&p.y, &p.x)
+       YminusX.Subtract(&p.y, &p.x)
 
-       if FeIsNegative(&p.X) != (s[31] >> 7) {
-               FeNeg(&p.X, &p.X)
-       }
+       PP.Multiply(&YplusX, &q.YplusX)
+       MM.Multiply(&YminusX, &q.YminusX)
+       TT2d.Multiply(&p.t, &q.T2d)
+       ZZ2.Multiply(&p.z, &q.Z)
 
-       FeMul(&p.T, &p.X, &p.Y)
-       return true
-}
+       ZZ2.Add(&ZZ2, &ZZ2)
 
-func (p *CompletedGroupElement) ToProjective(r *ProjectiveGroupElement) {
-       FeMul(&r.X, &p.X, &p.T)
-       FeMul(&r.Y, &p.Y, &p.Z)
-       FeMul(&r.Z, &p.Z, &p.T)
+       v.X.Subtract(&PP, &MM)
+       v.Y.Add(&PP, &MM)
+       v.Z.Add(&ZZ2, &TT2d)
+       v.T.Subtract(&ZZ2, &TT2d)
+       return v
 }
 
-func (p *CompletedGroupElement) ToExtended(r *ExtendedGroupElement) {
-       FeMul(&r.X, &p.X, &p.T)
-       FeMul(&r.Y, &p.Y, &p.Z)
-       FeMul(&r.Z, &p.Z, &p.T)
-       FeMul(&r.T, &p.X, &p.Y)
-}
+func (v *projP1xP1) Sub(p *Point, q *projCached) *projP1xP1 {
+       var YplusX, YminusX, PP, MM, TT2d, ZZ2 field.Element
 
-func (p *PreComputedGroupElement) Zero() {
-       FeOne(&p.yPlusX)
-       FeOne(&p.yMinusX)
-       FeZero(&p.xy2d)
-}
+       YplusX.Add(&p.y, &p.x)
+       YminusX.Subtract(&p.y, &p.x)
 
-func geAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) {
-       var t0 FieldElement
-
-       FeAdd(&r.X, &p.Y, &p.X)
-       FeSub(&r.Y, &p.Y, &p.X)
-       FeMul(&r.Z, &r.X, &q.yPlusX)
-       FeMul(&r.Y, &r.Y, &q.yMinusX)
-       FeMul(&r.T, &q.T2d, &p.T)
-       FeMul(&r.X, &p.Z, &q.Z)
-       FeAdd(&t0, &r.X, &r.X)
-       FeSub(&r.X, &r.Z, &r.Y)
-       FeAdd(&r.Y, &r.Z, &r.Y)
-       FeAdd(&r.Z, &t0, &r.T)
-       FeSub(&r.T, &t0, &r.T)
-}
+       PP.Multiply(&YplusX, &q.YminusX) // flipped sign
+       MM.Multiply(&YminusX, &q.YplusX) // flipped sign
+       TT2d.Multiply(&p.t, &q.T2d)
+       ZZ2.Multiply(&p.z, &q.Z)
 
-func geSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) {
-       var t0 FieldElement
-
-       FeAdd(&r.X, &p.Y, &p.X)
-       FeSub(&r.Y, &p.Y, &p.X)
-       FeMul(&r.Z, &r.X, &q.yMinusX)
-       FeMul(&r.Y, &r.Y, &q.yPlusX)
-       FeMul(&r.T, &q.T2d, &p.T)
-       FeMul(&r.X, &p.Z, &q.Z)
-       FeAdd(&t0, &r.X, &r.X)
-       FeSub(&r.X, &r.Z, &r.Y)
-       FeAdd(&r.Y, &r.Z, &r.Y)
-       FeSub(&r.Z, &t0, &r.T)
-       FeAdd(&r.T, &t0, &r.T)
-}
+       ZZ2.Add(&ZZ2, &ZZ2)
 
-func geMixedAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) {
-       var t0 FieldElement
-
-       FeAdd(&r.X, &p.Y, &p.X)
-       FeSub(&r.Y, &p.Y, &p.X)
-       FeMul(&r.Z, &r.X, &q.yPlusX)
-       FeMul(&r.Y, &r.Y, &q.yMinusX)
-       FeMul(&r.T, &q.xy2d, &p.T)
-       FeAdd(&t0, &p.Z, &p.Z)
-       FeSub(&r.X, &r.Z, &r.Y)
-       FeAdd(&r.Y, &r.Z, &r.Y)
-       FeAdd(&r.Z, &t0, &r.T)
-       FeSub(&r.T, &t0, &r.T)
+       v.X.Subtract(&PP, &MM)
+       v.Y.Add(&PP, &MM)
+       v.Z.Subtract(&ZZ2, &TT2d) // flipped sign
+       v.T.Add(&ZZ2, &TT2d)      // flipped sign
+       return v
 }
 
-func geMixedSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) {
-       var t0 FieldElement
-
-       FeAdd(&r.X, &p.Y, &p.X)
-       FeSub(&r.Y, &p.Y, &p.X)
-       FeMul(&r.Z, &r.X, &q.yMinusX)
-       FeMul(&r.Y, &r.Y, &q.yPlusX)
-       FeMul(&r.T, &q.xy2d, &p.T)
-       FeAdd(&t0, &p.Z, &p.Z)
-       FeSub(&r.X, &r.Z, &r.Y)
-       FeAdd(&r.Y, &r.Z, &r.Y)
-       FeSub(&r.Z, &t0, &r.T)
-       FeAdd(&r.T, &t0, &r.T)
-}
+func (v *projP1xP1) AddAffine(p *Point, q *affineCached) *projP1xP1 {
+       var YplusX, YminusX, PP, MM, TT2d, Z2 field.Element
 
-func slide(r *[256]int8, a *[32]byte) {
-       for i := range r {
-               r[i] = int8(1 & (a[i>>3] >> uint(i&7)))
-       }
+       YplusX.Add(&p.y, &p.x)
+       YminusX.Subtract(&p.y, &p.x)
 
-       for i := range r {
-               if r[i] != 0 {
-                       for b := 1; b <= 6 && i+b < 256; b++ {
-                               if r[i+b] != 0 {
-                                       if r[i]+(r[i+b]<<uint(b)) <= 15 {
-                                               r[i] += r[i+b] << uint(b)
-                                               r[i+b] = 0
-                                       } else if r[i]-(r[i+b]<<uint(b)) >= -15 {
-                                               r[i] -= r[i+b] << uint(b)
-                                               for k := i + b; k < 256; k++ {
-                                                       if r[k] == 0 {
-                                                               r[k] = 1
-                                                               break
-                                                       }
-                                                       r[k] = 0
-                                               }
-                                       } else {
-                                               break
-                                       }
-                               }
-                       }
-               }
-       }
-}
+       PP.Multiply(&YplusX, &q.YplusX)
+       MM.Multiply(&YminusX, &q.YminusX)
+       TT2d.Multiply(&p.t, &q.T2d)
 
-// GeDoubleScalarMultVartime sets r = a*A + b*B
-// where a = a[0]+256*a[1]+...+256^31 a[31].
-// and b = b[0]+256*b[1]+...+256^31 b[31].
-// B is the Ed25519 base point (x,4/5) with x positive.
-func GeDoubleScalarMultVartime(r *ProjectiveGroupElement, a *[32]byte, A *ExtendedGroupElement, b *[32]byte) {
-       var aSlide, bSlide [256]int8
-       var Ai [8]CachedGroupElement // A,3A,5A,7A,9A,11A,13A,15A
-       var t CompletedGroupElement
-       var u, A2 ExtendedGroupElement
-       var i int
-
-       slide(&aSlide, a)
-       slide(&bSlide, b)
-
-       A.ToCached(&Ai[0])
-       A.Double(&t)
-       t.ToExtended(&A2)
-
-       for i := 0; i < 7; i++ {
-               geAdd(&t, &A2, &Ai[i])
-               t.ToExtended(&u)
-               u.ToCached(&Ai[i+1])
-       }
+       Z2.Add(&p.z, &p.z)
 
-       r.Zero()
+       v.X.Subtract(&PP, &MM)
+       v.Y.Add(&PP, &MM)
+       v.Z.Add(&Z2, &TT2d)
+       v.T.Subtract(&Z2, &TT2d)
+       return v
+}
 
-       for i = 255; i >= 0; i-- {
-               if aSlide[i] != 0 || bSlide[i] != 0 {
-                       break
-               }
-       }
+func (v *projP1xP1) SubAffine(p *Point, q *affineCached) *projP1xP1 {
+       var YplusX, YminusX, PP, MM, TT2d, Z2 field.Element
 
-       for ; i >= 0; i-- {
-               r.Double(&t)
+       YplusX.Add(&p.y, &p.x)
+       YminusX.Subtract(&p.y, &p.x)
 
-               if aSlide[i] > 0 {
-                       t.ToExtended(&u)
-                       geAdd(&t, &u, &Ai[aSlide[i]/2])
-               } else if aSlide[i] < 0 {
-                       t.ToExtended(&u)
-                       geSub(&t, &u, &Ai[(-aSlide[i])/2])
-               }
+       PP.Multiply(&YplusX, &q.YminusX) // flipped sign
+       MM.Multiply(&YminusX, &q.YplusX) // flipped sign
+       TT2d.Multiply(&p.t, &q.T2d)
 
-               if bSlide[i] > 0 {
-                       t.ToExtended(&u)
-                       geMixedAdd(&t, &u, &bi[bSlide[i]/2])
-               } else if bSlide[i] < 0 {
-                       t.ToExtended(&u)
-                       geMixedSub(&t, &u, &bi[(-bSlide[i])/2])
-               }
+       Z2.Add(&p.z, &p.z)
 
-               t.ToProjective(r)
-       }
+       v.X.Subtract(&PP, &MM)
+       v.Y.Add(&PP, &MM)
+       v.Z.Subtract(&Z2, &TT2d) // flipped sign
+       v.T.Add(&Z2, &TT2d)      // flipped sign
+       return v
 }
 
-// equal returns 1 if b == c and 0 otherwise, assuming that b and c are
-// non-negative.
-func equal(b, c int32) int32 {
-       x := uint32(b ^ c)
-       x--
-       return int32(x >> 31)
-}
+// Doubling.
 
-// negative returns 1 if b < 0 and 0 otherwise.
-func negative(b int32) int32 {
-       return (b >> 31) & 1
-}
+func (v *projP1xP1) Double(p *projP2) *projP1xP1 {
+       var XX, YY, ZZ2, XplusYsq field.Element
 
-func PreComputedGroupElementCMove(t, u *PreComputedGroupElement, b int32) {
-       FeCMove(&t.yPlusX, &u.yPlusX, b)
-       FeCMove(&t.yMinusX, &u.yMinusX, b)
-       FeCMove(&t.xy2d, &u.xy2d, b)
-}
+       XX.Square(&p.X)
+       YY.Square(&p.Y)
+       ZZ2.Square(&p.Z)
+       ZZ2.Add(&ZZ2, &ZZ2)
+       XplusYsq.Add(&p.X, &p.Y)
+       XplusYsq.Square(&XplusYsq)
 
-func selectPoint(t *PreComputedGroupElement, pos int32, b int32) {
-       var minusT PreComputedGroupElement
-       bNegative := negative(b)
-       bAbs := b - (((-bNegative) & b) << 1)
+       v.Y.Add(&YY, &XX)
+       v.Z.Subtract(&YY, &XX)
 
-       t.Zero()
-       for i := int32(0); i < 8; i++ {
-               PreComputedGroupElementCMove(t, &base[pos][i], equal(bAbs, i+1))
-       }
-       FeCopy(&minusT.yPlusX, &t.yMinusX)
-       FeCopy(&minusT.yMinusX, &t.yPlusX)
-       FeNeg(&minusT.xy2d, &t.xy2d)
-       PreComputedGroupElementCMove(t, &minusT, bNegative)
+       v.X.Subtract(&XplusYsq, &v.Y)
+       v.T.Subtract(&ZZ2, &v.Z)
+       return v
 }
 
-// GeScalarMultBase computes h = a*B, where
-//   a = a[0]+256*a[1]+...+256^31 a[31]
-//   B is the Ed25519 base point (x,4/5) with x positive.
-//
-// Preconditions:
-//   a[31] <= 127
-func GeScalarMultBase(h *ExtendedGroupElement, a *[32]byte) {
-       var e [64]int8
-
-       for i, v := range a {
-               e[2*i] = int8(v & 15)
-               e[2*i+1] = int8((v >> 4) & 15)
-       }
+// Negation.
 
-       // each e[i] is between 0 and 15 and e[63] is between 0 and 7.
+// Negate sets v = -p, and returns v.
+func (v *Point) Negate(p *Point) *Point {
+       checkInitialized(p)
+       v.x.Negate(&p.x)
+       v.y.Set(&p.y)
+       v.z.Set(&p.z)
+       v.t.Negate(&p.t)
+       return v
+}
 
-       carry := int8(0)
-       for i := 0; i < 63; i++ {
-               e[i] += carry
-               carry = (e[i] + 8) >> 4
-               e[i] -= carry << 4
-       }
-       e[63] += carry
-       // each e[i] is between -8 and 8.
-
-       h.Zero()
-       var t PreComputedGroupElement
-       var r CompletedGroupElement
-       for i := int32(1); i < 64; i += 2 {
-               selectPoint(&t, i/2, int32(e[i]))
-               geMixedAdd(&r, h, &t)
-               r.ToExtended(h)
-       }
+// Equal returns 1 if v is equivalent to u, and 0 otherwise.
+func (v *Point) Equal(u *Point) int {
+       checkInitialized(v, u)
 
-       var s ProjectiveGroupElement
-
-       h.Double(&r)
-       r.ToProjective(&s)
-       s.Double(&r)
-       r.ToProjective(&s)
-       s.Double(&r)
-       r.ToProjective(&s)
-       s.Double(&r)
-       r.ToExtended(h)
-
-       for i := int32(0); i < 64; i += 2 {
-               selectPoint(&t, i/2, int32(e[i]))
-               geMixedAdd(&r, h, &t)
-               r.ToExtended(h)
-       }
+       var t1, t2, t3, t4 field.Element
+       t1.Multiply(&v.x, &u.z)
+       t2.Multiply(&u.x, &v.z)
+       t3.Multiply(&v.y, &u.z)
+       t4.Multiply(&u.y, &v.z)
+
+       return t1.Equal(&t2) & t3.Equal(&t4)
 }
 
-// The scalars are GF(2^252 + 27742317777372353535851937790883648493).
+// Constant-time operations
 
-// Input:
-//   a[0]+256*a[1]+...+256^31*a[31] = a
-//   b[0]+256*b[1]+...+256^31*b[31] = b
-//   c[0]+256*c[1]+...+256^31*c[31] = c
-//
-// Output:
-//   s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
-//   where l = 2^252 + 27742317777372353535851937790883648493.
-func ScMulAdd(s, a, b, c *[32]byte) {
-       a0 := 2097151 & load3(a[:])
-       a1 := 2097151 & (load4(a[2:]) >> 5)
-       a2 := 2097151 & (load3(a[5:]) >> 2)
-       a3 := 2097151 & (load4(a[7:]) >> 7)
-       a4 := 2097151 & (load4(a[10:]) >> 4)
-       a5 := 2097151 & (load3(a[13:]) >> 1)
-       a6 := 2097151 & (load4(a[15:]) >> 6)
-       a7 := 2097151 & (load3(a[18:]) >> 3)
-       a8 := 2097151 & load3(a[21:])
-       a9 := 2097151 & (load4(a[23:]) >> 5)
-       a10 := 2097151 & (load3(a[26:]) >> 2)
-       a11 := (load4(a[28:]) >> 7)
-       b0 := 2097151 & load3(b[:])
-       b1 := 2097151 & (load4(b[2:]) >> 5)
-       b2 := 2097151 & (load3(b[5:]) >> 2)
-       b3 := 2097151 & (load4(b[7:]) >> 7)
-       b4 := 2097151 & (load4(b[10:]) >> 4)
-       b5 := 2097151 & (load3(b[13:]) >> 1)
-       b6 := 2097151 & (load4(b[15:]) >> 6)
-       b7 := 2097151 & (load3(b[18:]) >> 3)
-       b8 := 2097151 & load3(b[21:])
-       b9 := 2097151 & (load4(b[23:]) >> 5)
-       b10 := 2097151 & (load3(b[26:]) >> 2)
-       b11 := (load4(b[28:]) >> 7)
-       c0 := 2097151 & load3(c[:])
-       c1 := 2097151 & (load4(c[2:]) >> 5)
-       c2 := 2097151 & (load3(c[5:]) >> 2)
-       c3 := 2097151 & (load4(c[7:]) >> 7)
-       c4 := 2097151 & (load4(c[10:]) >> 4)
-       c5 := 2097151 & (load3(c[13:]) >> 1)
-       c6 := 2097151 & (load4(c[15:]) >> 6)
-       c7 := 2097151 & (load3(c[18:]) >> 3)
-       c8 := 2097151 & load3(c[21:])
-       c9 := 2097151 & (load4(c[23:]) >> 5)
-       c10 := 2097151 & (load3(c[26:]) >> 2)
-       c11 := (load4(c[28:]) >> 7)
-       var carry [23]int64
-
-       s0 := c0 + a0*b0
-       s1 := c1 + a0*b1 + a1*b0
-       s2 := c2 + a0*b2 + a1*b1 + a2*b0
-       s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0
-       s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0
-       s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0
-       s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0
-       s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0
-       s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0
-       s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0
-       s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0
-       s11 := c11 + a0*b11 + a1*b10 + a2*b9 + a3*b8 + a4*b7 + a5*b6 + a6*b5 + a7*b4 + a8*b3 + a9*b2 + a10*b1 + a11*b0
-       s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1
-       s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2
-       s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3
-       s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4
-       s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5
-       s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6
-       s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7
-       s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8
-       s20 := a9*b11 + a10*b10 + a11*b9
-       s21 := a10*b11 + a11*b10
-       s22 := a11 * b11
-       s23 := int64(0)
-
-       carry[0] = (s0 + (1 << 20)) >> 21
-       s1 += carry[0]
-       s0 -= carry[0] << 21
-       carry[2] = (s2 + (1 << 20)) >> 21
-       s3 += carry[2]
-       s2 -= carry[2] << 21
-       carry[4] = (s4 + (1 << 20)) >> 21
-       s5 += carry[4]
-       s4 -= carry[4] << 21
-       carry[6] = (s6 + (1 << 20)) >> 21
-       s7 += carry[6]
-       s6 -= carry[6] << 21
-       carry[8] = (s8 + (1 << 20)) >> 21
-       s9 += carry[8]
-       s8 -= carry[8] << 21
-       carry[10] = (s10 + (1 << 20)) >> 21
-       s11 += carry[10]
-       s10 -= carry[10] << 21
-       carry[12] = (s12 + (1 << 20)) >> 21
-       s13 += carry[12]
-       s12 -= carry[12] << 21
-       carry[14] = (s14 + (1 << 20)) >> 21
-       s15 += carry[14]
-       s14 -= carry[14] << 21
-       carry[16] = (s16 + (1 << 20)) >> 21
-       s17 += carry[16]
-       s16 -= carry[16] << 21
-       carry[18] = (s18 + (1 << 20)) >> 21
-       s19 += carry[18]
-       s18 -= carry[18] << 21
-       carry[20] = (s20 + (1 << 20)) >> 21
-       s21 += carry[20]
-       s20 -= carry[20] << 21
-       carry[22] = (s22 + (1 << 20)) >> 21
-       s23 += carry[22]
-       s22 -= carry[22] << 21
-
-       carry[1] = (s1 + (1 << 20)) >> 21
-       s2 += carry[1]
-       s1 -= carry[1] << 21
-       carry[3] = (s3 + (1 << 20)) >> 21
-       s4 += carry[3]
-       s3 -= carry[3] << 21
-       carry[5] = (s5 + (1 << 20)) >> 21
-       s6 += carry[5]
-       s5 -= carry[5] << 21
-       carry[7] = (s7 + (1 << 20)) >> 21
-       s8 += carry[7]
-       s7 -= carry[7] << 21
-       carry[9] = (s9 + (1 << 20)) >> 21
-       s10 += carry[9]
-       s9 -= carry[9] << 21
-       carry[11] = (s11 + (1 << 20)) >> 21
-       s12 += carry[11]
-       s11 -= carry[11] << 21
-       carry[13] = (s13 + (1 << 20)) >> 21
-       s14 += carry[13]
-       s13 -= carry[13] << 21
-       carry[15] = (s15 + (1 << 20)) >> 21
-       s16 += carry[15]
-       s15 -= carry[15] << 21
-       carry[17] = (s17 + (1 << 20)) >> 21
-       s18 += carry[17]
-       s17 -= carry[17] << 21
-       carry[19] = (s19 + (1 << 20)) >> 21
-       s20 += carry[19]
-       s19 -= carry[19] << 21
-       carry[21] = (s21 + (1 << 20)) >> 21
-       s22 += carry[21]
-       s21 -= carry[21] << 21
-
-       s11 += s23 * 666643
-       s12 += s23 * 470296
-       s13 += s23 * 654183
-       s14 -= s23 * 997805
-       s15 += s23 * 136657
-       s16 -= s23 * 683901
-       s23 = 0
-
-       s10 += s22 * 666643
-       s11 += s22 * 470296
-       s12 += s22 * 654183
-       s13 -= s22 * 997805
-       s14 += s22 * 136657
-       s15 -= s22 * 683901
-       s22 = 0
-
-       s9 += s21 * 666643
-       s10 += s21 * 470296
-       s11 += s21 * 654183
-       s12 -= s21 * 997805
-       s13 += s21 * 136657
-       s14 -= s21 * 683901
-       s21 = 0
-
-       s8 += s20 * 666643
-       s9 += s20 * 470296
-       s10 += s20 * 654183
-       s11 -= s20 * 997805
-       s12 += s20 * 136657
-       s13 -= s20 * 683901
-       s20 = 0
-
-       s7 += s19 * 666643
-       s8 += s19 * 470296
-       s9 += s19 * 654183
-       s10 -= s19 * 997805
-       s11 += s19 * 136657
-       s12 -= s19 * 683901
-       s19 = 0
-
-       s6 += s18 * 666643
-       s7 += s18 * 470296
-       s8 += s18 * 654183
-       s9 -= s18 * 997805
-       s10 += s18 * 136657
-       s11 -= s18 * 683901
-       s18 = 0
-
-       carry[6] = (s6 + (1 << 20)) >> 21
-       s7 += carry[6]
-       s6 -= carry[6] << 21
-       carry[8] = (s8 + (1 << 20)) >> 21
-       s9 += carry[8]
-       s8 -= carry[8] << 21
-       carry[10] = (s10 + (1 << 20)) >> 21
-       s11 += carry[10]
-       s10 -= carry[10] << 21
-       carry[12] = (s12 + (1 << 20)) >> 21
-       s13 += carry[12]
-       s12 -= carry[12] << 21
-       carry[14] = (s14 + (1 << 20)) >> 21
-       s15 += carry[14]
-       s14 -= carry[14] << 21
-       carry[16] = (s16 + (1 << 20)) >> 21
-       s17 += carry[16]
-       s16 -= carry[16] << 21
-
-       carry[7] = (s7 + (1 << 20)) >> 21
-       s8 += carry[7]
-       s7 -= carry[7] << 21
-       carry[9] = (s9 + (1 << 20)) >> 21
-       s10 += carry[9]
-       s9 -= carry[9] << 21
-       carry[11] = (s11 + (1 << 20)) >> 21
-       s12 += carry[11]
-       s11 -= carry[11] << 21
-       carry[13] = (s13 + (1 << 20)) >> 21
-       s14 += carry[13]
-       s13 -= carry[13] << 21
-       carry[15] = (s15 + (1 << 20)) >> 21
-       s16 += carry[15]
-       s15 -= carry[15] << 21
-
-       s5 += s17 * 666643
-       s6 += s17 * 470296
-       s7 += s17 * 654183
-       s8 -= s17 * 997805
-       s9 += s17 * 136657
-       s10 -= s17 * 683901
-       s17 = 0
-
-       s4 += s16 * 666643
-       s5 += s16 * 470296
-       s6 += s16 * 654183
-       s7 -= s16 * 997805
-       s8 += s16 * 136657
-       s9 -= s16 * 683901
-       s16 = 0
-
-       s3 += s15 * 666643
-       s4 += s15 * 470296
-       s5 += s15 * 654183
-       s6 -= s15 * 997805
-       s7 += s15 * 136657
-       s8 -= s15 * 683901
-       s15 = 0
-
-       s2 += s14 * 666643
-       s3 += s14 * 470296
-       s4 += s14 * 654183
-       s5 -= s14 * 997805
-       s6 += s14 * 136657
-       s7 -= s14 * 683901
-       s14 = 0
-
-       s1 += s13 * 666643
-       s2 += s13 * 470296
-       s3 += s13 * 654183
-       s4 -= s13 * 997805
-       s5 += s13 * 136657
-       s6 -= s13 * 683901
-       s13 = 0
-
-       s0 += s12 * 666643
-       s1 += s12 * 470296
-       s2 += s12 * 654183
-       s3 -= s12 * 997805
-       s4 += s12 * 136657
-       s5 -= s12 * 683901
-       s12 = 0
-
-       carry[0] = (s0 + (1 << 20)) >> 21
-       s1 += carry[0]
-       s0 -= carry[0] << 21
-       carry[2] = (s2 + (1 << 20)) >> 21
-       s3 += carry[2]
-       s2 -= carry[2] << 21
-       carry[4] = (s4 + (1 << 20)) >> 21
-       s5 += carry[4]
-       s4 -= carry[4] << 21
-       carry[6] = (s6 + (1 << 20)) >> 21
-       s7 += carry[6]
-       s6 -= carry[6] << 21
-       carry[8] = (s8 + (1 << 20)) >> 21
-       s9 += carry[8]
-       s8 -= carry[8] << 21
-       carry[10] = (s10 + (1 << 20)) >> 21
-       s11 += carry[10]
-       s10 -= carry[10] << 21
-
-       carry[1] = (s1 + (1 << 20)) >> 21
-       s2 += carry[1]
-       s1 -= carry[1] << 21
-       carry[3] = (s3 + (1 << 20)) >> 21
-       s4 += carry[3]
-       s3 -= carry[3] << 21
-       carry[5] = (s5 + (1 << 20)) >> 21
-       s6 += carry[5]
-       s5 -= carry[5] << 21
-       carry[7] = (s7 + (1 << 20)) >> 21
-       s8 += carry[7]
-       s7 -= carry[7] << 21
-       carry[9] = (s9 + (1 << 20)) >> 21
-       s10 += carry[9]
-       s9 -= carry[9] << 21
-       carry[11] = (s11 + (1 << 20)) >> 21
-       s12 += carry[11]
-       s11 -= carry[11] << 21
-
-       s0 += s12 * 666643
-       s1 += s12 * 470296
-       s2 += s12 * 654183
-       s3 -= s12 * 997805
-       s4 += s12 * 136657
-       s5 -= s12 * 683901
-       s12 = 0
-
-       carry[0] = s0 >> 21
-       s1 += carry[0]
-       s0 -= carry[0] << 21
-       carry[1] = s1 >> 21
-       s2 += carry[1]
-       s1 -= carry[1] << 21
-       carry[2] = s2 >> 21
-       s3 += carry[2]
-       s2 -= carry[2] << 21
-       carry[3] = s3 >> 21
-       s4 += carry[3]
-       s3 -= carry[3] << 21
-       carry[4] = s4 >> 21
-       s5 += carry[4]
-       s4 -= carry[4] << 21
-       carry[5] = s5 >> 21
-       s6 += carry[5]
-       s5 -= carry[5] << 21
-       carry[6] = s6 >> 21
-       s7 += carry[6]
-       s6 -= carry[6] << 21
-       carry[7] = s7 >> 21
-       s8 += carry[7]
-       s7 -= carry[7] << 21
-       carry[8] = s8 >> 21
-       s9 += carry[8]
-       s8 -= carry[8] << 21
-       carry[9] = s9 >> 21
-       s10 += carry[9]
-       s9 -= carry[9] << 21
-       carry[10] = s10 >> 21
-       s11 += carry[10]
-       s10 -= carry[10] << 21
-       carry[11] = s11 >> 21
-       s12 += carry[11]
-       s11 -= carry[11] << 21
-
-       s0 += s12 * 666643
-       s1 += s12 * 470296
-       s2 += s12 * 654183
-       s3 -= s12 * 997805
-       s4 += s12 * 136657
-       s5 -= s12 * 683901
-       s12 = 0
-
-       carry[0] = s0 >> 21
-       s1 += carry[0]
-       s0 -= carry[0] << 21
-       carry[1] = s1 >> 21
-       s2 += carry[1]
-       s1 -= carry[1] << 21
-       carry[2] = s2 >> 21
-       s3 += carry[2]
-       s2 -= carry[2] << 21
-       carry[3] = s3 >> 21
-       s4 += carry[3]
-       s3 -= carry[3] << 21
-       carry[4] = s4 >> 21
-       s5 += carry[4]
-       s4 -= carry[4] << 21
-       carry[5] = s5 >> 21
-       s6 += carry[5]
-       s5 -= carry[5] << 21
-       carry[6] = s6 >> 21
-       s7 += carry[6]
-       s6 -= carry[6] << 21
-       carry[7] = s7 >> 21
-       s8 += carry[7]
-       s7 -= carry[7] << 21
-       carry[8] = s8 >> 21
-       s9 += carry[8]
-       s8 -= carry[8] << 21
-       carry[9] = s9 >> 21
-       s10 += carry[9]
-       s9 -= carry[9] << 21
-       carry[10] = s10 >> 21
-       s11 += carry[10]
-       s10 -= carry[10] << 21
-
-       s[0] = byte(s0 >> 0)
-       s[1] = byte(s0 >> 8)
-       s[2] = byte((s0 >> 16) | (s1 << 5))
-       s[3] = byte(s1 >> 3)
-       s[4] = byte(s1 >> 11)
-       s[5] = byte((s1 >> 19) | (s2 << 2))
-       s[6] = byte(s2 >> 6)
-       s[7] = byte((s2 >> 14) | (s3 << 7))
-       s[8] = byte(s3 >> 1)
-       s[9] = byte(s3 >> 9)
-       s[10] = byte((s3 >> 17) | (s4 << 4))
-       s[11] = byte(s4 >> 4)
-       s[12] = byte(s4 >> 12)
-       s[13] = byte((s4 >> 20) | (s5 << 1))
-       s[14] = byte(s5 >> 7)
-       s[15] = byte((s5 >> 15) | (s6 << 6))
-       s[16] = byte(s6 >> 2)
-       s[17] = byte(s6 >> 10)
-       s[18] = byte((s6 >> 18) | (s7 << 3))
-       s[19] = byte(s7 >> 5)
-       s[20] = byte(s7 >> 13)
-       s[21] = byte(s8 >> 0)
-       s[22] = byte(s8 >> 8)
-       s[23] = byte((s8 >> 16) | (s9 << 5))
-       s[24] = byte(s9 >> 3)
-       s[25] = byte(s9 >> 11)
-       s[26] = byte((s9 >> 19) | (s10 << 2))
-       s[27] = byte(s10 >> 6)
-       s[28] = byte((s10 >> 14) | (s11 << 7))
-       s[29] = byte(s11 >> 1)
-       s[30] = byte(s11 >> 9)
-       s[31] = byte(s11 >> 17)
+// Select sets v to a if cond == 1 and to b if cond == 0.
+func (v *projCached) Select(a, b *projCached, cond int) *projCached {
+       v.YplusX.Select(&a.YplusX, &b.YplusX, cond)
+       v.YminusX.Select(&a.YminusX, &b.YminusX, cond)
+       v.Z.Select(&a.Z, &b.Z, cond)
+       v.T2d.Select(&a.T2d, &b.T2d, cond)
+       return v
 }
 
-// Input:
-//   s[0]+256*s[1]+...+256^63*s[63] = s
-//
-// Output:
-//   s[0]+256*s[1]+...+256^31*s[31] = s mod l
-//   where l = 2^252 + 27742317777372353535851937790883648493.
-func ScReduce(out *[32]byte, s *[64]byte) {
-       s0 := 2097151 & load3(s[:])
-       s1 := 2097151 & (load4(s[2:]) >> 5)
-       s2 := 2097151 & (load3(s[5:]) >> 2)
-       s3 := 2097151 & (load4(s[7:]) >> 7)
-       s4 := 2097151 & (load4(s[10:]) >> 4)
-       s5 := 2097151 & (load3(s[13:]) >> 1)
-       s6 := 2097151 & (load4(s[15:]) >> 6)
-       s7 := 2097151 & (load3(s[18:]) >> 3)
-       s8 := 2097151 & load3(s[21:])
-       s9 := 2097151 & (load4(s[23:]) >> 5)
-       s10 := 2097151 & (load3(s[26:]) >> 2)
-       s11 := 2097151 & (load4(s[28:]) >> 7)
-       s12 := 2097151 & (load4(s[31:]) >> 4)
-       s13 := 2097151 & (load3(s[34:]) >> 1)
-       s14 := 2097151 & (load4(s[36:]) >> 6)
-       s15 := 2097151 & (load3(s[39:]) >> 3)
-       s16 := 2097151 & load3(s[42:])
-       s17 := 2097151 & (load4(s[44:]) >> 5)
-       s18 := 2097151 & (load3(s[47:]) >> 2)
-       s19 := 2097151 & (load4(s[49:]) >> 7)
-       s20 := 2097151 & (load4(s[52:]) >> 4)
-       s21 := 2097151 & (load3(s[55:]) >> 1)
-       s22 := 2097151 & (load4(s[57:]) >> 6)
-       s23 := (load4(s[60:]) >> 3)
-
-       s11 += s23 * 666643
-       s12 += s23 * 470296
-       s13 += s23 * 654183
-       s14 -= s23 * 997805
-       s15 += s23 * 136657
-       s16 -= s23 * 683901
-       s23 = 0
-
-       s10 += s22 * 666643
-       s11 += s22 * 470296
-       s12 += s22 * 654183
-       s13 -= s22 * 997805
-       s14 += s22 * 136657
-       s15 -= s22 * 683901
-       s22 = 0
-
-       s9 += s21 * 666643
-       s10 += s21 * 470296
-       s11 += s21 * 654183
-       s12 -= s21 * 997805
-       s13 += s21 * 136657
-       s14 -= s21 * 683901
-       s21 = 0
-
-       s8 += s20 * 666643
-       s9 += s20 * 470296
-       s10 += s20 * 654183
-       s11 -= s20 * 997805
-       s12 += s20 * 136657
-       s13 -= s20 * 683901
-       s20 = 0
-
-       s7 += s19 * 666643
-       s8 += s19 * 470296
-       s9 += s19 * 654183
-       s10 -= s19 * 997805
-       s11 += s19 * 136657
-       s12 -= s19 * 683901
-       s19 = 0
-
-       s6 += s18 * 666643
-       s7 += s18 * 470296
-       s8 += s18 * 654183
-       s9 -= s18 * 997805
-       s10 += s18 * 136657
-       s11 -= s18 * 683901
-       s18 = 0
-
-       var carry [17]int64
-
-       carry[6] = (s6 + (1 << 20)) >> 21
-       s7 += carry[6]
-       s6 -= carry[6] << 21
-       carry[8] = (s8 + (1 << 20)) >> 21
-       s9 += carry[8]
-       s8 -= carry[8] << 21
-       carry[10] = (s10 + (1 << 20)) >> 21
-       s11 += carry[10]
-       s10 -= carry[10] << 21
-       carry[12] = (s12 + (1 << 20)) >> 21
-       s13 += carry[12]
-       s12 -= carry[12] << 21
-       carry[14] = (s14 + (1 << 20)) >> 21
-       s15 += carry[14]
-       s14 -= carry[14] << 21
-       carry[16] = (s16 + (1 << 20)) >> 21
-       s17 += carry[16]
-       s16 -= carry[16] << 21
-
-       carry[7] = (s7 + (1 << 20)) >> 21
-       s8 += carry[7]
-       s7 -= carry[7] << 21
-       carry[9] = (s9 + (1 << 20)) >> 21
-       s10 += carry[9]
-       s9 -= carry[9] << 21
-       carry[11] = (s11 + (1 << 20)) >> 21
-       s12 += carry[11]
-       s11 -= carry[11] << 21
-       carry[13] = (s13 + (1 << 20)) >> 21
-       s14 += carry[13]
-       s13 -= carry[13] << 21
-       carry[15] = (s15 + (1 << 20)) >> 21
-       s16 += carry[15]
-       s15 -= carry[15] << 21
-
-       s5 += s17 * 666643
-       s6 += s17 * 470296
-       s7 += s17 * 654183
-       s8 -= s17 * 997805
-       s9 += s17 * 136657
-       s10 -= s17 * 683901
-       s17 = 0
-
-       s4 += s16 * 666643
-       s5 += s16 * 470296
-       s6 += s16 * 654183
-       s7 -= s16 * 997805
-       s8 += s16 * 136657
-       s9 -= s16 * 683901
-       s16 = 0
-
-       s3 += s15 * 666643
-       s4 += s15 * 470296
-       s5 += s15 * 654183
-       s6 -= s15 * 997805
-       s7 += s15 * 136657
-       s8 -= s15 * 683901
-       s15 = 0
-
-       s2 += s14 * 666643
-       s3 += s14 * 470296
-       s4 += s14 * 654183
-       s5 -= s14 * 997805
-       s6 += s14 * 136657
-       s7 -= s14 * 683901
-       s14 = 0
-
-       s1 += s13 * 666643
-       s2 += s13 * 470296
-       s3 += s13 * 654183
-       s4 -= s13 * 997805
-       s5 += s13 * 136657
-       s6 -= s13 * 683901
-       s13 = 0
-
-       s0 += s12 * 666643
-       s1 += s12 * 470296
-       s2 += s12 * 654183
-       s3 -= s12 * 997805
-       s4 += s12 * 136657
-       s5 -= s12 * 683901
-       s12 = 0
-
-       carry[0] = (s0 + (1 << 20)) >> 21
-       s1 += carry[0]
-       s0 -= carry[0] << 21
-       carry[2] = (s2 + (1 << 20)) >> 21
-       s3 += carry[2]
-       s2 -= carry[2] << 21
-       carry[4] = (s4 + (1 << 20)) >> 21
-       s5 += carry[4]
-       s4 -= carry[4] << 21
-       carry[6] = (s6 + (1 << 20)) >> 21
-       s7 += carry[6]
-       s6 -= carry[6] << 21
-       carry[8] = (s8 + (1 << 20)) >> 21
-       s9 += carry[8]
-       s8 -= carry[8] << 21
-       carry[10] = (s10 + (1 << 20)) >> 21
-       s11 += carry[10]
-       s10 -= carry[10] << 21
-
-       carry[1] = (s1 + (1 << 20)) >> 21
-       s2 += carry[1]
-       s1 -= carry[1] << 21
-       carry[3] = (s3 + (1 << 20)) >> 21
-       s4 += carry[3]
-       s3 -= carry[3] << 21
-       carry[5] = (s5 + (1 << 20)) >> 21
-       s6 += carry[5]
-       s5 -= carry[5] << 21
-       carry[7] = (s7 + (1 << 20)) >> 21
-       s8 += carry[7]
-       s7 -= carry[7] << 21
-       carry[9] = (s9 + (1 << 20)) >> 21
-       s10 += carry[9]
-       s9 -= carry[9] << 21
-       carry[11] = (s11 + (1 << 20)) >> 21
-       s12 += carry[11]
-       s11 -= carry[11] << 21
-
-       s0 += s12 * 666643
-       s1 += s12 * 470296
-       s2 += s12 * 654183
-       s3 -= s12 * 997805
-       s4 += s12 * 136657
-       s5 -= s12 * 683901
-       s12 = 0
-
-       carry[0] = s0 >> 21
-       s1 += carry[0]
-       s0 -= carry[0] << 21
-       carry[1] = s1 >> 21
-       s2 += carry[1]
-       s1 -= carry[1] << 21
-       carry[2] = s2 >> 21
-       s3 += carry[2]
-       s2 -= carry[2] << 21
-       carry[3] = s3 >> 21
-       s4 += carry[3]
-       s3 -= carry[3] << 21
-       carry[4] = s4 >> 21
-       s5 += carry[4]
-       s4 -= carry[4] << 21
-       carry[5] = s5 >> 21
-       s6 += carry[5]
-       s5 -= carry[5] << 21
-       carry[6] = s6 >> 21
-       s7 += carry[6]
-       s6 -= carry[6] << 21
-       carry[7] = s7 >> 21
-       s8 += carry[7]
-       s7 -= carry[7] << 21
-       carry[8] = s8 >> 21
-       s9 += carry[8]
-       s8 -= carry[8] << 21
-       carry[9] = s9 >> 21
-       s10 += carry[9]
-       s9 -= carry[9] << 21
-       carry[10] = s10 >> 21
-       s11 += carry[10]
-       s10 -= carry[10] << 21
-       carry[11] = s11 >> 21
-       s12 += carry[11]
-       s11 -= carry[11] << 21
-
-       s0 += s12 * 666643
-       s1 += s12 * 470296
-       s2 += s12 * 654183
-       s3 -= s12 * 997805
-       s4 += s12 * 136657
-       s5 -= s12 * 683901
-       s12 = 0
-
-       carry[0] = s0 >> 21
-       s1 += carry[0]
-       s0 -= carry[0] << 21
-       carry[1] = s1 >> 21
-       s2 += carry[1]
-       s1 -= carry[1] << 21
-       carry[2] = s2 >> 21
-       s3 += carry[2]
-       s2 -= carry[2] << 21
-       carry[3] = s3 >> 21
-       s4 += carry[3]
-       s3 -= carry[3] << 21
-       carry[4] = s4 >> 21
-       s5 += carry[4]
-       s4 -= carry[4] << 21
-       carry[5] = s5 >> 21
-       s6 += carry[5]
-       s5 -= carry[5] << 21
-       carry[6] = s6 >> 21
-       s7 += carry[6]
-       s6 -= carry[6] << 21
-       carry[7] = s7 >> 21
-       s8 += carry[7]
-       s7 -= carry[7] << 21
-       carry[8] = s8 >> 21
-       s9 += carry[8]
-       s8 -= carry[8] << 21
-       carry[9] = s9 >> 21
-       s10 += carry[9]
-       s9 -= carry[9] << 21
-       carry[10] = s10 >> 21
-       s11 += carry[10]
-       s10 -= carry[10] << 21
-
-       out[0] = byte(s0 >> 0)
-       out[1] = byte(s0 >> 8)
-       out[2] = byte((s0 >> 16) | (s1 << 5))
-       out[3] = byte(s1 >> 3)
-       out[4] = byte(s1 >> 11)
-       out[5] = byte((s1 >> 19) | (s2 << 2))
-       out[6] = byte(s2 >> 6)
-       out[7] = byte((s2 >> 14) | (s3 << 7))
-       out[8] = byte(s3 >> 1)
-       out[9] = byte(s3 >> 9)
-       out[10] = byte((s3 >> 17) | (s4 << 4))
-       out[11] = byte(s4 >> 4)
-       out[12] = byte(s4 >> 12)
-       out[13] = byte((s4 >> 20) | (s5 << 1))
-       out[14] = byte(s5 >> 7)
-       out[15] = byte((s5 >> 15) | (s6 << 6))
-       out[16] = byte(s6 >> 2)
-       out[17] = byte(s6 >> 10)
-       out[18] = byte((s6 >> 18) | (s7 << 3))
-       out[19] = byte(s7 >> 5)
-       out[20] = byte(s7 >> 13)
-       out[21] = byte(s8 >> 0)
-       out[22] = byte(s8 >> 8)
-       out[23] = byte((s8 >> 16) | (s9 << 5))
-       out[24] = byte(s9 >> 3)
-       out[25] = byte(s9 >> 11)
-       out[26] = byte((s9 >> 19) | (s10 << 2))
-       out[27] = byte(s10 >> 6)
-       out[28] = byte((s10 >> 14) | (s11 << 7))
-       out[29] = byte(s11 >> 1)
-       out[30] = byte(s11 >> 9)
-       out[31] = byte(s11 >> 17)
+// Select sets v to a if cond == 1 and to b if cond == 0.
+func (v *affineCached) Select(a, b *affineCached, cond int) *affineCached {
+       v.YplusX.Select(&a.YplusX, &b.YplusX, cond)
+       v.YminusX.Select(&a.YminusX, &b.YminusX, cond)
+       v.T2d.Select(&a.T2d, &b.T2d, cond)
+       return v
 }
 
-// order is the order of Curve25519 in little-endian form.
-var order = [4]uint64{0x5812631a5cf5d3ed, 0x14def9dea2f79cd6, 0, 0x1000000000000000}
-
-// ScMinimal returns true if the given scalar is less than the order of the
-// curve.
-func ScMinimal(scalar *[32]byte) bool {
-       for i := 3; ; i-- {
-               v := binary.LittleEndian.Uint64(scalar[i*8:])
-               if v > order[i] {
-                       return false
-               } else if v < order[i] {
-                       break
-               } else if i == 0 {
-                       return false
-               }
-       }
+// CondNeg negates v if cond == 1 and leaves it unchanged if cond == 0.
+func (v *projCached) CondNeg(cond int) *projCached {
+       v.YplusX.Swap(&v.YminusX, cond)
+       v.T2d.Select(new(field.Element).Negate(&v.T2d), &v.T2d, cond)
+       return v
+}
 
-       return true
+// CondNeg negates v if cond == 1 and leaves it unchanged if cond == 0.
+func (v *affineCached) CondNeg(cond int) *affineCached {
+       v.YplusX.Swap(&v.YminusX, cond)
+       v.T2d.Select(new(field.Element).Negate(&v.T2d), &v.T2d, cond)
+       return v
 }

diff --git a/src/crypto/ed25519/internal/edwards25519/edwards25519_test.go b/src/crypto/ed25519/internal/edwards25519/edwards25519_test.go
new file mode 100644 (file)
index 0000000..ac7c78c
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/edwards25519_test.go
@@ -0,0 +1,299 @@
+// Copyright (c) 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package edwards25519
+
+import (
+       "crypto/ed25519/internal/edwards25519/field"
+       "encoding/hex"
+       "reflect"
+       "testing"
+)
+
+var B = NewGeneratorPoint()
+var I = NewIdentityPoint()
+
+func checkOnCurve(t *testing.T, points ...*Point) {
+       t.Helper()
+       for i, p := range points {
+               var XX, YY, ZZ, ZZZZ field.Element
+               XX.Square(&p.x)
+               YY.Square(&p.y)
+               ZZ.Square(&p.z)
+               ZZZZ.Square(&ZZ)
+               // -x² + y² = 1 + dx²y²
+               // -(X/Z)² + (Y/Z)² = 1 + d(X/Z)²(Y/Z)²
+               // (-X² + Y²)/Z² = 1 + (dX²Y²)/Z⁴
+               // (-X² + Y²)*Z² = Z⁴ + dX²Y²
+               var lhs, rhs field.Element
+               lhs.Subtract(&YY, &XX).Multiply(&lhs, &ZZ)
+               rhs.Multiply(d, &XX).Multiply(&rhs, &YY).Add(&rhs, &ZZZZ)
+               if lhs.Equal(&rhs) != 1 {
+                       t.Errorf("X, Y, and Z do not specify a point on the curve\nX = %v\nY = %v\nZ = %v", p.x, p.y, p.z)
+               }
+               // xy = T/Z
+               lhs.Multiply(&p.x, &p.y)
+               rhs.Multiply(&p.z, &p.t)
+               if lhs.Equal(&rhs) != 1 {
+                       t.Errorf("point %d is not valid\nX = %v\nY = %v\nZ = %v", i, p.x, p.y, p.z)
+               }
+       }
+}
+
+func TestGenerator(t *testing.T) {
+       // These are the coordinates of B from RFC 8032, Section 5.1, converted to
+       // little endian hex.
+       x := "1ad5258f602d56c9b2a7259560c72c695cdcd6fd31e2a4c0fe536ecdd3366921"
+       y := "5866666666666666666666666666666666666666666666666666666666666666"
+       if got := hex.EncodeToString(B.x.Bytes()); got != x {
+               t.Errorf("wrong B.x: got %s, expected %s", got, x)
+       }
+       if got := hex.EncodeToString(B.y.Bytes()); got != y {
+               t.Errorf("wrong B.y: got %s, expected %s", got, y)
+       }
+       if B.z.Equal(feOne) != 1 {
+               t.Errorf("wrong B.z: got %v, expected 1", B.z)
+       }
+       // Check that t is correct.
+       checkOnCurve(t, B)
+}
+
+func TestAddSubNegOnBasePoint(t *testing.T) {
+       checkLhs, checkRhs := &Point{}, &Point{}
+
+       checkLhs.Add(B, B)
+       tmpP2 := new(projP2).FromP3(B)
+       tmpP1xP1 := new(projP1xP1).Double(tmpP2)
+       checkRhs.fromP1xP1(tmpP1xP1)
+       if checkLhs.Equal(checkRhs) != 1 {
+               t.Error("B + B != [2]B")
+       }
+       checkOnCurve(t, checkLhs, checkRhs)
+
+       checkLhs.Subtract(B, B)
+       Bneg := new(Point).Negate(B)
+       checkRhs.Add(B, Bneg)
+       if checkLhs.Equal(checkRhs) != 1 {
+               t.Error("B - B != B + (-B)")
+       }
+       if I.Equal(checkLhs) != 1 {
+               t.Error("B - B != 0")
+       }
+       if I.Equal(checkRhs) != 1 {
+               t.Error("B + (-B) != 0")
+       }
+       checkOnCurve(t, checkLhs, checkRhs, Bneg)
+}
+
+func TestComparable(t *testing.T) {
+       if reflect.TypeOf(Point{}).Comparable() {
+               t.Error("Point is unexpectedly comparable")
+       }
+}
+
+func TestInvalidEncodings(t *testing.T) {
+       // An invalid point, that also happens to have y > p.
+       invalid := "efffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f"
+       p := NewGeneratorPoint()
+       if out, err := p.SetBytes(decodeHex(invalid)); err == nil {
+               t.Error("expected error for invalid point")
+       } else if out != nil {
+               t.Error("SetBytes did not return nil on an invalid encoding")
+       } else if p.Equal(B) != 1 {
+               t.Error("the Point was modified while decoding an invalid encoding")
+       }
+       checkOnCurve(t, p)
+}
+
+func TestNonCanonicalPoints(t *testing.T) {
+       type test struct {
+               name                string
+               encoding, canonical string
+       }
+       tests := []test{
+               // Points with x = 0 and the sign bit set. With x = 0 the curve equation
+               // gives y² = 1, so y = ±1. 1 has two valid encodings.
+               {
+                       "y=1,sign-",
+                       "0100000000000000000000000000000000000000000000000000000000000080",
+                       "0100000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p+1,sign-",
+                       "eeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "0100000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p-1,sign-",
+                       "ecffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "ecffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+               },
+
+               // Non-canonical y encodings with values 2²⁵⁵-19 (p) to 2²⁵⁵-1 (p+18).
+               {
+                       "y=p,sign+",
+                       "edffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "0000000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p,sign-",
+                       "edffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "0000000000000000000000000000000000000000000000000000000000000080",
+               },
+               {
+                       "y=p+1,sign+",
+                       "eeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "0100000000000000000000000000000000000000000000000000000000000000",
+               },
+               // "y=p+1,sign-" is already tested above.
+               // p+2 is not a valid y-coordinate.
+               {
+                       "y=p+3,sign+",
+                       "f0ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "0300000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p+3,sign-",
+                       "f0ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "0300000000000000000000000000000000000000000000000000000000000080",
+               },
+               {
+                       "y=p+4,sign+",
+                       "f1ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "0400000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p+4,sign-",
+                       "f1ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "0400000000000000000000000000000000000000000000000000000000000080",
+               },
+               {
+                       "y=p+5,sign+",
+                       "f2ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "0500000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p+5,sign-",
+                       "f2ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "0500000000000000000000000000000000000000000000000000000000000080",
+               },
+               {
+                       "y=p+6,sign+",
+                       "f3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "0600000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p+6,sign-",
+                       "f3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "0600000000000000000000000000000000000000000000000000000000000080",
+               },
+               // p+7 is not a valid y-coordinate.
+               // p+8 is not a valid y-coordinate.
+               {
+                       "y=p+9,sign+",
+                       "f6ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "0900000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p+9,sign-",
+                       "f6ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "0900000000000000000000000000000000000000000000000000000000000080",
+               },
+               {
+                       "y=p+10,sign+",
+                       "f7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "0a00000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p+10,sign-",
+                       "f7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "0a00000000000000000000000000000000000000000000000000000000000080",
+               },
+               // p+11 is not a valid y-coordinate.
+               // p+12 is not a valid y-coordinate.
+               // p+13 is not a valid y-coordinate.
+               {
+                       "y=p+14,sign+",
+                       "fbffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "0e00000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p+14,sign-",
+                       "fbffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "0e00000000000000000000000000000000000000000000000000000000000080",
+               },
+               {
+                       "y=p+15,sign+",
+                       "fcffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "0f00000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p+15,sign-",
+                       "fcffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "0f00000000000000000000000000000000000000000000000000000000000080",
+               },
+               {
+                       "y=p+16,sign+",
+                       "fdffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "1000000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p+16,sign-",
+                       "fdffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "1000000000000000000000000000000000000000000000000000000000000080",
+               },
+               // p+17 is not a valid y-coordinate.
+               {
+                       "y=p+18,sign+",
+                       "ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f",
+                       "1200000000000000000000000000000000000000000000000000000000000000",
+               },
+               {
+                       "y=p+18,sign-",
+                       "ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+                       "1200000000000000000000000000000000000000000000000000000000000080",
+               },
+       }
+       for _, tt := range tests {
+               t.Run(tt.name, func(t *testing.T) {
+                       p1, err := new(Point).SetBytes(decodeHex(tt.encoding))
+                       if err != nil {
+                               t.Fatalf("error decoding non-canonical point: %v", err)
+                       }
+                       p2, err := new(Point).SetBytes(decodeHex(tt.canonical))
+                       if err != nil {
+                               t.Fatalf("error decoding canonical point: %v", err)
+                       }
+                       if p1.Equal(p2) != 1 {
+                               t.Errorf("equivalent points are not equal: %v, %v", p1, p2)
+                       }
+                       if encoding := hex.EncodeToString(p1.Bytes()); encoding != tt.canonical {
+                               t.Errorf("re-encoding does not match canonical; got %q, expected %q", encoding, tt.canonical)
+                       }
+                       checkOnCurve(t, p1, p2)
+               })
+       }
+}
+
+var testAllocationsSink byte
+
+func TestAllocations(t *testing.T) {
+       if allocs := testing.AllocsPerRun(100, func() {
+               p := NewIdentityPoint()
+               p.Add(p, NewGeneratorPoint())
+               s := NewScalar()
+               testAllocationsSink ^= s.Bytes()[0]
+               testAllocationsSink ^= p.Bytes()[0]
+       }); allocs > 0 {
+               t.Errorf("expected zero allocations, got %0.1v", allocs)
+       }
+}
+
+func decodeHex(s string) []byte {
+       b, err := hex.DecodeString(s)
+       if err != nil {
+               panic(err)
+       }
+       return b
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/field/_asm/fe_amd64_asm.go b/src/crypto/ed25519/internal/edwards25519/field/_asm/fe_amd64_asm.go
new file mode 100644 (file)
index 0000000..fbc0cce
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/_asm/fe_amd64_asm.go
@@ -0,0 +1,294 @@
+// Copyright (c) 2021 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package main
+
+import (
+       "fmt"
+
+       . "github.com/mmcloughlin/avo/build"
+       . "github.com/mmcloughlin/avo/gotypes"
+       . "github.com/mmcloughlin/avo/operand"
+       . "github.com/mmcloughlin/avo/reg"
+)
+
+//go:generate go run . -out ../fe_amd64.s -stubs ../fe_amd64.go -pkg field
+
+func main() {
+       Package("crypto/ed25519/internal/edwards25519/field")
+       ConstraintExpr("amd64,gc,!purego")
+       feMul()
+       feSquare()
+       Generate()
+}
+
+type namedComponent struct {
+       Component
+       name string
+}
+
+func (c namedComponent) String() string { return c.name }
+
+type uint128 struct {
+       name   string
+       hi, lo GPVirtual
+}
+
+func (c uint128) String() string { return c.name }
+
+func feSquare() {
+       TEXT("feSquare", NOSPLIT, "func(out, a *Element)")
+       Doc("feSquare sets out = a * a. It works like feSquareGeneric.")
+       Pragma("noescape")
+
+       a := Dereference(Param("a"))
+       l0 := namedComponent{a.Field("l0"), "l0"}
+       l1 := namedComponent{a.Field("l1"), "l1"}
+       l2 := namedComponent{a.Field("l2"), "l2"}
+       l3 := namedComponent{a.Field("l3"), "l3"}
+       l4 := namedComponent{a.Field("l4"), "l4"}
+
+       // r0 = l0×l0 + 19×2×(l1×l4 + l2×l3)
+       r0 := uint128{"r0", GP64(), GP64()}
+       mul64(r0, 1, l0, l0)
+       addMul64(r0, 38, l1, l4)
+       addMul64(r0, 38, l2, l3)
+
+       // r1 = 2×l0×l1 + 19×2×l2×l4 + 19×l3×l3
+       r1 := uint128{"r1", GP64(), GP64()}
+       mul64(r1, 2, l0, l1)
+       addMul64(r1, 38, l2, l4)
+       addMul64(r1, 19, l3, l3)
+
+       // r2 = = 2×l0×l2 + l1×l1 + 19×2×l3×l4
+       r2 := uint128{"r2", GP64(), GP64()}
+       mul64(r2, 2, l0, l2)
+       addMul64(r2, 1, l1, l1)
+       addMul64(r2, 38, l3, l4)
+
+       // r3 = = 2×l0×l3 + 2×l1×l2 + 19×l4×l4
+       r3 := uint128{"r3", GP64(), GP64()}
+       mul64(r3, 2, l0, l3)
+       addMul64(r3, 2, l1, l2)
+       addMul64(r3, 19, l4, l4)
+
+       // r4 = = 2×l0×l4 + 2×l1×l3 + l2×l2
+       r4 := uint128{"r4", GP64(), GP64()}
+       mul64(r4, 2, l0, l4)
+       addMul64(r4, 2, l1, l3)
+       addMul64(r4, 1, l2, l2)
+
+       Comment("First reduction chain")
+       maskLow51Bits := GP64()
+       MOVQ(Imm((1<<51)-1), maskLow51Bits)
+       c0, r0lo := shiftRightBy51(&r0)
+       c1, r1lo := shiftRightBy51(&r1)
+       c2, r2lo := shiftRightBy51(&r2)
+       c3, r3lo := shiftRightBy51(&r3)
+       c4, r4lo := shiftRightBy51(&r4)
+       maskAndAdd(r0lo, maskLow51Bits, c4, 19)
+       maskAndAdd(r1lo, maskLow51Bits, c0, 1)
+       maskAndAdd(r2lo, maskLow51Bits, c1, 1)
+       maskAndAdd(r3lo, maskLow51Bits, c2, 1)
+       maskAndAdd(r4lo, maskLow51Bits, c3, 1)
+
+       Comment("Second reduction chain (carryPropagate)")
+       // c0 = r0 >> 51
+       MOVQ(r0lo, c0)
+       SHRQ(Imm(51), c0)
+       // c1 = r1 >> 51
+       MOVQ(r1lo, c1)
+       SHRQ(Imm(51), c1)
+       // c2 = r2 >> 51
+       MOVQ(r2lo, c2)
+       SHRQ(Imm(51), c2)
+       // c3 = r3 >> 51
+       MOVQ(r3lo, c3)
+       SHRQ(Imm(51), c3)
+       // c4 = r4 >> 51
+       MOVQ(r4lo, c4)
+       SHRQ(Imm(51), c4)
+       maskAndAdd(r0lo, maskLow51Bits, c4, 19)
+       maskAndAdd(r1lo, maskLow51Bits, c0, 1)
+       maskAndAdd(r2lo, maskLow51Bits, c1, 1)
+       maskAndAdd(r3lo, maskLow51Bits, c2, 1)
+       maskAndAdd(r4lo, maskLow51Bits, c3, 1)
+
+       Comment("Store output")
+       out := Dereference(Param("out"))
+       Store(r0lo, out.Field("l0"))
+       Store(r1lo, out.Field("l1"))
+       Store(r2lo, out.Field("l2"))
+       Store(r3lo, out.Field("l3"))
+       Store(r4lo, out.Field("l4"))
+
+       RET()
+}
+
+func feMul() {
+       TEXT("feMul", NOSPLIT, "func(out, a, b *Element)")
+       Doc("feMul sets out = a * b. It works like feMulGeneric.")
+       Pragma("noescape")
+
+       a := Dereference(Param("a"))
+       a0 := namedComponent{a.Field("l0"), "a0"}
+       a1 := namedComponent{a.Field("l1"), "a1"}
+       a2 := namedComponent{a.Field("l2"), "a2"}
+       a3 := namedComponent{a.Field("l3"), "a3"}
+       a4 := namedComponent{a.Field("l4"), "a4"}
+
+       b := Dereference(Param("b"))
+       b0 := namedComponent{b.Field("l0"), "b0"}
+       b1 := namedComponent{b.Field("l1"), "b1"}
+       b2 := namedComponent{b.Field("l2"), "b2"}
+       b3 := namedComponent{b.Field("l3"), "b3"}
+       b4 := namedComponent{b.Field("l4"), "b4"}
+
+       // r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1)
+       r0 := uint128{"r0", GP64(), GP64()}
+       mul64(r0, 1, a0, b0)
+       addMul64(r0, 19, a1, b4)
+       addMul64(r0, 19, a2, b3)
+       addMul64(r0, 19, a3, b2)
+       addMul64(r0, 19, a4, b1)
+
+       // r1 = a0×b1 + a1×b0 + 19×(a2×b4 + a3×b3 + a4×b2)
+       r1 := uint128{"r1", GP64(), GP64()}
+       mul64(r1, 1, a0, b1)
+       addMul64(r1, 1, a1, b0)
+       addMul64(r1, 19, a2, b4)
+       addMul64(r1, 19, a3, b3)
+       addMul64(r1, 19, a4, b2)
+
+       // r2 = a0×b2 + a1×b1 + a2×b0 + 19×(a3×b4 + a4×b3)
+       r2 := uint128{"r2", GP64(), GP64()}
+       mul64(r2, 1, a0, b2)
+       addMul64(r2, 1, a1, b1)
+       addMul64(r2, 1, a2, b0)
+       addMul64(r2, 19, a3, b4)
+       addMul64(r2, 19, a4, b3)
+
+       // r3 = a0×b3 + a1×b2 + a2×b1 + a3×b0 + 19×a4×b4
+       r3 := uint128{"r3", GP64(), GP64()}
+       mul64(r3, 1, a0, b3)
+       addMul64(r3, 1, a1, b2)
+       addMul64(r3, 1, a2, b1)
+       addMul64(r3, 1, a3, b0)
+       addMul64(r3, 19, a4, b4)
+
+       // r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0
+       r4 := uint128{"r4", GP64(), GP64()}
+       mul64(r4, 1, a0, b4)
+       addMul64(r4, 1, a1, b3)
+       addMul64(r4, 1, a2, b2)
+       addMul64(r4, 1, a3, b1)
+       addMul64(r4, 1, a4, b0)
+
+       Comment("First reduction chain")
+       maskLow51Bits := GP64()
+       MOVQ(Imm((1<<51)-1), maskLow51Bits)
+       c0, r0lo := shiftRightBy51(&r0)
+       c1, r1lo := shiftRightBy51(&r1)
+       c2, r2lo := shiftRightBy51(&r2)
+       c3, r3lo := shiftRightBy51(&r3)
+       c4, r4lo := shiftRightBy51(&r4)
+       maskAndAdd(r0lo, maskLow51Bits, c4, 19)
+       maskAndAdd(r1lo, maskLow51Bits, c0, 1)
+       maskAndAdd(r2lo, maskLow51Bits, c1, 1)
+       maskAndAdd(r3lo, maskLow51Bits, c2, 1)
+       maskAndAdd(r4lo, maskLow51Bits, c3, 1)
+
+       Comment("Second reduction chain (carryPropagate)")
+       // c0 = r0 >> 51
+       MOVQ(r0lo, c0)
+       SHRQ(Imm(51), c0)
+       // c1 = r1 >> 51
+       MOVQ(r1lo, c1)
+       SHRQ(Imm(51), c1)
+       // c2 = r2 >> 51
+       MOVQ(r2lo, c2)
+       SHRQ(Imm(51), c2)
+       // c3 = r3 >> 51
+       MOVQ(r3lo, c3)
+       SHRQ(Imm(51), c3)
+       // c4 = r4 >> 51
+       MOVQ(r4lo, c4)
+       SHRQ(Imm(51), c4)
+       maskAndAdd(r0lo, maskLow51Bits, c4, 19)
+       maskAndAdd(r1lo, maskLow51Bits, c0, 1)
+       maskAndAdd(r2lo, maskLow51Bits, c1, 1)
+       maskAndAdd(r3lo, maskLow51Bits, c2, 1)
+       maskAndAdd(r4lo, maskLow51Bits, c3, 1)
+
+       Comment("Store output")
+       out := Dereference(Param("out"))
+       Store(r0lo, out.Field("l0"))
+       Store(r1lo, out.Field("l1"))
+       Store(r2lo, out.Field("l2"))
+       Store(r3lo, out.Field("l3"))
+       Store(r4lo, out.Field("l4"))
+
+       RET()
+}
+
+// mul64 sets r to i * aX * bX.
+func mul64(r uint128, i int, aX, bX namedComponent) {
+       switch i {
+       case 1:
+               Comment(fmt.Sprintf("%s = %s×%s", r, aX, bX))
+               Load(aX, RAX)
+       case 2:
+               Comment(fmt.Sprintf("%s = 2×%s×%s", r, aX, bX))
+               Load(aX, RAX)
+               SHLQ(Imm(1), RAX)
+       default:
+               panic("unsupported i value")
+       }
+       MULQ(mustAddr(bX)) // RDX, RAX = RAX * bX
+       MOVQ(RAX, r.lo)
+       MOVQ(RDX, r.hi)
+}
+
+// addMul64 sets r to r + i * aX * bX.
+func addMul64(r uint128, i uint64, aX, bX namedComponent) {
+       switch i {
+       case 1:
+               Comment(fmt.Sprintf("%s += %s×%s", r, aX, bX))
+               Load(aX, RAX)
+       default:
+               Comment(fmt.Sprintf("%s += %d×%s×%s", r, i, aX, bX))
+               IMUL3Q(Imm(i), Load(aX, GP64()), RAX)
+       }
+       MULQ(mustAddr(bX)) // RDX, RAX = RAX * bX
+       ADDQ(RAX, r.lo)
+       ADCQ(RDX, r.hi)
+}
+
+// shiftRightBy51 returns r >> 51 and r.lo.
+//
+// After this function is called, the uint128 may not be used anymore.
+func shiftRightBy51(r *uint128) (out, lo GPVirtual) {
+       out = r.hi
+       lo = r.lo
+       SHLQ(Imm(64-51), r.lo, r.hi)
+       r.lo, r.hi = nil, nil // make sure the uint128 is unusable
+       return
+}
+
+// maskAndAdd sets r = r&mask + c*i.
+func maskAndAdd(r, mask, c GPVirtual, i uint64) {
+       ANDQ(mask, r)
+       if i != 1 {
+               IMUL3Q(Imm(i), c, c)
+       }
+       ADDQ(c, r)
+}
+
+func mustAddr(c Component) Op {
+       b, err := c.Resolve()
+       if err != nil {
+               panic(err)
+       }
+       return b.Addr
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/field/_asm/go.mod b/src/crypto/ed25519/internal/edwards25519/field/_asm/go.mod
new file mode 100644 (file)
index 0000000..1127ade
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/_asm/go.mod
@@ -0,0 +1,5 @@
+module asm
+
+go 1.16
+
+require github.com/mmcloughlin/avo v0.2.0

diff --git a/src/crypto/ed25519/internal/edwards25519/field/_asm/go.sum b/src/crypto/ed25519/internal/edwards25519/field/_asm/go.sum
new file mode 100644 (file)
index 0000000..dae4777
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/_asm/go.sum
@@ -0,0 +1,31 @@
+github.com/mmcloughlin/avo v0.2.0 h1:6vhoSaKtxb6f4RiH+LK2qL6GSMpFzhEwJYTTSZNy09w=
+github.com/mmcloughlin/avo v0.2.0/go.mod h1:5tidO2Z9Z7N6X7UMcGg+1KTj51O8OxYDCMHxCZTVpEA=
+github.com/yuin/goldmark v1.2.1/go.mod h1:3hX8gzYuyVAZsxl0MRgGTJEmQBFcNTphYh9decYSb74=
+golang.org/x/arch v0.0.0-20210405154355-08b684f594a5/go.mod h1:flIaEI6LNU6xOCD5PaJvn9wGP0agmIOqjrtsKGRguv4=
+golang.org/x/crypto v0.0.0-20190308221718-c2843e01d9a2/go.mod h1:djNgcEr1/C05ACkg1iLfiJU5Ep61QUkGW8qpdssI0+w=
+golang.org/x/crypto v0.0.0-20191011191535-87dc89f01550/go.mod h1:yigFU9vqHzYiE8UmvKecakEJjdnWj3jj499lnFckfCI=
+golang.org/x/crypto v0.0.0-20200622213623-75b288015ac9/go.mod h1:LzIPMQfyMNhhGPhUkYOs5KpL4U8rLKemX1yGLhDgUto=
+golang.org/x/mod v0.3.0 h1:RM4zey1++hCTbCVQfnWeKs9/IEsaBLA8vTkd0WVtmH4=
+golang.org/x/mod v0.3.0/go.mod h1:s0Qsj1ACt9ePp/hMypM3fl4fZqREWJwdYDEqhRiZZUA=
+golang.org/x/net v0.0.0-20190404232315-eb5bcb51f2a3/go.mod h1:t9HGtf8HONx5eT2rtn7q6eTqICYqUVnKs3thJo3Qplg=
+golang.org/x/net v0.0.0-20190620200207-3b0461eec859/go.mod h1:z5CRVTTTmAJ677TzLLGU+0bjPO0LkuOLi4/5GtJWs/s=
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diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe.go b/src/crypto/ed25519/internal/edwards25519/field/fe.go
new file mode 100644 (file)
index 0000000..dbe8659
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/fe.go
@@ -0,0 +1,416 @@
+// Copyright (c) 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package field implements fast arithmetic modulo 2^255-19.
+package field
+
+import (
+       "crypto/subtle"
+       "encoding/binary"
+       "math/bits"
+)
+
+// Element represents an element of the field GF(2^255-19). Note that this
+// is not a cryptographically secure group, and should only be used to interact
+// with edwards25519.Point coordinates.
+//
+// This type works similarly to math/big.Int, and all arguments and receivers
+// are allowed to alias.
+//
+// The zero value is a valid zero element.
+type Element struct {
+       // An element t represents the integer
+       //     t.l0 + t.l1*2^51 + t.l2*2^102 + t.l3*2^153 + t.l4*2^204
+       //
+       // Between operations, all limbs are expected to be lower than 2^52.
+       l0 uint64
+       l1 uint64
+       l2 uint64
+       l3 uint64
+       l4 uint64
+}
+
+const maskLow51Bits uint64 = (1 << 51) - 1
+
+var feZero = &Element{0, 0, 0, 0, 0}
+
+// Zero sets v = 0, and returns v.
+func (v *Element) Zero() *Element {
+       *v = *feZero
+       return v
+}
+
+var feOne = &Element{1, 0, 0, 0, 0}
+
+// One sets v = 1, and returns v.
+func (v *Element) One() *Element {
+       *v = *feOne
+       return v
+}
+
+// reduce reduces v modulo 2^255 - 19 and returns it.
+func (v *Element) reduce() *Element {
+       v.carryPropagate()
+
+       // After the light reduction we now have a field element representation
+       // v < 2^255 + 2^13 * 19, but need v < 2^255 - 19.
+
+       // If v >= 2^255 - 19, then v + 19 >= 2^255, which would overflow 2^255 - 1,
+       // generating a carry. That is, c will be 0 if v < 2^255 - 19, and 1 otherwise.
+       c := (v.l0 + 19) >> 51
+       c = (v.l1 + c) >> 51
+       c = (v.l2 + c) >> 51
+       c = (v.l3 + c) >> 51
+       c = (v.l4 + c) >> 51
+
+       // If v < 2^255 - 19 and c = 0, this will be a no-op. Otherwise, it's
+       // effectively applying the reduction identity to the carry.
+       v.l0 += 19 * c
+
+       v.l1 += v.l0 >> 51
+       v.l0 = v.l0 & maskLow51Bits
+       v.l2 += v.l1 >> 51
+       v.l1 = v.l1 & maskLow51Bits
+       v.l3 += v.l2 >> 51
+       v.l2 = v.l2 & maskLow51Bits
+       v.l4 += v.l3 >> 51
+       v.l3 = v.l3 & maskLow51Bits
+       // no additional carry
+       v.l4 = v.l4 & maskLow51Bits
+
+       return v
+}
+
+// Add sets v = a + b, and returns v.
+func (v *Element) Add(a, b *Element) *Element {
+       v.l0 = a.l0 + b.l0
+       v.l1 = a.l1 + b.l1
+       v.l2 = a.l2 + b.l2
+       v.l3 = a.l3 + b.l3
+       v.l4 = a.l4 + b.l4
+       // Using the generic implementation here is actually faster than the
+       // assembly. Probably because the body of this function is so simple that
+       // the compiler can figure out better optimizations by inlining the carry
+       // propagation.
+       return v.carryPropagateGeneric()
+}
+
+// Subtract sets v = a - b, and returns v.
+func (v *Element) Subtract(a, b *Element) *Element {
+       // We first add 2 * p, to guarantee the subtraction won't underflow, and
+       // then subtract b (which can be up to 2^255 + 2^13 * 19).
+       v.l0 = (a.l0 + 0xFFFFFFFFFFFDA) - b.l0
+       v.l1 = (a.l1 + 0xFFFFFFFFFFFFE) - b.l1
+       v.l2 = (a.l2 + 0xFFFFFFFFFFFFE) - b.l2
+       v.l3 = (a.l3 + 0xFFFFFFFFFFFFE) - b.l3
+       v.l4 = (a.l4 + 0xFFFFFFFFFFFFE) - b.l4
+       return v.carryPropagate()
+}
+
+// Negate sets v = -a, and returns v.
+func (v *Element) Negate(a *Element) *Element {
+       return v.Subtract(feZero, a)
+}
+
+// Invert sets v = 1/z mod p, and returns v.
+//
+// If z == 0, Invert returns v = 0.
+func (v *Element) Invert(z *Element) *Element {
+       // Inversion is implemented as exponentiation with exponent p − 2. It uses the
+       // same sequence of 255 squarings and 11 multiplications as [Curve25519].
+       var z2, z9, z11, z2_5_0, z2_10_0, z2_20_0, z2_50_0, z2_100_0, t Element
+
+       z2.Square(z)             // 2
+       t.Square(&z2)            // 4
+       t.Square(&t)             // 8
+       z9.Multiply(&t, z)       // 9
+       z11.Multiply(&z9, &z2)   // 11
+       t.Square(&z11)           // 22
+       z2_5_0.Multiply(&t, &z9) // 31 = 2^5 - 2^0
+
+       t.Square(&z2_5_0) // 2^6 - 2^1
+       for i := 0; i < 4; i++ {
+               t.Square(&t) // 2^10 - 2^5
+       }
+       z2_10_0.Multiply(&t, &z2_5_0) // 2^10 - 2^0
+
+       t.Square(&z2_10_0) // 2^11 - 2^1
+       for i := 0; i < 9; i++ {
+               t.Square(&t) // 2^20 - 2^10
+       }
+       z2_20_0.Multiply(&t, &z2_10_0) // 2^20 - 2^0
+
+       t.Square(&z2_20_0) // 2^21 - 2^1
+       for i := 0; i < 19; i++ {
+               t.Square(&t) // 2^40 - 2^20
+       }
+       t.Multiply(&t, &z2_20_0) // 2^40 - 2^0
+
+       t.Square(&t) // 2^41 - 2^1
+       for i := 0; i < 9; i++ {
+               t.Square(&t) // 2^50 - 2^10
+       }
+       z2_50_0.Multiply(&t, &z2_10_0) // 2^50 - 2^0
+
+       t.Square(&z2_50_0) // 2^51 - 2^1
+       for i := 0; i < 49; i++ {
+               t.Square(&t) // 2^100 - 2^50
+       }
+       z2_100_0.Multiply(&t, &z2_50_0) // 2^100 - 2^0
+
+       t.Square(&z2_100_0) // 2^101 - 2^1
+       for i := 0; i < 99; i++ {
+               t.Square(&t) // 2^200 - 2^100
+       }
+       t.Multiply(&t, &z2_100_0) // 2^200 - 2^0
+
+       t.Square(&t) // 2^201 - 2^1
+       for i := 0; i < 49; i++ {
+               t.Square(&t) // 2^250 - 2^50
+       }
+       t.Multiply(&t, &z2_50_0) // 2^250 - 2^0
+
+       t.Square(&t) // 2^251 - 2^1
+       t.Square(&t) // 2^252 - 2^2
+       t.Square(&t) // 2^253 - 2^3
+       t.Square(&t) // 2^254 - 2^4
+       t.Square(&t) // 2^255 - 2^5
+
+       return v.Multiply(&t, &z11) // 2^255 - 21
+}
+
+// Set sets v = a, and returns v.
+func (v *Element) Set(a *Element) *Element {
+       *v = *a
+       return v
+}
+
+// SetBytes sets v to x, which must be a 32-byte little-endian encoding.
+//
+// Consistent with RFC 7748, the most significant bit (the high bit of the
+// last byte) is ignored, and non-canonical values (2^255-19 through 2^255-1)
+// are accepted. Note that this is laxer than specified by RFC 8032.
+func (v *Element) SetBytes(x []byte) *Element {
+       if len(x) != 32 {
+               panic("edwards25519: invalid field element input size")
+       }
+
+       // Bits 0:51 (bytes 0:8, bits 0:64, shift 0, mask 51).
+       v.l0 = binary.LittleEndian.Uint64(x[0:8])
+       v.l0 &= maskLow51Bits
+       // Bits 51:102 (bytes 6:14, bits 48:112, shift 3, mask 51).
+       v.l1 = binary.LittleEndian.Uint64(x[6:14]) >> 3
+       v.l1 &= maskLow51Bits
+       // Bits 102:153 (bytes 12:20, bits 96:160, shift 6, mask 51).
+       v.l2 = binary.LittleEndian.Uint64(x[12:20]) >> 6
+       v.l2 &= maskLow51Bits
+       // Bits 153:204 (bytes 19:27, bits 152:216, shift 1, mask 51).
+       v.l3 = binary.LittleEndian.Uint64(x[19:27]) >> 1
+       v.l3 &= maskLow51Bits
+       // Bits 204:251 (bytes 24:32, bits 192:256, shift 12, mask 51).
+       // Note: not bytes 25:33, shift 4, to avoid overread.
+       v.l4 = binary.LittleEndian.Uint64(x[24:32]) >> 12
+       v.l4 &= maskLow51Bits
+
+       return v
+}
+
+// Bytes returns the canonical 32-byte little-endian encoding of v.
+func (v *Element) Bytes() []byte {
+       // This function is outlined to make the allocations inline in the caller
+       // rather than happen on the heap.
+       var out [32]byte
+       return v.bytes(&out)
+}
+
+func (v *Element) bytes(out *[32]byte) []byte {
+       t := *v
+       t.reduce()
+
+       var buf [8]byte
+       for i, l := range [5]uint64{t.l0, t.l1, t.l2, t.l3, t.l4} {
+               bitsOffset := i * 51
+               binary.LittleEndian.PutUint64(buf[:], l<<uint(bitsOffset%8))
+               for i, bb := range buf {
+                       off := bitsOffset/8 + i
+                       if off >= len(out) {
+                               break
+                       }
+                       out[off] |= bb
+               }
+       }
+
+       return out[:]
+}
+
+// Equal returns 1 if v and u are equal, and 0 otherwise.
+func (v *Element) Equal(u *Element) int {
+       sa, sv := u.Bytes(), v.Bytes()
+       return subtle.ConstantTimeCompare(sa, sv)
+}
+
+// mask64Bits returns 0xffffffff if cond is 1, and 0 otherwise.
+func mask64Bits(cond int) uint64 { return ^(uint64(cond) - 1) }
+
+// Select sets v to a if cond == 1, and to b if cond == 0.
+func (v *Element) Select(a, b *Element, cond int) *Element {
+       m := mask64Bits(cond)
+       v.l0 = (m & a.l0) | (^m & b.l0)
+       v.l1 = (m & a.l1) | (^m & b.l1)
+       v.l2 = (m & a.l2) | (^m & b.l2)
+       v.l3 = (m & a.l3) | (^m & b.l3)
+       v.l4 = (m & a.l4) | (^m & b.l4)
+       return v
+}
+
+// Swap swaps v and u if cond == 1 or leaves them unchanged if cond == 0, and returns v.
+func (v *Element) Swap(u *Element, cond int) {
+       m := mask64Bits(cond)
+       t := m & (v.l0 ^ u.l0)
+       v.l0 ^= t
+       u.l0 ^= t
+       t = m & (v.l1 ^ u.l1)
+       v.l1 ^= t
+       u.l1 ^= t
+       t = m & (v.l2 ^ u.l2)
+       v.l2 ^= t
+       u.l2 ^= t
+       t = m & (v.l3 ^ u.l3)
+       v.l3 ^= t
+       u.l3 ^= t
+       t = m & (v.l4 ^ u.l4)
+       v.l4 ^= t
+       u.l4 ^= t
+}
+
+// IsNegative returns 1 if v is negative, and 0 otherwise.
+func (v *Element) IsNegative() int {
+       return int(v.Bytes()[0] & 1)
+}
+
+// Absolute sets v to |u|, and returns v.
+func (v *Element) Absolute(u *Element) *Element {
+       return v.Select(new(Element).Negate(u), u, u.IsNegative())
+}
+
+// Multiply sets v = x * y, and returns v.
+func (v *Element) Multiply(x, y *Element) *Element {
+       feMul(v, x, y)
+       return v
+}
+
+// Square sets v = x * x, and returns v.
+func (v *Element) Square(x *Element) *Element {
+       feSquare(v, x)
+       return v
+}
+
+// Mult32 sets v = x * y, and returns v.
+func (v *Element) Mult32(x *Element, y uint32) *Element {
+       x0lo, x0hi := mul51(x.l0, y)
+       x1lo, x1hi := mul51(x.l1, y)
+       x2lo, x2hi := mul51(x.l2, y)
+       x3lo, x3hi := mul51(x.l3, y)
+       x4lo, x4hi := mul51(x.l4, y)
+       v.l0 = x0lo + 19*x4hi // carried over per the reduction identity
+       v.l1 = x1lo + x0hi
+       v.l2 = x2lo + x1hi
+       v.l3 = x3lo + x2hi
+       v.l4 = x4lo + x3hi
+       // The hi portions are going to be only 32 bits, plus any previous excess,
+       // so we can skip the carry propagation.
+       return v
+}
+
+// mul51 returns lo + hi * 2⁵¹ = a * b.
+func mul51(a uint64, b uint32) (lo uint64, hi uint64) {
+       mh, ml := bits.Mul64(a, uint64(b))
+       lo = ml & maskLow51Bits
+       hi = (mh << 13) | (ml >> 51)
+       return
+}
+
+// Pow22523 set v = x^((p-5)/8), and returns v. (p-5)/8 is 2^252-3.
+func (v *Element) Pow22523(x *Element) *Element {
+       var t0, t1, t2 Element
+
+       t0.Square(x)             // x^2
+       t1.Square(&t0)           // x^4
+       t1.Square(&t1)           // x^8
+       t1.Multiply(x, &t1)      // x^9
+       t0.Multiply(&t0, &t1)    // x^11
+       t0.Square(&t0)           // x^22
+       t0.Multiply(&t1, &t0)    // x^31
+       t1.Square(&t0)           // x^62
+       for i := 1; i < 5; i++ { // x^992
+               t1.Square(&t1)
+       }
+       t0.Multiply(&t1, &t0)     // x^1023 -> 1023 = 2^10 - 1
+       t1.Square(&t0)            // 2^11 - 2
+       for i := 1; i < 10; i++ { // 2^20 - 2^10
+               t1.Square(&t1)
+       }
+       t1.Multiply(&t1, &t0)     // 2^20 - 1
+       t2.Square(&t1)            // 2^21 - 2
+       for i := 1; i < 20; i++ { // 2^40 - 2^20
+               t2.Square(&t2)
+       }
+       t1.Multiply(&t2, &t1)     // 2^40 - 1
+       t1.Square(&t1)            // 2^41 - 2
+       for i := 1; i < 10; i++ { // 2^50 - 2^10
+               t1.Square(&t1)
+       }
+       t0.Multiply(&t1, &t0)     // 2^50 - 1
+       t1.Square(&t0)            // 2^51 - 2
+       for i := 1; i < 50; i++ { // 2^100 - 2^50
+               t1.Square(&t1)
+       }
+       t1.Multiply(&t1, &t0)      // 2^100 - 1
+       t2.Square(&t1)             // 2^101 - 2
+       for i := 1; i < 100; i++ { // 2^200 - 2^100
+               t2.Square(&t2)
+       }
+       t1.Multiply(&t2, &t1)     // 2^200 - 1
+       t1.Square(&t1)            // 2^201 - 2
+       for i := 1; i < 50; i++ { // 2^250 - 2^50
+               t1.Square(&t1)
+       }
+       t0.Multiply(&t1, &t0)     // 2^250 - 1
+       t0.Square(&t0)            // 2^251 - 2
+       t0.Square(&t0)            // 2^252 - 4
+       return v.Multiply(&t0, x) // 2^252 - 3 -> x^(2^252-3)
+}
+
+// sqrtM1 is 2^((p-1)/4), which squared is equal to -1 by Euler's Criterion.
+var sqrtM1 = &Element{1718705420411056, 234908883556509,
+       2233514472574048, 2117202627021982, 765476049583133}
+
+// SqrtRatio sets r to the non-negative square root of the ratio of u and v.
+//
+// If u/v is square, SqrtRatio returns r and 1. If u/v is not square, SqrtRatio
+// sets r according to Section 4.3 of draft-irtf-cfrg-ristretto255-decaf448-00,
+// and returns r and 0.
+func (r *Element) SqrtRatio(u, v *Element) (rr *Element, wasSquare int) {
+       var a, b Element
+
+       // r = (u * v3) * (u * v7)^((p-5)/8)
+       v2 := a.Square(v)
+       uv3 := b.Multiply(u, b.Multiply(v2, v))
+       uv7 := a.Multiply(uv3, a.Square(v2))
+       r.Multiply(uv3, r.Pow22523(uv7))
+
+       check := a.Multiply(v, a.Square(r)) // check = v * r^2
+
+       uNeg := b.Negate(u)
+       correctSignSqrt := check.Equal(u)
+       flippedSignSqrt := check.Equal(uNeg)
+       flippedSignSqrtI := check.Equal(uNeg.Multiply(uNeg, sqrtM1))
+
+       rPrime := b.Multiply(r, sqrtM1) // r_prime = SQRT_M1 * r
+       // r = CT_SELECT(r_prime IF flipped_sign_sqrt | flipped_sign_sqrt_i ELSE r)
+       r.Select(rPrime, r, flippedSignSqrt|flippedSignSqrtI)
+
+       r.Absolute(r) // Choose the nonnegative square root.
+       return r, correctSignSqrt | flippedSignSqrt
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_alias_test.go b/src/crypto/ed25519/internal/edwards25519/field/fe_alias_test.go
new file mode 100644 (file)
index 0000000..5ad81df
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/fe_alias_test.go
@@ -0,0 +1,126 @@
+// Copyright (c) 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package field
+
+import (
+       "testing"
+       "testing/quick"
+)
+
+func checkAliasingOneArg(f func(v, x *Element) *Element) func(v, x Element) bool {
+       return func(v, x Element) bool {
+               x1, v1 := x, x
+
+               // Calculate a reference f(x) without aliasing.
+               if out := f(&v, &x); out != &v && isInBounds(out) {
+                       return false
+               }
+
+               // Test aliasing the argument and the receiver.
+               if out := f(&v1, &v1); out != &v1 || v1 != v {
+                       return false
+               }
+
+               // Ensure the arguments was not modified.
+               return x == x1
+       }
+}
+
+func checkAliasingTwoArgs(f func(v, x, y *Element) *Element) func(v, x, y Element) bool {
+       return func(v, x, y Element) bool {
+               x1, y1, v1 := x, y, Element{}
+
+               // Calculate a reference f(x, y) without aliasing.
+               if out := f(&v, &x, &y); out != &v && isInBounds(out) {
+                       return false
+               }
+
+               // Test aliasing the first argument and the receiver.
+               v1 = x
+               if out := f(&v1, &v1, &y); out != &v1 || v1 != v {
+                       return false
+               }
+               // Test aliasing the second argument and the receiver.
+               v1 = y
+               if out := f(&v1, &x, &v1); out != &v1 || v1 != v {
+                       return false
+               }
+
+               // Calculate a reference f(x, x) without aliasing.
+               if out := f(&v, &x, &x); out != &v {
+                       return false
+               }
+
+               // Test aliasing the first argument and the receiver.
+               v1 = x
+               if out := f(&v1, &v1, &x); out != &v1 || v1 != v {
+                       return false
+               }
+               // Test aliasing the second argument and the receiver.
+               v1 = x
+               if out := f(&v1, &x, &v1); out != &v1 || v1 != v {
+                       return false
+               }
+               // Test aliasing both arguments and the receiver.
+               v1 = x
+               if out := f(&v1, &v1, &v1); out != &v1 || v1 != v {
+                       return false
+               }
+
+               // Ensure the arguments were not modified.
+               return x == x1 && y == y1
+       }
+}
+
+// TestAliasing checks that receivers and arguments can alias each other without
+// leading to incorrect results. That is, it ensures that it's safe to write
+//
+//     v.Invert(v)
+//
+// or
+//
+//     v.Add(v, v)
+//
+// without any of the inputs getting clobbered by the output being written.
+func TestAliasing(t *testing.T) {
+       type target struct {
+               name     string
+               oneArgF  func(v, x *Element) *Element
+               twoArgsF func(v, x, y *Element) *Element
+       }
+       for _, tt := range []target{
+               {name: "Absolute", oneArgF: (*Element).Absolute},
+               {name: "Invert", oneArgF: (*Element).Invert},
+               {name: "Negate", oneArgF: (*Element).Negate},
+               {name: "Set", oneArgF: (*Element).Set},
+               {name: "Square", oneArgF: (*Element).Square},
+               {name: "Multiply", twoArgsF: (*Element).Multiply},
+               {name: "Add", twoArgsF: (*Element).Add},
+               {name: "Subtract", twoArgsF: (*Element).Subtract},
+               {
+                       name: "Select0",
+                       twoArgsF: func(v, x, y *Element) *Element {
+                               return (*Element).Select(v, x, y, 0)
+                       },
+               },
+               {
+                       name: "Select1",
+                       twoArgsF: func(v, x, y *Element) *Element {
+                               return (*Element).Select(v, x, y, 1)
+                       },
+               },
+       } {
+               var err error
+               switch {
+               case tt.oneArgF != nil:
+                       err = quick.Check(checkAliasingOneArg(tt.oneArgF), &quick.Config{MaxCountScale: 1 << 8})
+               case tt.twoArgsF != nil:
+                       err = quick.Check(checkAliasingTwoArgs(tt.twoArgsF), &quick.Config{MaxCountScale: 1 << 8})
+               }
+               if err != nil {
+                       t.Errorf("%v: %v", tt.name, err)
+               }
+       }
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.go b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.go
new file mode 100644 (file)
index 0000000..44dc8e8
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.go
@@ -0,0 +1,13 @@
+// Code generated by command: go run fe_amd64_asm.go -out ../fe_amd64.s -stubs ../fe_amd64.go -pkg field. DO NOT EDIT.
+
+// +build amd64,gc,!purego
+
+package field
+
+// feMul sets out = a * b. It works like feMulGeneric.
+//go:noescape
+func feMul(out *Element, a *Element, b *Element)
+
+// feSquare sets out = a * a. It works like feSquareGeneric.
+//go:noescape
+func feSquare(out *Element, a *Element)

diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.s b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.s
new file mode 100644 (file)
index 0000000..0aa1e86
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.s
@@ -0,0 +1,378 @@
+// Code generated by command: go run fe_amd64_asm.go -out ../fe_amd64.s -stubs ../fe_amd64.go -pkg field. DO NOT EDIT.
+
+// +build amd64,gc,!purego
+
+#include "textflag.h"
+
+// func feMul(out *Element, a *Element, b *Element)
+TEXT ·feMul(SB), NOSPLIT, $0-24
+       MOVQ a+8(FP), CX
+       MOVQ b+16(FP), BX
+
+       // r0 = a0×b0
+       MOVQ (CX), AX
+       MULQ (BX)
+       MOVQ AX, DI
+       MOVQ DX, SI
+
+       // r0 += 19×a1×b4
+       MOVQ   8(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   32(BX)
+       ADDQ   AX, DI
+       ADCQ   DX, SI
+
+       // r0 += 19×a2×b3
+       MOVQ   16(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   24(BX)
+       ADDQ   AX, DI
+       ADCQ   DX, SI
+
+       // r0 += 19×a3×b2
+       MOVQ   24(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   16(BX)
+       ADDQ   AX, DI
+       ADCQ   DX, SI
+
+       // r0 += 19×a4×b1
+       MOVQ   32(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   8(BX)
+       ADDQ   AX, DI
+       ADCQ   DX, SI
+
+       // r1 = a0×b1
+       MOVQ (CX), AX
+       MULQ 8(BX)
+       MOVQ AX, R9
+       MOVQ DX, R8
+
+       // r1 += a1×b0
+       MOVQ 8(CX), AX
+       MULQ (BX)
+       ADDQ AX, R9
+       ADCQ DX, R8
+
+       // r1 += 19×a2×b4
+       MOVQ   16(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   32(BX)
+       ADDQ   AX, R9
+       ADCQ   DX, R8
+
+       // r1 += 19×a3×b3
+       MOVQ   24(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   24(BX)
+       ADDQ   AX, R9
+       ADCQ   DX, R8
+
+       // r1 += 19×a4×b2
+       MOVQ   32(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   16(BX)
+       ADDQ   AX, R9
+       ADCQ   DX, R8
+
+       // r2 = a0×b2
+       MOVQ (CX), AX
+       MULQ 16(BX)
+       MOVQ AX, R11
+       MOVQ DX, R10
+
+       // r2 += a1×b1
+       MOVQ 8(CX), AX
+       MULQ 8(BX)
+       ADDQ AX, R11
+       ADCQ DX, R10
+
+       // r2 += a2×b0
+       MOVQ 16(CX), AX
+       MULQ (BX)
+       ADDQ AX, R11
+       ADCQ DX, R10
+
+       // r2 += 19×a3×b4
+       MOVQ   24(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   32(BX)
+       ADDQ   AX, R11
+       ADCQ   DX, R10
+
+       // r2 += 19×a4×b3
+       MOVQ   32(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   24(BX)
+       ADDQ   AX, R11
+       ADCQ   DX, R10
+
+       // r3 = a0×b3
+       MOVQ (CX), AX
+       MULQ 24(BX)
+       MOVQ AX, R13
+       MOVQ DX, R12
+
+       // r3 += a1×b2
+       MOVQ 8(CX), AX
+       MULQ 16(BX)
+       ADDQ AX, R13
+       ADCQ DX, R12
+
+       // r3 += a2×b1
+       MOVQ 16(CX), AX
+       MULQ 8(BX)
+       ADDQ AX, R13
+       ADCQ DX, R12
+
+       // r3 += a3×b0
+       MOVQ 24(CX), AX
+       MULQ (BX)
+       ADDQ AX, R13
+       ADCQ DX, R12
+
+       // r3 += 19×a4×b4
+       MOVQ   32(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   32(BX)
+       ADDQ   AX, R13
+       ADCQ   DX, R12
+
+       // r4 = a0×b4
+       MOVQ (CX), AX
+       MULQ 32(BX)
+       MOVQ AX, R15
+       MOVQ DX, R14
+
+       // r4 += a1×b3
+       MOVQ 8(CX), AX
+       MULQ 24(BX)
+       ADDQ AX, R15
+       ADCQ DX, R14
+
+       // r4 += a2×b2
+       MOVQ 16(CX), AX
+       MULQ 16(BX)
+       ADDQ AX, R15
+       ADCQ DX, R14
+
+       // r4 += a3×b1
+       MOVQ 24(CX), AX
+       MULQ 8(BX)
+       ADDQ AX, R15
+       ADCQ DX, R14
+
+       // r4 += a4×b0
+       MOVQ 32(CX), AX
+       MULQ (BX)
+       ADDQ AX, R15
+       ADCQ DX, R14
+
+       // First reduction chain
+       MOVQ   $0x0007ffffffffffff, AX
+       SHLQ   $0x0d, DI, SI
+       SHLQ   $0x0d, R9, R8
+       SHLQ   $0x0d, R11, R10
+       SHLQ   $0x0d, R13, R12
+       SHLQ   $0x0d, R15, R14
+       ANDQ   AX, DI
+       IMUL3Q $0x13, R14, R14
+       ADDQ   R14, DI
+       ANDQ   AX, R9
+       ADDQ   SI, R9
+       ANDQ   AX, R11
+       ADDQ   R8, R11
+       ANDQ   AX, R13
+       ADDQ   R10, R13
+       ANDQ   AX, R15
+       ADDQ   R12, R15
+
+       // Second reduction chain (carryPropagate)
+       MOVQ   DI, SI
+       SHRQ   $0x33, SI
+       MOVQ   R9, R8
+       SHRQ   $0x33, R8
+       MOVQ   R11, R10
+       SHRQ   $0x33, R10
+       MOVQ   R13, R12
+       SHRQ   $0x33, R12
+       MOVQ   R15, R14
+       SHRQ   $0x33, R14
+       ANDQ   AX, DI
+       IMUL3Q $0x13, R14, R14
+       ADDQ   R14, DI
+       ANDQ   AX, R9
+       ADDQ   SI, R9
+       ANDQ   AX, R11
+       ADDQ   R8, R11
+       ANDQ   AX, R13
+       ADDQ   R10, R13
+       ANDQ   AX, R15
+       ADDQ   R12, R15
+
+       // Store output
+       MOVQ out+0(FP), AX
+       MOVQ DI, (AX)
+       MOVQ R9, 8(AX)
+       MOVQ R11, 16(AX)
+       MOVQ R13, 24(AX)
+       MOVQ R15, 32(AX)
+       RET
+
+// func feSquare(out *Element, a *Element)
+TEXT ·feSquare(SB), NOSPLIT, $0-16
+       MOVQ a+8(FP), CX
+
+       // r0 = l0×l0
+       MOVQ (CX), AX
+       MULQ (CX)
+       MOVQ AX, SI
+       MOVQ DX, BX
+
+       // r0 += 38×l1×l4
+       MOVQ   8(CX), AX
+       IMUL3Q $0x26, AX, AX
+       MULQ   32(CX)
+       ADDQ   AX, SI
+       ADCQ   DX, BX
+
+       // r0 += 38×l2×l3
+       MOVQ   16(CX), AX
+       IMUL3Q $0x26, AX, AX
+       MULQ   24(CX)
+       ADDQ   AX, SI
+       ADCQ   DX, BX
+
+       // r1 = 2×l0×l1
+       MOVQ (CX), AX
+       SHLQ $0x01, AX
+       MULQ 8(CX)
+       MOVQ AX, R8
+       MOVQ DX, DI
+
+       // r1 += 38×l2×l4
+       MOVQ   16(CX), AX
+       IMUL3Q $0x26, AX, AX
+       MULQ   32(CX)
+       ADDQ   AX, R8
+       ADCQ   DX, DI
+
+       // r1 += 19×l3×l3
+       MOVQ   24(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   24(CX)
+       ADDQ   AX, R8
+       ADCQ   DX, DI
+
+       // r2 = 2×l0×l2
+       MOVQ (CX), AX
+       SHLQ $0x01, AX
+       MULQ 16(CX)
+       MOVQ AX, R10
+       MOVQ DX, R9
+
+       // r2 += l1×l1
+       MOVQ 8(CX), AX
+       MULQ 8(CX)
+       ADDQ AX, R10
+       ADCQ DX, R9
+
+       // r2 += 38×l3×l4
+       MOVQ   24(CX), AX
+       IMUL3Q $0x26, AX, AX
+       MULQ   32(CX)
+       ADDQ   AX, R10
+       ADCQ   DX, R9
+
+       // r3 = 2×l0×l3
+       MOVQ (CX), AX
+       SHLQ $0x01, AX
+       MULQ 24(CX)
+       MOVQ AX, R12
+       MOVQ DX, R11
+
+       // r3 += 2×l1×l2
+       MOVQ   8(CX), AX
+       IMUL3Q $0x02, AX, AX
+       MULQ   16(CX)
+       ADDQ   AX, R12
+       ADCQ   DX, R11
+
+       // r3 += 19×l4×l4
+       MOVQ   32(CX), AX
+       IMUL3Q $0x13, AX, AX
+       MULQ   32(CX)
+       ADDQ   AX, R12
+       ADCQ   DX, R11
+
+       // r4 = 2×l0×l4
+       MOVQ (CX), AX
+       SHLQ $0x01, AX
+       MULQ 32(CX)
+       MOVQ AX, R14
+       MOVQ DX, R13
+
+       // r4 += 2×l1×l3
+       MOVQ   8(CX), AX
+       IMUL3Q $0x02, AX, AX
+       MULQ   24(CX)
+       ADDQ   AX, R14
+       ADCQ   DX, R13
+
+       // r4 += l2×l2
+       MOVQ 16(CX), AX
+       MULQ 16(CX)
+       ADDQ AX, R14
+       ADCQ DX, R13
+
+       // First reduction chain
+       MOVQ   $0x0007ffffffffffff, AX
+       SHLQ   $0x0d, SI, BX
+       SHLQ   $0x0d, R8, DI
+       SHLQ   $0x0d, R10, R9
+       SHLQ   $0x0d, R12, R11
+       SHLQ   $0x0d, R14, R13
+       ANDQ   AX, SI
+       IMUL3Q $0x13, R13, R13
+       ADDQ   R13, SI
+       ANDQ   AX, R8
+       ADDQ   BX, R8
+       ANDQ   AX, R10
+       ADDQ   DI, R10
+       ANDQ   AX, R12
+       ADDQ   R9, R12
+       ANDQ   AX, R14
+       ADDQ   R11, R14
+
+       // Second reduction chain (carryPropagate)
+       MOVQ   SI, BX
+       SHRQ   $0x33, BX
+       MOVQ   R8, DI
+       SHRQ   $0x33, DI
+       MOVQ   R10, R9
+       SHRQ   $0x33, R9
+       MOVQ   R12, R11
+       SHRQ   $0x33, R11
+       MOVQ   R14, R13
+       SHRQ   $0x33, R13
+       ANDQ   AX, SI
+       IMUL3Q $0x13, R13, R13
+       ADDQ   R13, SI
+       ANDQ   AX, R8
+       ADDQ   BX, R8
+       ANDQ   AX, R10
+       ADDQ   DI, R10
+       ANDQ   AX, R12
+       ADDQ   R9, R12
+       ANDQ   AX, R14
+       ADDQ   R11, R14
+
+       // Store output
+       MOVQ out+0(FP), AX
+       MOVQ SI, (AX)
+       MOVQ R8, 8(AX)
+       MOVQ R10, 16(AX)
+       MOVQ R12, 24(AX)
+       MOVQ R14, 32(AX)
+       RET

diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_amd64_noasm.go b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64_noasm.go
new file mode 100644 (file)
index 0000000..ddb6c9b
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64_noasm.go
@@ -0,0 +1,12 @@
+// Copyright (c) 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+//go:build !amd64 || !gc || purego
+// +build !amd64 !gc purego
+
+package field
+
+func feMul(v, x, y *Element) { feMulGeneric(v, x, y) }
+
+func feSquare(v, x *Element) { feSquareGeneric(v, x) }

diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.go b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.go
new file mode 100644 (file)
index 0000000..af459ef
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.go
@@ -0,0 +1,16 @@
+// Copyright (c) 2020 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+//go:build arm64 && gc && !purego
+// +build arm64,gc,!purego
+
+package field
+
+//go:noescape
+func carryPropagate(v *Element)
+
+func (v *Element) carryPropagate() *Element {
+       carryPropagate(v)
+       return v
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.s b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.s
new file mode 100644 (file)
index 0000000..751ab2a
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.s
@@ -0,0 +1,42 @@
+// Copyright (c) 2020 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// +build arm64,gc,!purego
+
+#include "textflag.h"
+
+// carryPropagate works exactly like carryPropagateGeneric and uses the
+// same AND, ADD, and LSR+MADD instructions emitted by the compiler, but
+// avoids loading R0-R4 twice and uses LDP and STP.
+//
+// See https://golang.org/issues/43145 for the main compiler issue.
+//
+// func carryPropagate(v *Element)
+TEXT ·carryPropagate(SB),NOFRAME|NOSPLIT,$0-8
+       MOVD v+0(FP), R20
+
+       LDP 0(R20), (R0, R1)
+       LDP 16(R20), (R2, R3)
+       MOVD 32(R20), R4
+
+       AND $0x7ffffffffffff, R0, R10
+       AND $0x7ffffffffffff, R1, R11
+       AND $0x7ffffffffffff, R2, R12
+       AND $0x7ffffffffffff, R3, R13
+       AND $0x7ffffffffffff, R4, R14
+
+       ADD R0>>51, R11, R11
+       ADD R1>>51, R12, R12
+       ADD R2>>51, R13, R13
+       ADD R3>>51, R14, R14
+       // R4>>51 * 19 + R10 -> R10
+       LSR $51, R4, R21
+       MOVD $19, R22
+       MADD R22, R10, R21, R10
+
+       STP (R10, R11), 0(R20)
+       STP (R12, R13), 16(R20)
+       MOVD R14, 32(R20)
+
+       RET

diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_arm64_noasm.go b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64_noasm.go
new file mode 100644 (file)
index 0000000..234a5b2
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64_noasm.go
@@ -0,0 +1,12 @@
+// Copyright (c) 2021 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+//go:build !arm64 || !gc || purego
+// +build !arm64 !gc purego
+
+package field
+
+func (v *Element) carryPropagate() *Element {
+       return v.carryPropagateGeneric()
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_bench_test.go b/src/crypto/ed25519/internal/edwards25519/field/fe_bench_test.go
new file mode 100644 (file)
index 0000000..77dc06c
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/fe_bench_test.go
@@ -0,0 +1,36 @@
+// Copyright (c) 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package field
+
+import "testing"
+
+func BenchmarkAdd(b *testing.B) {
+       var x, y Element
+       x.One()
+       y.Add(feOne, feOne)
+       b.ResetTimer()
+       for i := 0; i < b.N; i++ {
+               x.Add(&x, &y)
+       }
+}
+
+func BenchmarkMultiply(b *testing.B) {
+       var x, y Element
+       x.One()
+       y.Add(feOne, feOne)
+       b.ResetTimer()
+       for i := 0; i < b.N; i++ {
+               x.Multiply(&x, &y)
+       }
+}
+
+func BenchmarkMult32(b *testing.B) {
+       var x Element
+       x.One()
+       b.ResetTimer()
+       for i := 0; i < b.N; i++ {
+               x.Mult32(&x, 0xaa42aa42)
+       }
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_generic.go b/src/crypto/ed25519/internal/edwards25519/field/fe_generic.go
new file mode 100644 (file)
index 0000000..bccf851
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/fe_generic.go
@@ -0,0 +1,264 @@
+// Copyright (c) 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package field
+
+import "math/bits"
+
+// uint128 holds a 128-bit number as two 64-bit limbs, for use with the
+// bits.Mul64 and bits.Add64 intrinsics.
+type uint128 struct {
+       lo, hi uint64
+}
+
+// mul64 returns a * b.
+func mul64(a, b uint64) uint128 {
+       hi, lo := bits.Mul64(a, b)
+       return uint128{lo, hi}
+}
+
+// addMul64 returns v + a * b.
+func addMul64(v uint128, a, b uint64) uint128 {
+       hi, lo := bits.Mul64(a, b)
+       lo, c := bits.Add64(lo, v.lo, 0)
+       hi, _ = bits.Add64(hi, v.hi, c)
+       return uint128{lo, hi}
+}
+
+// shiftRightBy51 returns a >> 51. a is assumed to be at most 115 bits.
+func shiftRightBy51(a uint128) uint64 {
+       return (a.hi << (64 - 51)) | (a.lo >> 51)
+}
+
+func feMulGeneric(v, a, b *Element) {
+       a0 := a.l0
+       a1 := a.l1
+       a2 := a.l2
+       a3 := a.l3
+       a4 := a.l4
+
+       b0 := b.l0
+       b1 := b.l1
+       b2 := b.l2
+       b3 := b.l3
+       b4 := b.l4
+
+       // Limb multiplication works like pen-and-paper columnar multiplication, but
+       // with 51-bit limbs instead of digits.
+       //
+       //                          a4   a3   a2   a1   a0  x
+       //                          b4   b3   b2   b1   b0  =
+       //                         ------------------------
+       //                        a4b0 a3b0 a2b0 a1b0 a0b0  +
+       //                   a4b1 a3b1 a2b1 a1b1 a0b1       +
+       //              a4b2 a3b2 a2b2 a1b2 a0b2            +
+       //         a4b3 a3b3 a2b3 a1b3 a0b3                 +
+       //    a4b4 a3b4 a2b4 a1b4 a0b4                      =
+       //   ----------------------------------------------
+       //      r8   r7   r6   r5   r4   r3   r2   r1   r0
+       //
+       // We can then use the reduction identity (a * 2²⁵⁵ + b = a * 19 + b) to
+       // reduce the limbs that would overflow 255 bits. r5 * 2²⁵⁵ becomes 19 * r5,
+       // r6 * 2³⁰⁶ becomes 19 * r6 * 2⁵¹, etc.
+       //
+       // Reduction can be carried out simultaneously to multiplication. For
+       // example, we do not compute r5: whenever the result of a multiplication
+       // belongs to r5, like a1b4, we multiply it by 19 and add the result to r0.
+       //
+       //            a4b0    a3b0    a2b0    a1b0    a0b0  +
+       //            a3b1    a2b1    a1b1    a0b1 19×a4b1  +
+       //            a2b2    a1b2    a0b2 19×a4b2 19×a3b2  +
+       //            a1b3    a0b3 19×a4b3 19×a3b3 19×a2b3  +
+       //            a0b4 19×a4b4 19×a3b4 19×a2b4 19×a1b4  =
+       //           --------------------------------------
+       //              r4      r3      r2      r1      r0
+       //
+       // Finally we add up the columns into wide, overlapping limbs.
+
+       a1_19 := a1 * 19
+       a2_19 := a2 * 19
+       a3_19 := a3 * 19
+       a4_19 := a4 * 19
+
+       // r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1)
+       r0 := mul64(a0, b0)
+       r0 = addMul64(r0, a1_19, b4)
+       r0 = addMul64(r0, a2_19, b3)
+       r0 = addMul64(r0, a3_19, b2)
+       r0 = addMul64(r0, a4_19, b1)
+
+       // r1 = a0×b1 + a1×b0 + 19×(a2×b4 + a3×b3 + a4×b2)
+       r1 := mul64(a0, b1)
+       r1 = addMul64(r1, a1, b0)
+       r1 = addMul64(r1, a2_19, b4)
+       r1 = addMul64(r1, a3_19, b3)
+       r1 = addMul64(r1, a4_19, b2)
+
+       // r2 = a0×b2 + a1×b1 + a2×b0 + 19×(a3×b4 + a4×b3)
+       r2 := mul64(a0, b2)
+       r2 = addMul64(r2, a1, b1)
+       r2 = addMul64(r2, a2, b0)
+       r2 = addMul64(r2, a3_19, b4)
+       r2 = addMul64(r2, a4_19, b3)
+
+       // r3 = a0×b3 + a1×b2 + a2×b1 + a3×b0 + 19×a4×b4
+       r3 := mul64(a0, b3)
+       r3 = addMul64(r3, a1, b2)
+       r3 = addMul64(r3, a2, b1)
+       r3 = addMul64(r3, a3, b0)
+       r3 = addMul64(r3, a4_19, b4)
+
+       // r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0
+       r4 := mul64(a0, b4)
+       r4 = addMul64(r4, a1, b3)
+       r4 = addMul64(r4, a2, b2)
+       r4 = addMul64(r4, a3, b1)
+       r4 = addMul64(r4, a4, b0)
+
+       // After the multiplication, we need to reduce (carry) the five coefficients
+       // to obtain a result with limbs that are at most slightly larger than 2⁵¹,
+       // to respect the Element invariant.
+       //
+       // Overall, the reduction works the same as carryPropagate, except with
+       // wider inputs: we take the carry for each coefficient by shifting it right
+       // by 51, and add it to the limb above it. The top carry is multiplied by 19
+       // according to the reduction identity and added to the lowest limb.
+       //
+       // The largest coefficient (r0) will be at most 111 bits, which guarantees
+       // that all carries are at most 111 - 51 = 60 bits, which fits in a uint64.
+       //
+       //     r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1)
+       //     r0 < 2⁵²×2⁵² + 19×(2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵²)
+       //     r0 < (1 + 19 × 4) × 2⁵² × 2⁵²
+       //     r0 < 2⁷ × 2⁵² × 2⁵²
+       //     r0 < 2¹¹¹
+       //
+       // Moreover, the top coefficient (r4) is at most 107 bits, so c4 is at most
+       // 56 bits, and c4 * 19 is at most 61 bits, which again fits in a uint64 and
+       // allows us to easily apply the reduction identity.
+       //
+       //     r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0
+       //     r4 < 5 × 2⁵² × 2⁵²
+       //     r4 < 2¹⁰⁷
+       //
+
+       c0 := shiftRightBy51(r0)
+       c1 := shiftRightBy51(r1)
+       c2 := shiftRightBy51(r2)
+       c3 := shiftRightBy51(r3)
+       c4 := shiftRightBy51(r4)
+
+       rr0 := r0.lo&maskLow51Bits + c4*19
+       rr1 := r1.lo&maskLow51Bits + c0
+       rr2 := r2.lo&maskLow51Bits + c1
+       rr3 := r3.lo&maskLow51Bits + c2
+       rr4 := r4.lo&maskLow51Bits + c3
+
+       // Now all coefficients fit into 64-bit registers but are still too large to
+       // be passed around as a Element. We therefore do one last carry chain,
+       // where the carries will be small enough to fit in the wiggle room above 2⁵¹.
+       *v = Element{rr0, rr1, rr2, rr3, rr4}
+       v.carryPropagate()
+}
+
+func feSquareGeneric(v, a *Element) {
+       l0 := a.l0
+       l1 := a.l1
+       l2 := a.l2
+       l3 := a.l3
+       l4 := a.l4
+
+       // Squaring works precisely like multiplication above, but thanks to its
+       // symmetry we get to group a few terms together.
+       //
+       //                          l4   l3   l2   l1   l0  x
+       //                          l4   l3   l2   l1   l0  =
+       //                         ------------------------
+       //                        l4l0 l3l0 l2l0 l1l0 l0l0  +
+       //                   l4l1 l3l1 l2l1 l1l1 l0l1       +
+       //              l4l2 l3l2 l2l2 l1l2 l0l2            +
+       //         l4l3 l3l3 l2l3 l1l3 l0l3                 +
+       //    l4l4 l3l4 l2l4 l1l4 l0l4                      =
+       //   ----------------------------------------------
+       //      r8   r7   r6   r5   r4   r3   r2   r1   r0
+       //
+       //            l4l0    l3l0    l2l0    l1l0    l0l0  +
+       //            l3l1    l2l1    l1l1    l0l1 19×l4l1  +
+       //            l2l2    l1l2    l0l2 19×l4l2 19×l3l2  +
+       //            l1l3    l0l3 19×l4l3 19×l3l3 19×l2l3  +
+       //            l0l4 19×l4l4 19×l3l4 19×l2l4 19×l1l4  =
+       //           --------------------------------------
+       //              r4      r3      r2      r1      r0
+       //
+       // With precomputed 2×, 19×, and 2×19× terms, we can compute each limb with
+       // only three Mul64 and four Add64, instead of five and eight.
+
+       l0_2 := l0 * 2
+       l1_2 := l1 * 2
+
+       l1_38 := l1 * 38
+       l2_38 := l2 * 38
+       l3_38 := l3 * 38
+
+       l3_19 := l3 * 19
+       l4_19 := l4 * 19
+
+       // r0 = l0×l0 + 19×(l1×l4 + l2×l3 + l3×l2 + l4×l1) = l0×l0 + 19×2×(l1×l4 + l2×l3)
+       r0 := mul64(l0, l0)
+       r0 = addMul64(r0, l1_38, l4)
+       r0 = addMul64(r0, l2_38, l3)
+
+       // r1 = l0×l1 + l1×l0 + 19×(l2×l4 + l3×l3 + l4×l2) = 2×l0×l1 + 19×2×l2×l4 + 19×l3×l3
+       r1 := mul64(l0_2, l1)
+       r1 = addMul64(r1, l2_38, l4)
+       r1 = addMul64(r1, l3_19, l3)
+
+       // r2 = l0×l2 + l1×l1 + l2×l0 + 19×(l3×l4 + l4×l3) = 2×l0×l2 + l1×l1 + 19×2×l3×l4
+       r2 := mul64(l0_2, l2)
+       r2 = addMul64(r2, l1, l1)
+       r2 = addMul64(r2, l3_38, l4)
+
+       // r3 = l0×l3 + l1×l2 + l2×l1 + l3×l0 + 19×l4×l4 = 2×l0×l3 + 2×l1×l2 + 19×l4×l4
+       r3 := mul64(l0_2, l3)
+       r3 = addMul64(r3, l1_2, l2)
+       r3 = addMul64(r3, l4_19, l4)
+
+       // r4 = l0×l4 + l1×l3 + l2×l2 + l3×l1 + l4×l0 = 2×l0×l4 + 2×l1×l3 + l2×l2
+       r4 := mul64(l0_2, l4)
+       r4 = addMul64(r4, l1_2, l3)
+       r4 = addMul64(r4, l2, l2)
+
+       c0 := shiftRightBy51(r0)
+       c1 := shiftRightBy51(r1)
+       c2 := shiftRightBy51(r2)
+       c3 := shiftRightBy51(r3)
+       c4 := shiftRightBy51(r4)
+
+       rr0 := r0.lo&maskLow51Bits + c4*19
+       rr1 := r1.lo&maskLow51Bits + c0
+       rr2 := r2.lo&maskLow51Bits + c1
+       rr3 := r3.lo&maskLow51Bits + c2
+       rr4 := r4.lo&maskLow51Bits + c3
+
+       *v = Element{rr0, rr1, rr2, rr3, rr4}
+       v.carryPropagate()
+}
+
+// carryPropagate brings the limbs below 52 bits by applying the reduction
+// identity (a * 2²⁵⁵ + b = a * 19 + b) to the l4 carry.
+func (v *Element) carryPropagateGeneric() *Element {
+       c0 := v.l0 >> 51
+       c1 := v.l1 >> 51
+       c2 := v.l2 >> 51
+       c3 := v.l3 >> 51
+       c4 := v.l4 >> 51
+
+       v.l0 = v.l0&maskLow51Bits + c4*19
+       v.l1 = v.l1&maskLow51Bits + c0
+       v.l2 = v.l2&maskLow51Bits + c1
+       v.l3 = v.l3&maskLow51Bits + c2
+       v.l4 = v.l4&maskLow51Bits + c3
+
+       return v
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_test.go b/src/crypto/ed25519/internal/edwards25519/field/fe_test.go
new file mode 100644 (file)
index 0000000..b484459
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/field/fe_test.go
@@ -0,0 +1,558 @@
+// Copyright (c) 2017 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package field
+
+import (
+       "bytes"
+       "crypto/rand"
+       "encoding/hex"
+       "io"
+       "math/big"
+       "math/bits"
+       mathrand "math/rand"
+       "reflect"
+       "testing"
+       "testing/quick"
+)
+
+func (v Element) String() string {
+       return hex.EncodeToString(v.Bytes())
+}
+
+// quickCheckConfig1024 will make each quickcheck test run (1024 * -quickchecks)
+// times. The default value of -quickchecks is 100.
+var quickCheckConfig1024 = &quick.Config{MaxCountScale: 1 << 10}
+
+func generateFieldElement(rand *mathrand.Rand) Element {
+       const maskLow52Bits = (1 << 52) - 1
+       return Element{
+               rand.Uint64() & maskLow52Bits,
+               rand.Uint64() & maskLow52Bits,
+               rand.Uint64() & maskLow52Bits,
+               rand.Uint64() & maskLow52Bits,
+               rand.Uint64() & maskLow52Bits,
+       }
+}
+
+// weirdLimbs can be combined to generate a range of edge-case field elements.
+// 0 and -1 are intentionally more weighted, as they combine well.
+var (
+       weirdLimbs51 = []uint64{
+               0, 0, 0, 0,
+               1,
+               19 - 1,
+               19,
+               0x2aaaaaaaaaaaa,
+               0x5555555555555,
+               (1 << 51) - 20,
+               (1 << 51) - 19,
+               (1 << 51) - 1, (1 << 51) - 1,
+               (1 << 51) - 1, (1 << 51) - 1,
+       }
+       weirdLimbs52 = []uint64{
+               0, 0, 0, 0, 0, 0,
+               1,
+               19 - 1,
+               19,
+               0x2aaaaaaaaaaaa,
+               0x5555555555555,
+               (1 << 51) - 20,
+               (1 << 51) - 19,
+               (1 << 51) - 1, (1 << 51) - 1,
+               (1 << 51) - 1, (1 << 51) - 1,
+               (1 << 51) - 1, (1 << 51) - 1,
+               1 << 51,
+               (1 << 51) + 1,
+               (1 << 52) - 19,
+               (1 << 52) - 1,
+       }
+)
+
+func generateWeirdFieldElement(rand *mathrand.Rand) Element {
+       return Element{
+               weirdLimbs52[rand.Intn(len(weirdLimbs52))],
+               weirdLimbs51[rand.Intn(len(weirdLimbs51))],
+               weirdLimbs51[rand.Intn(len(weirdLimbs51))],
+               weirdLimbs51[rand.Intn(len(weirdLimbs51))],
+               weirdLimbs51[rand.Intn(len(weirdLimbs51))],
+       }
+}
+
+func (Element) Generate(rand *mathrand.Rand, size int) reflect.Value {
+       if rand.Intn(2) == 0 {
+               return reflect.ValueOf(generateWeirdFieldElement(rand))
+       }
+       return reflect.ValueOf(generateFieldElement(rand))
+}
+
+// isInBounds returns whether the element is within the expected bit size bounds
+// after a light reduction.
+func isInBounds(x *Element) bool {
+       return bits.Len64(x.l0) <= 52 &&
+               bits.Len64(x.l1) <= 52 &&
+               bits.Len64(x.l2) <= 52 &&
+               bits.Len64(x.l3) <= 52 &&
+               bits.Len64(x.l4) <= 52
+}
+
+func TestMultiplyDistributesOverAdd(t *testing.T) {
+       multiplyDistributesOverAdd := func(x, y, z Element) bool {
+               // Compute t1 = (x+y)*z
+               t1 := new(Element)
+               t1.Add(&x, &y)
+               t1.Multiply(t1, &z)
+
+               // Compute t2 = x*z + y*z
+               t2 := new(Element)
+               t3 := new(Element)
+               t2.Multiply(&x, &z)
+               t3.Multiply(&y, &z)
+               t2.Add(t2, t3)
+
+               return t1.Equal(t2) == 1 && isInBounds(t1) && isInBounds(t2)
+       }
+
+       if err := quick.Check(multiplyDistributesOverAdd, quickCheckConfig1024); err != nil {
+               t.Error(err)
+       }
+}
+
+func TestMul64to128(t *testing.T) {
+       a := uint64(5)
+       b := uint64(5)
+       r := mul64(a, b)
+       if r.lo != 0x19 || r.hi != 0 {
+               t.Errorf("lo-range wide mult failed, got %d + %d*(2**64)", r.lo, r.hi)
+       }
+
+       a = uint64(18014398509481983) // 2^54 - 1
+       b = uint64(18014398509481983) // 2^54 - 1
+       r = mul64(a, b)
+       if r.lo != 0xff80000000000001 || r.hi != 0xfffffffffff {
+               t.Errorf("hi-range wide mult failed, got %d + %d*(2**64)", r.lo, r.hi)
+       }
+
+       a = uint64(1125899906842661)
+       b = uint64(2097155)
+       r = mul64(a, b)
+       r = addMul64(r, a, b)
+       r = addMul64(r, a, b)
+       r = addMul64(r, a, b)
+       r = addMul64(r, a, b)
+       if r.lo != 16888498990613035 || r.hi != 640 {
+               t.Errorf("wrong answer: %d + %d*(2**64)", r.lo, r.hi)
+       }
+}
+
+func TestSetBytesRoundTrip(t *testing.T) {
+       f1 := func(in [32]byte, fe Element) bool {
+               fe.SetBytes(in[:])
+
+               // Mask the most significant bit as it's ignored by SetBytes. (Now
+               // instead of earlier so we check the masking in SetBytes is working.)
+               in[len(in)-1] &= (1 << 7) - 1
+
+               return bytes.Equal(in[:], fe.Bytes()) && isInBounds(&fe)
+       }
+       if err := quick.Check(f1, nil); err != nil {
+               t.Errorf("failed bytes->FE->bytes round-trip: %v", err)
+       }
+
+       f2 := func(fe, r Element) bool {
+               r.SetBytes(fe.Bytes())
+
+               // Intentionally not using Equal not to go through Bytes again.
+               // Calling reduce because both Generate and SetBytes can produce
+               // non-canonical representations.
+               fe.reduce()
+               r.reduce()
+               return fe == r
+       }
+       if err := quick.Check(f2, nil); err != nil {
+               t.Errorf("failed FE->bytes->FE round-trip: %v", err)
+       }
+
+       // Check some fixed vectors from dalek
+       type feRTTest struct {
+               fe Element
+               b  []byte
+       }
+       var tests = []feRTTest{
+               {
+                       fe: Element{358744748052810, 1691584618240980, 977650209285361, 1429865912637724, 560044844278676},
+                       b:  []byte{74, 209, 69, 197, 70, 70, 161, 222, 56, 226, 229, 19, 112, 60, 25, 92, 187, 74, 222, 56, 50, 153, 51, 233, 40, 74, 57, 6, 160, 185, 213, 31},
+               },
+               {
+                       fe: Element{84926274344903, 473620666599931, 365590438845504, 1028470286882429, 2146499180330972},
+                       b:  []byte{199, 23, 106, 112, 61, 77, 216, 79, 186, 60, 11, 118, 13, 16, 103, 15, 42, 32, 83, 250, 44, 57, 204, 198, 78, 199, 253, 119, 146, 172, 3, 122},
+               },
+       }
+
+       for _, tt := range tests {
+               b := tt.fe.Bytes()
+               if !bytes.Equal(b, tt.b) || new(Element).SetBytes(tt.b).Equal(&tt.fe) != 1 {
+                       t.Errorf("Failed fixed roundtrip: %v", tt)
+               }
+       }
+}
+
+func swapEndianness(buf []byte) []byte {
+       for i := 0; i < len(buf)/2; i++ {
+               buf[i], buf[len(buf)-i-1] = buf[len(buf)-i-1], buf[i]
+       }
+       return buf
+}
+
+func TestBytesBigEquivalence(t *testing.T) {
+       f1 := func(in [32]byte, fe, fe1 Element) bool {
+               fe.SetBytes(in[:])
+
+               in[len(in)-1] &= (1 << 7) - 1 // mask the most significant bit
+               b := new(big.Int).SetBytes(swapEndianness(in[:]))
+               fe1.fromBig(b)
+
+               if fe != fe1 {
+                       return false
+               }
+
+               buf := make([]byte, 32) // pad with zeroes
+               copy(buf, swapEndianness(fe1.toBig().Bytes()))
+
+               return bytes.Equal(fe.Bytes(), buf) && isInBounds(&fe) && isInBounds(&fe1)
+       }
+       if err := quick.Check(f1, nil); err != nil {
+               t.Error(err)
+       }
+}
+
+// fromBig sets v = n, and returns v. The bit length of n must not exceed 256.
+func (v *Element) fromBig(n *big.Int) *Element {
+       if n.BitLen() > 32*8 {
+               panic("edwards25519: invalid field element input size")
+       }
+
+       buf := make([]byte, 0, 32)
+       for _, word := range n.Bits() {
+               for i := 0; i < bits.UintSize; i += 8 {
+                       if len(buf) >= cap(buf) {
+                               break
+                       }
+                       buf = append(buf, byte(word))
+                       word >>= 8
+               }
+       }
+
+       return v.SetBytes(buf[:32])
+}
+
+func (v *Element) fromDecimal(s string) *Element {
+       n, ok := new(big.Int).SetString(s, 10)
+       if !ok {
+               panic("not a valid decimal: " + s)
+       }
+       return v.fromBig(n)
+}
+
+// toBig returns v as a big.Int.
+func (v *Element) toBig() *big.Int {
+       buf := v.Bytes()
+
+       words := make([]big.Word, 32*8/bits.UintSize)
+       for n := range words {
+               for i := 0; i < bits.UintSize; i += 8 {
+                       if len(buf) == 0 {
+                               break
+                       }
+                       words[n] |= big.Word(buf[0]) << big.Word(i)
+                       buf = buf[1:]
+               }
+       }
+
+       return new(big.Int).SetBits(words)
+}
+
+func TestDecimalConstants(t *testing.T) {
+       sqrtM1String := "19681161376707505956807079304988542015446066515923890162744021073123829784752"
+       if exp := new(Element).fromDecimal(sqrtM1String); sqrtM1.Equal(exp) != 1 {
+               t.Errorf("sqrtM1 is %v, expected %v", sqrtM1, exp)
+       }
+       // d is in the parent package, and we don't want to expose d or fromDecimal.
+       // dString := "37095705934669439343138083508754565189542113879843219016388785533085940283555"
+       // if exp := new(Element).fromDecimal(dString); d.Equal(exp) != 1 {
+       //      t.Errorf("d is %v, expected %v", d, exp)
+       // }
+}
+
+func TestSetBytesRoundTripEdgeCases(t *testing.T) {
+       // TODO: values close to 0, close to 2^255-19, between 2^255-19 and 2^255-1,
+       // and between 2^255 and 2^256-1. Test both the documented SetBytes
+       // behavior, and that Bytes reduces them.
+}
+
+// Tests self-consistency between Multiply and Square.
+func TestConsistency(t *testing.T) {
+       var x Element
+       var x2, x2sq Element
+
+       x = Element{1, 1, 1, 1, 1}
+       x2.Multiply(&x, &x)
+       x2sq.Square(&x)
+
+       if x2 != x2sq {
+               t.Fatalf("all ones failed\nmul: %x\nsqr: %x\n", x2, x2sq)
+       }
+
+       var bytes [32]byte
+
+       _, err := io.ReadFull(rand.Reader, bytes[:])
+       if err != nil {
+               t.Fatal(err)
+       }
+       x.SetBytes(bytes[:])
+
+       x2.Multiply(&x, &x)
+       x2sq.Square(&x)
+
+       if x2 != x2sq {
+               t.Fatalf("all ones failed\nmul: %x\nsqr: %x\n", x2, x2sq)
+       }
+}
+
+func TestEqual(t *testing.T) {
+       x := Element{1, 1, 1, 1, 1}
+       y := Element{5, 4, 3, 2, 1}
+
+       eq := x.Equal(&x)
+       if eq != 1 {
+               t.Errorf("wrong about equality")
+       }
+
+       eq = x.Equal(&y)
+       if eq != 0 {
+               t.Errorf("wrong about inequality")
+       }
+}
+
+func TestInvert(t *testing.T) {
+       x := Element{1, 1, 1, 1, 1}
+       one := Element{1, 0, 0, 0, 0}
+       var xinv, r Element
+
+       xinv.Invert(&x)
+       r.Multiply(&x, &xinv)
+       r.reduce()
+
+       if one != r {
+               t.Errorf("inversion identity failed, got: %x", r)
+       }
+
+       var bytes [32]byte
+
+       _, err := io.ReadFull(rand.Reader, bytes[:])
+       if err != nil {
+               t.Fatal(err)
+       }
+       x.SetBytes(bytes[:])
+
+       xinv.Invert(&x)
+       r.Multiply(&x, &xinv)
+       r.reduce()
+
+       if one != r {
+               t.Errorf("random inversion identity failed, got: %x for field element %x", r, x)
+       }
+
+       zero := Element{}
+       x.Set(&zero)
+       if xx := xinv.Invert(&x); xx != &xinv {
+               t.Errorf("inverting zero did not return the receiver")
+       } else if xinv.Equal(&zero) != 1 {
+               t.Errorf("inverting zero did not return zero")
+       }
+}
+
+func TestSelectSwap(t *testing.T) {
+       a := Element{358744748052810, 1691584618240980, 977650209285361, 1429865912637724, 560044844278676}
+       b := Element{84926274344903, 473620666599931, 365590438845504, 1028470286882429, 2146499180330972}
+
+       var c, d Element
+
+       c.Select(&a, &b, 1)
+       d.Select(&a, &b, 0)
+
+       if c.Equal(&a) != 1 || d.Equal(&b) != 1 {
+               t.Errorf("Select failed")
+       }
+
+       c.Swap(&d, 0)
+
+       if c.Equal(&a) != 1 || d.Equal(&b) != 1 {
+               t.Errorf("Swap failed")
+       }
+
+       c.Swap(&d, 1)
+
+       if c.Equal(&b) != 1 || d.Equal(&a) != 1 {
+               t.Errorf("Swap failed")
+       }
+}
+
+func TestMult32(t *testing.T) {
+       mult32EquivalentToMul := func(x Element, y uint32) bool {
+               t1 := new(Element)
+               for i := 0; i < 100; i++ {
+                       t1.Mult32(&x, y)
+               }
+
+               ty := new(Element)
+               ty.l0 = uint64(y)
+
+               t2 := new(Element)
+               for i := 0; i < 100; i++ {
+                       t2.Multiply(&x, ty)
+               }
+
+               return t1.Equal(t2) == 1 && isInBounds(t1) && isInBounds(t2)
+       }
+
+       if err := quick.Check(mult32EquivalentToMul, quickCheckConfig1024); err != nil {
+               t.Error(err)
+       }
+}
+
+func TestSqrtRatio(t *testing.T) {
+       // From draft-irtf-cfrg-ristretto255-decaf448-00, Appendix A.4.
+       type test struct {
+               u, v      string
+               wasSquare int
+               r         string
+       }
+       var tests = []test{
+               // If u is 0, the function is defined to return (0, TRUE), even if v
+               // is zero. Note that where used in this package, the denominator v
+               // is never zero.
+               {
+                       "0000000000000000000000000000000000000000000000000000000000000000",
+                       "0000000000000000000000000000000000000000000000000000000000000000",
+                       1, "0000000000000000000000000000000000000000000000000000000000000000",
+               },
+               // 0/1 == 0²
+               {
+                       "0000000000000000000000000000000000000000000000000000000000000000",
+                       "0100000000000000000000000000000000000000000000000000000000000000",
+                       1, "0000000000000000000000000000000000000000000000000000000000000000",
+               },
+               // If u is non-zero and v is zero, defined to return (0, FALSE).
+               {
+                       "0100000000000000000000000000000000000000000000000000000000000000",
+                       "0000000000000000000000000000000000000000000000000000000000000000",
+                       0, "0000000000000000000000000000000000000000000000000000000000000000",
+               },
+               // 2/1 is not square in this field.
+               {
+                       "0200000000000000000000000000000000000000000000000000000000000000",
+                       "0100000000000000000000000000000000000000000000000000000000000000",
+                       0, "3c5ff1b5d8e4113b871bd052f9e7bcd0582804c266ffb2d4f4203eb07fdb7c54",
+               },
+               // 4/1 == 2²
+               {
+                       "0400000000000000000000000000000000000000000000000000000000000000",
+                       "0100000000000000000000000000000000000000000000000000000000000000",
+                       1, "0200000000000000000000000000000000000000000000000000000000000000",
+               },
+               // 1/4 == (2⁻¹)² == (2^(p-2))² per Euler's theorem
+               {
+                       "0100000000000000000000000000000000000000000000000000000000000000",
+                       "0400000000000000000000000000000000000000000000000000000000000000",
+                       1, "f6ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff3f",
+               },
+       }
+
+       for i, tt := range tests {
+               u := new(Element).SetBytes(decodeHex(tt.u))
+               v := new(Element).SetBytes(decodeHex(tt.v))
+               want := new(Element).SetBytes(decodeHex(tt.r))
+               got, wasSquare := new(Element).SqrtRatio(u, v)
+               if got.Equal(want) == 0 || wasSquare != tt.wasSquare {
+                       t.Errorf("%d: got (%v, %v), want (%v, %v)", i, got, wasSquare, want, tt.wasSquare)
+               }
+       }
+}
+
+func TestCarryPropagate(t *testing.T) {
+       asmLikeGeneric := func(a [5]uint64) bool {
+               t1 := &Element{a[0], a[1], a[2], a[3], a[4]}
+               t2 := &Element{a[0], a[1], a[2], a[3], a[4]}
+
+               t1.carryPropagate()
+               t2.carryPropagateGeneric()
+
+               if *t1 != *t2 {
+                       t.Logf("got: %#v,\nexpected: %#v", t1, t2)
+               }
+
+               return *t1 == *t2 && isInBounds(t2)
+       }
+
+       if err := quick.Check(asmLikeGeneric, quickCheckConfig1024); err != nil {
+               t.Error(err)
+       }
+
+       if !asmLikeGeneric([5]uint64{0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff}) {
+               t.Errorf("failed for {0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff}")
+       }
+}
+
+func TestFeSquare(t *testing.T) {
+       asmLikeGeneric := func(a Element) bool {
+               t1 := a
+               t2 := a
+
+               feSquareGeneric(&t1, &t1)
+               feSquare(&t2, &t2)
+
+               if t1 != t2 {
+                       t.Logf("got: %#v,\nexpected: %#v", t1, t2)
+               }
+
+               return t1 == t2 && isInBounds(&t2)
+       }
+
+       if err := quick.Check(asmLikeGeneric, quickCheckConfig1024); err != nil {
+               t.Error(err)
+       }
+}
+
+func TestFeMul(t *testing.T) {
+       asmLikeGeneric := func(a, b Element) bool {
+               a1 := a
+               a2 := a
+               b1 := b
+               b2 := b
+
+               feMulGeneric(&a1, &a1, &b1)
+               feMul(&a2, &a2, &b2)
+
+               if a1 != a2 || b1 != b2 {
+                       t.Logf("got: %#v,\nexpected: %#v", a1, a2)
+                       t.Logf("got: %#v,\nexpected: %#v", b1, b2)
+               }
+
+               return a1 == a2 && isInBounds(&a2) &&
+                       b1 == b2 && isInBounds(&b2)
+       }
+
+       if err := quick.Check(asmLikeGeneric, quickCheckConfig1024); err != nil {
+               t.Error(err)
+       }
+}
+
+func decodeHex(s string) []byte {
+       b, err := hex.DecodeString(s)
+       if err != nil {
+               panic(err)
+       }
+       return b
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/scalar.go b/src/crypto/ed25519/internal/edwards25519/scalar.go
new file mode 100644 (file)
index 0000000..889acaa
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/scalar.go
@@ -0,0 +1,1025 @@
+// Copyright (c) 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package edwards25519
+
+import (
+       "crypto/subtle"
+       "encoding/binary"
+       "errors"
+)
+
+// A Scalar is an integer modulo
+//
+//     l = 2^252 + 27742317777372353535851937790883648493
+//
+// which is the prime order of the edwards25519 group.
+//
+// This type works similarly to math/big.Int, and all arguments and
+// receivers are allowed to alias.
+//
+// The zero value is a valid zero element.
+type Scalar struct {
+       // s is the Scalar value in little-endian. The value is always reduced
+       // between operations.
+       s [32]byte
+}
+
+var (
+       scZero = Scalar{[32]byte{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}
+
+       scOne = Scalar{[32]byte{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}
+
+       scMinusOne = Scalar{[32]byte{236, 211, 245, 92, 26, 99, 18, 88, 214, 156, 247, 162, 222, 249, 222, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16}}
+)
+
+// NewScalar returns a new zero Scalar.
+func NewScalar() *Scalar {
+       return &Scalar{}
+}
+
+// MultiplyAdd sets s = x * y + z mod l, and returns s.
+func (s *Scalar) MultiplyAdd(x, y, z *Scalar) *Scalar {
+       scMulAdd(&s.s, &x.s, &y.s, &z.s)
+       return s
+}
+
+// Add sets s = x + y mod l, and returns s.
+func (s *Scalar) Add(x, y *Scalar) *Scalar {
+       // s = 1 * x + y mod l
+       scMulAdd(&s.s, &scOne.s, &x.s, &y.s)
+       return s
+}
+
+// Subtract sets s = x - y mod l, and returns s.
+func (s *Scalar) Subtract(x, y *Scalar) *Scalar {
+       // s = -1 * y + x mod l
+       scMulAdd(&s.s, &scMinusOne.s, &y.s, &x.s)
+       return s
+}
+
+// Negate sets s = -x mod l, and returns s.
+func (s *Scalar) Negate(x *Scalar) *Scalar {
+       // s = -1 * x + 0 mod l
+       scMulAdd(&s.s, &scMinusOne.s, &x.s, &scZero.s)
+       return s
+}
+
+// Multiply sets s = x * y mod l, and returns s.
+func (s *Scalar) Multiply(x, y *Scalar) *Scalar {
+       // s = x * y + 0 mod l
+       scMulAdd(&s.s, &x.s, &y.s, &scZero.s)
+       return s
+}
+
+// Set sets s = x, and returns s.
+func (s *Scalar) Set(x *Scalar) *Scalar {
+       *s = *x
+       return s
+}
+
+// SetUniformBytes sets s to an uniformly distributed value given 64 uniformly
+// distributed random bytes.
+func (s *Scalar) SetUniformBytes(x []byte) *Scalar {
+       if len(x) != 64 {
+               panic("edwards25519: invalid SetUniformBytes input length")
+       }
+       var wideBytes [64]byte
+       copy(wideBytes[:], x[:])
+       scReduce(&s.s, &wideBytes)
+       return s
+}
+
+// SetCanonicalBytes sets s = x, where x is a 32-byte little-endian encoding of
+// s, and returns s. If x is not a canonical encoding of s, SetCanonicalBytes
+// returns nil and an error, and the receiver is unchanged.
+func (s *Scalar) SetCanonicalBytes(x []byte) (*Scalar, error) {
+       if len(x) != 32 {
+               return nil, errors.New("invalid scalar length")
+       }
+       ss := &Scalar{}
+       copy(ss.s[:], x)
+       if !isReduced(ss) {
+               return nil, errors.New("invalid scalar encoding")
+       }
+       s.s = ss.s
+       return s, nil
+}
+
+// isReduced returns whether the given scalar is reduced modulo l.
+func isReduced(s *Scalar) bool {
+       for i := len(s.s) - 1; i >= 0; i-- {
+               switch {
+               case s.s[i] > scMinusOne.s[i]:
+                       return false
+               case s.s[i] < scMinusOne.s[i]:
+                       return true
+               }
+       }
+       return true
+}
+
+// SetBytesWithClamping applies the buffer pruning described in RFC 8032,
+// Section 5.1.5 (also known as clamping) and sets s to the result. The input
+// must be 32 bytes, and it is not modified.
+//
+// Note that since Scalar values are always reduced modulo the prime order of
+// the curve, the resulting value will not preserve any of the cofactor-clearing
+// properties that clamping is meant to provide. It will however work as
+// expected as long as it is applied to points on the prime order subgroup, like
+// in Ed25519. In fact, it is lost to history why RFC 8032 adopted the
+// irrelevant RFC 7748 clamping, but it is now required for compatibility.
+func (s *Scalar) SetBytesWithClamping(x []byte) *Scalar {
+       // The description above omits the purpose of the high bits of the clamping
+       // for brevity, but those are also lost to reductions, and are also
+       // irrelevant to edwards25519 as they protect against a specific
+       // implementation bug that was once observed in a generic Montgomery ladder.
+       if len(x) != 32 {
+               panic("edwards25519: invalid SetBytesWithClamping input length")
+       }
+       var wideBytes [64]byte
+       copy(wideBytes[:], x[:])
+       wideBytes[0] &= 248
+       wideBytes[31] &= 63
+       wideBytes[31] |= 64
+       scReduce(&s.s, &wideBytes)
+       return s
+}
+
+// Bytes returns the canonical 32-byte little-endian encoding of s.
+func (s *Scalar) Bytes() []byte {
+       buf := make([]byte, 32)
+       copy(buf, s.s[:])
+       return buf
+}
+
+// Equal returns 1 if s and t are equal, and 0 otherwise.
+func (s *Scalar) Equal(t *Scalar) int {
+       return subtle.ConstantTimeCompare(s.s[:], t.s[:])
+}
+
+// scMulAdd and scReduce are ported from the public domain, “ref10”
+// implementation of ed25519 from SUPERCOP.
+
+func load3(in []byte) int64 {
+       r := int64(in[0])
+       r |= int64(in[1]) << 8
+       r |= int64(in[2]) << 16
+       return r
+}
+
+func load4(in []byte) int64 {
+       r := int64(in[0])
+       r |= int64(in[1]) << 8
+       r |= int64(in[2]) << 16
+       r |= int64(in[3]) << 24
+       return r
+}
+
+// Input:
+//   a[0]+256*a[1]+...+256^31*a[31] = a
+//   b[0]+256*b[1]+...+256^31*b[31] = b
+//   c[0]+256*c[1]+...+256^31*c[31] = c
+//
+// Output:
+//   s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
+//   where l = 2^252 + 27742317777372353535851937790883648493.
+func scMulAdd(s, a, b, c *[32]byte) {
+       a0 := 2097151 & load3(a[:])
+       a1 := 2097151 & (load4(a[2:]) >> 5)
+       a2 := 2097151 & (load3(a[5:]) >> 2)
+       a3 := 2097151 & (load4(a[7:]) >> 7)
+       a4 := 2097151 & (load4(a[10:]) >> 4)
+       a5 := 2097151 & (load3(a[13:]) >> 1)
+       a6 := 2097151 & (load4(a[15:]) >> 6)
+       a7 := 2097151 & (load3(a[18:]) >> 3)
+       a8 := 2097151 & load3(a[21:])
+       a9 := 2097151 & (load4(a[23:]) >> 5)
+       a10 := 2097151 & (load3(a[26:]) >> 2)
+       a11 := (load4(a[28:]) >> 7)
+       b0 := 2097151 & load3(b[:])
+       b1 := 2097151 & (load4(b[2:]) >> 5)
+       b2 := 2097151 & (load3(b[5:]) >> 2)
+       b3 := 2097151 & (load4(b[7:]) >> 7)
+       b4 := 2097151 & (load4(b[10:]) >> 4)
+       b5 := 2097151 & (load3(b[13:]) >> 1)
+       b6 := 2097151 & (load4(b[15:]) >> 6)
+       b7 := 2097151 & (load3(b[18:]) >> 3)
+       b8 := 2097151 & load3(b[21:])
+       b9 := 2097151 & (load4(b[23:]) >> 5)
+       b10 := 2097151 & (load3(b[26:]) >> 2)
+       b11 := (load4(b[28:]) >> 7)
+       c0 := 2097151 & load3(c[:])
+       c1 := 2097151 & (load4(c[2:]) >> 5)
+       c2 := 2097151 & (load3(c[5:]) >> 2)
+       c3 := 2097151 & (load4(c[7:]) >> 7)
+       c4 := 2097151 & (load4(c[10:]) >> 4)
+       c5 := 2097151 & (load3(c[13:]) >> 1)
+       c6 := 2097151 & (load4(c[15:]) >> 6)
+       c7 := 2097151 & (load3(c[18:]) >> 3)
+       c8 := 2097151 & load3(c[21:])
+       c9 := 2097151 & (load4(c[23:]) >> 5)
+       c10 := 2097151 & (load3(c[26:]) >> 2)
+       c11 := (load4(c[28:]) >> 7)
+       var carry [23]int64
+
+       s0 := c0 + a0*b0
+       s1 := c1 + a0*b1 + a1*b0
+       s2 := c2 + a0*b2 + a1*b1 + a2*b0
+       s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0
+       s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0
+       s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0
+       s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0
+       s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0
+       s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0
+       s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0
+       s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0
+       s11 := c11 + a0*b11 + a1*b10 + a2*b9 + a3*b8 + a4*b7 + a5*b6 + a6*b5 + a7*b4 + a8*b3 + a9*b2 + a10*b1 + a11*b0
+       s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1
+       s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2
+       s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3
+       s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4
+       s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5
+       s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6
+       s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7
+       s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8
+       s20 := a9*b11 + a10*b10 + a11*b9
+       s21 := a10*b11 + a11*b10
+       s22 := a11 * b11
+       s23 := int64(0)
+
+       carry[0] = (s0 + (1 << 20)) >> 21
+       s1 += carry[0]
+       s0 -= carry[0] << 21
+       carry[2] = (s2 + (1 << 20)) >> 21
+       s3 += carry[2]
+       s2 -= carry[2] << 21
+       carry[4] = (s4 + (1 << 20)) >> 21
+       s5 += carry[4]
+       s4 -= carry[4] << 21
+       carry[6] = (s6 + (1 << 20)) >> 21
+       s7 += carry[6]
+       s6 -= carry[6] << 21
+       carry[8] = (s8 + (1 << 20)) >> 21
+       s9 += carry[8]
+       s8 -= carry[8] << 21
+       carry[10] = (s10 + (1 << 20)) >> 21
+       s11 += carry[10]
+       s10 -= carry[10] << 21
+       carry[12] = (s12 + (1 << 20)) >> 21
+       s13 += carry[12]
+       s12 -= carry[12] << 21
+       carry[14] = (s14 + (1 << 20)) >> 21
+       s15 += carry[14]
+       s14 -= carry[14] << 21
+       carry[16] = (s16 + (1 << 20)) >> 21
+       s17 += carry[16]
+       s16 -= carry[16] << 21
+       carry[18] = (s18 + (1 << 20)) >> 21
+       s19 += carry[18]
+       s18 -= carry[18] << 21
+       carry[20] = (s20 + (1 << 20)) >> 21
+       s21 += carry[20]
+       s20 -= carry[20] << 21
+       carry[22] = (s22 + (1 << 20)) >> 21
+       s23 += carry[22]
+       s22 -= carry[22] << 21
+
+       carry[1] = (s1 + (1 << 20)) >> 21
+       s2 += carry[1]
+       s1 -= carry[1] << 21
+       carry[3] = (s3 + (1 << 20)) >> 21
+       s4 += carry[3]
+       s3 -= carry[3] << 21
+       carry[5] = (s5 + (1 << 20)) >> 21
+       s6 += carry[5]
+       s5 -= carry[5] << 21
+       carry[7] = (s7 + (1 << 20)) >> 21
+       s8 += carry[7]
+       s7 -= carry[7] << 21
+       carry[9] = (s9 + (1 << 20)) >> 21
+       s10 += carry[9]
+       s9 -= carry[9] << 21
+       carry[11] = (s11 + (1 << 20)) >> 21
+       s12 += carry[11]
+       s11 -= carry[11] << 21
+       carry[13] = (s13 + (1 << 20)) >> 21
+       s14 += carry[13]
+       s13 -= carry[13] << 21
+       carry[15] = (s15 + (1 << 20)) >> 21
+       s16 += carry[15]
+       s15 -= carry[15] << 21
+       carry[17] = (s17 + (1 << 20)) >> 21
+       s18 += carry[17]
+       s17 -= carry[17] << 21
+       carry[19] = (s19 + (1 << 20)) >> 21
+       s20 += carry[19]
+       s19 -= carry[19] << 21
+       carry[21] = (s21 + (1 << 20)) >> 21
+       s22 += carry[21]
+       s21 -= carry[21] << 21
+
+       s11 += s23 * 666643
+       s12 += s23 * 470296
+       s13 += s23 * 654183
+       s14 -= s23 * 997805
+       s15 += s23 * 136657
+       s16 -= s23 * 683901
+       s23 = 0
+
+       s10 += s22 * 666643
+       s11 += s22 * 470296
+       s12 += s22 * 654183
+       s13 -= s22 * 997805
+       s14 += s22 * 136657
+       s15 -= s22 * 683901
+       s22 = 0
+
+       s9 += s21 * 666643
+       s10 += s21 * 470296
+       s11 += s21 * 654183
+       s12 -= s21 * 997805
+       s13 += s21 * 136657
+       s14 -= s21 * 683901
+       s21 = 0
+
+       s8 += s20 * 666643
+       s9 += s20 * 470296
+       s10 += s20 * 654183
+       s11 -= s20 * 997805
+       s12 += s20 * 136657
+       s13 -= s20 * 683901
+       s20 = 0
+
+       s7 += s19 * 666643
+       s8 += s19 * 470296
+       s9 += s19 * 654183
+       s10 -= s19 * 997805
+       s11 += s19 * 136657
+       s12 -= s19 * 683901
+       s19 = 0
+
+       s6 += s18 * 666643
+       s7 += s18 * 470296
+       s8 += s18 * 654183
+       s9 -= s18 * 997805
+       s10 += s18 * 136657
+       s11 -= s18 * 683901
+       s18 = 0
+
+       carry[6] = (s6 + (1 << 20)) >> 21
+       s7 += carry[6]
+       s6 -= carry[6] << 21
+       carry[8] = (s8 + (1 << 20)) >> 21
+       s9 += carry[8]
+       s8 -= carry[8] << 21
+       carry[10] = (s10 + (1 << 20)) >> 21
+       s11 += carry[10]
+       s10 -= carry[10] << 21
+       carry[12] = (s12 + (1 << 20)) >> 21
+       s13 += carry[12]
+       s12 -= carry[12] << 21
+       carry[14] = (s14 + (1 << 20)) >> 21
+       s15 += carry[14]
+       s14 -= carry[14] << 21
+       carry[16] = (s16 + (1 << 20)) >> 21
+       s17 += carry[16]
+       s16 -= carry[16] << 21
+
+       carry[7] = (s7 + (1 << 20)) >> 21
+       s8 += carry[7]
+       s7 -= carry[7] << 21
+       carry[9] = (s9 + (1 << 20)) >> 21
+       s10 += carry[9]
+       s9 -= carry[9] << 21
+       carry[11] = (s11 + (1 << 20)) >> 21
+       s12 += carry[11]
+       s11 -= carry[11] << 21
+       carry[13] = (s13 + (1 << 20)) >> 21
+       s14 += carry[13]
+       s13 -= carry[13] << 21
+       carry[15] = (s15 + (1 << 20)) >> 21
+       s16 += carry[15]
+       s15 -= carry[15] << 21
+
+       s5 += s17 * 666643
+       s6 += s17 * 470296
+       s7 += s17 * 654183
+       s8 -= s17 * 997805
+       s9 += s17 * 136657
+       s10 -= s17 * 683901
+       s17 = 0
+
+       s4 += s16 * 666643
+       s5 += s16 * 470296
+       s6 += s16 * 654183
+       s7 -= s16 * 997805
+       s8 += s16 * 136657
+       s9 -= s16 * 683901
+       s16 = 0
+
+       s3 += s15 * 666643
+       s4 += s15 * 470296
+       s5 += s15 * 654183
+       s6 -= s15 * 997805
+       s7 += s15 * 136657
+       s8 -= s15 * 683901
+       s15 = 0
+
+       s2 += s14 * 666643
+       s3 += s14 * 470296
+       s4 += s14 * 654183
+       s5 -= s14 * 997805
+       s6 += s14 * 136657
+       s7 -= s14 * 683901
+       s14 = 0
+
+       s1 += s13 * 666643
+       s2 += s13 * 470296
+       s3 += s13 * 654183
+       s4 -= s13 * 997805
+       s5 += s13 * 136657
+       s6 -= s13 * 683901
+       s13 = 0
+
+       s0 += s12 * 666643
+       s1 += s12 * 470296
+       s2 += s12 * 654183
+       s3 -= s12 * 997805
+       s4 += s12 * 136657
+       s5 -= s12 * 683901
+       s12 = 0
+
+       carry[0] = (s0 + (1 << 20)) >> 21
+       s1 += carry[0]
+       s0 -= carry[0] << 21
+       carry[2] = (s2 + (1 << 20)) >> 21
+       s3 += carry[2]
+       s2 -= carry[2] << 21
+       carry[4] = (s4 + (1 << 20)) >> 21
+       s5 += carry[4]
+       s4 -= carry[4] << 21
+       carry[6] = (s6 + (1 << 20)) >> 21
+       s7 += carry[6]
+       s6 -= carry[6] << 21
+       carry[8] = (s8 + (1 << 20)) >> 21
+       s9 += carry[8]
+       s8 -= carry[8] << 21
+       carry[10] = (s10 + (1 << 20)) >> 21
+       s11 += carry[10]
+       s10 -= carry[10] << 21
+
+       carry[1] = (s1 + (1 << 20)) >> 21
+       s2 += carry[1]
+       s1 -= carry[1] << 21
+       carry[3] = (s3 + (1 << 20)) >> 21
+       s4 += carry[3]
+       s3 -= carry[3] << 21
+       carry[5] = (s5 + (1 << 20)) >> 21
+       s6 += carry[5]
+       s5 -= carry[5] << 21
+       carry[7] = (s7 + (1 << 20)) >> 21
+       s8 += carry[7]
+       s7 -= carry[7] << 21
+       carry[9] = (s9 + (1 << 20)) >> 21
+       s10 += carry[9]
+       s9 -= carry[9] << 21
+       carry[11] = (s11 + (1 << 20)) >> 21
+       s12 += carry[11]
+       s11 -= carry[11] << 21
+
+       s0 += s12 * 666643
+       s1 += s12 * 470296
+       s2 += s12 * 654183
+       s3 -= s12 * 997805
+       s4 += s12 * 136657
+       s5 -= s12 * 683901
+       s12 = 0
+
+       carry[0] = s0 >> 21
+       s1 += carry[0]
+       s0 -= carry[0] << 21
+       carry[1] = s1 >> 21
+       s2 += carry[1]
+       s1 -= carry[1] << 21
+       carry[2] = s2 >> 21
+       s3 += carry[2]
+       s2 -= carry[2] << 21
+       carry[3] = s3 >> 21
+       s4 += carry[3]
+       s3 -= carry[3] << 21
+       carry[4] = s4 >> 21
+       s5 += carry[4]
+       s4 -= carry[4] << 21
+       carry[5] = s5 >> 21
+       s6 += carry[5]
+       s5 -= carry[5] << 21
+       carry[6] = s6 >> 21
+       s7 += carry[6]
+       s6 -= carry[6] << 21
+       carry[7] = s7 >> 21
+       s8 += carry[7]
+       s7 -= carry[7] << 21
+       carry[8] = s8 >> 21
+       s9 += carry[8]
+       s8 -= carry[8] << 21
+       carry[9] = s9 >> 21
+       s10 += carry[9]
+       s9 -= carry[9] << 21
+       carry[10] = s10 >> 21
+       s11 += carry[10]
+       s10 -= carry[10] << 21
+       carry[11] = s11 >> 21
+       s12 += carry[11]
+       s11 -= carry[11] << 21
+
+       s0 += s12 * 666643
+       s1 += s12 * 470296
+       s2 += s12 * 654183
+       s3 -= s12 * 997805
+       s4 += s12 * 136657
+       s5 -= s12 * 683901
+       s12 = 0
+
+       carry[0] = s0 >> 21
+       s1 += carry[0]
+       s0 -= carry[0] << 21
+       carry[1] = s1 >> 21
+       s2 += carry[1]
+       s1 -= carry[1] << 21
+       carry[2] = s2 >> 21
+       s3 += carry[2]
+       s2 -= carry[2] << 21
+       carry[3] = s3 >> 21
+       s4 += carry[3]
+       s3 -= carry[3] << 21
+       carry[4] = s4 >> 21
+       s5 += carry[4]
+       s4 -= carry[4] << 21
+       carry[5] = s5 >> 21
+       s6 += carry[5]
+       s5 -= carry[5] << 21
+       carry[6] = s6 >> 21
+       s7 += carry[6]
+       s6 -= carry[6] << 21
+       carry[7] = s7 >> 21
+       s8 += carry[7]
+       s7 -= carry[7] << 21
+       carry[8] = s8 >> 21
+       s9 += carry[8]
+       s8 -= carry[8] << 21
+       carry[9] = s9 >> 21
+       s10 += carry[9]
+       s9 -= carry[9] << 21
+       carry[10] = s10 >> 21
+       s11 += carry[10]
+       s10 -= carry[10] << 21
+
+       s[0] = byte(s0 >> 0)
+       s[1] = byte(s0 >> 8)
+       s[2] = byte((s0 >> 16) | (s1 << 5))
+       s[3] = byte(s1 >> 3)
+       s[4] = byte(s1 >> 11)
+       s[5] = byte((s1 >> 19) | (s2 << 2))
+       s[6] = byte(s2 >> 6)
+       s[7] = byte((s2 >> 14) | (s3 << 7))
+       s[8] = byte(s3 >> 1)
+       s[9] = byte(s3 >> 9)
+       s[10] = byte((s3 >> 17) | (s4 << 4))
+       s[11] = byte(s4 >> 4)
+       s[12] = byte(s4 >> 12)
+       s[13] = byte((s4 >> 20) | (s5 << 1))
+       s[14] = byte(s5 >> 7)
+       s[15] = byte((s5 >> 15) | (s6 << 6))
+       s[16] = byte(s6 >> 2)
+       s[17] = byte(s6 >> 10)
+       s[18] = byte((s6 >> 18) | (s7 << 3))
+       s[19] = byte(s7 >> 5)
+       s[20] = byte(s7 >> 13)
+       s[21] = byte(s8 >> 0)
+       s[22] = byte(s8 >> 8)
+       s[23] = byte((s8 >> 16) | (s9 << 5))
+       s[24] = byte(s9 >> 3)
+       s[25] = byte(s9 >> 11)
+       s[26] = byte((s9 >> 19) | (s10 << 2))
+       s[27] = byte(s10 >> 6)
+       s[28] = byte((s10 >> 14) | (s11 << 7))
+       s[29] = byte(s11 >> 1)
+       s[30] = byte(s11 >> 9)
+       s[31] = byte(s11 >> 17)
+}
+
+// Input:
+//   s[0]+256*s[1]+...+256^63*s[63] = s
+//
+// Output:
+//   s[0]+256*s[1]+...+256^31*s[31] = s mod l
+//   where l = 2^252 + 27742317777372353535851937790883648493.
+func scReduce(out *[32]byte, s *[64]byte) {
+       s0 := 2097151 & load3(s[:])
+       s1 := 2097151 & (load4(s[2:]) >> 5)
+       s2 := 2097151 & (load3(s[5:]) >> 2)
+       s3 := 2097151 & (load4(s[7:]) >> 7)
+       s4 := 2097151 & (load4(s[10:]) >> 4)
+       s5 := 2097151 & (load3(s[13:]) >> 1)
+       s6 := 2097151 & (load4(s[15:]) >> 6)
+       s7 := 2097151 & (load3(s[18:]) >> 3)
+       s8 := 2097151 & load3(s[21:])
+       s9 := 2097151 & (load4(s[23:]) >> 5)
+       s10 := 2097151 & (load3(s[26:]) >> 2)
+       s11 := 2097151 & (load4(s[28:]) >> 7)
+       s12 := 2097151 & (load4(s[31:]) >> 4)
+       s13 := 2097151 & (load3(s[34:]) >> 1)
+       s14 := 2097151 & (load4(s[36:]) >> 6)
+       s15 := 2097151 & (load3(s[39:]) >> 3)
+       s16 := 2097151 & load3(s[42:])
+       s17 := 2097151 & (load4(s[44:]) >> 5)
+       s18 := 2097151 & (load3(s[47:]) >> 2)
+       s19 := 2097151 & (load4(s[49:]) >> 7)
+       s20 := 2097151 & (load4(s[52:]) >> 4)
+       s21 := 2097151 & (load3(s[55:]) >> 1)
+       s22 := 2097151 & (load4(s[57:]) >> 6)
+       s23 := (load4(s[60:]) >> 3)
+
+       s11 += s23 * 666643
+       s12 += s23 * 470296
+       s13 += s23 * 654183
+       s14 -= s23 * 997805
+       s15 += s23 * 136657
+       s16 -= s23 * 683901
+       s23 = 0
+
+       s10 += s22 * 666643
+       s11 += s22 * 470296
+       s12 += s22 * 654183
+       s13 -= s22 * 997805
+       s14 += s22 * 136657
+       s15 -= s22 * 683901
+       s22 = 0
+
+       s9 += s21 * 666643
+       s10 += s21 * 470296
+       s11 += s21 * 654183
+       s12 -= s21 * 997805
+       s13 += s21 * 136657
+       s14 -= s21 * 683901
+       s21 = 0
+
+       s8 += s20 * 666643
+       s9 += s20 * 470296
+       s10 += s20 * 654183
+       s11 -= s20 * 997805
+       s12 += s20 * 136657
+       s13 -= s20 * 683901
+       s20 = 0
+
+       s7 += s19 * 666643
+       s8 += s19 * 470296
+       s9 += s19 * 654183
+       s10 -= s19 * 997805
+       s11 += s19 * 136657
+       s12 -= s19 * 683901
+       s19 = 0
+
+       s6 += s18 * 666643
+       s7 += s18 * 470296
+       s8 += s18 * 654183
+       s9 -= s18 * 997805
+       s10 += s18 * 136657
+       s11 -= s18 * 683901
+       s18 = 0
+
+       var carry [17]int64
+
+       carry[6] = (s6 + (1 << 20)) >> 21
+       s7 += carry[6]
+       s6 -= carry[6] << 21
+       carry[8] = (s8 + (1 << 20)) >> 21
+       s9 += carry[8]
+       s8 -= carry[8] << 21
+       carry[10] = (s10 + (1 << 20)) >> 21
+       s11 += carry[10]
+       s10 -= carry[10] << 21
+       carry[12] = (s12 + (1 << 20)) >> 21
+       s13 += carry[12]
+       s12 -= carry[12] << 21
+       carry[14] = (s14 + (1 << 20)) >> 21
+       s15 += carry[14]
+       s14 -= carry[14] << 21
+       carry[16] = (s16 + (1 << 20)) >> 21
+       s17 += carry[16]
+       s16 -= carry[16] << 21
+
+       carry[7] = (s7 + (1 << 20)) >> 21
+       s8 += carry[7]
+       s7 -= carry[7] << 21
+       carry[9] = (s9 + (1 << 20)) >> 21
+       s10 += carry[9]
+       s9 -= carry[9] << 21
+       carry[11] = (s11 + (1 << 20)) >> 21
+       s12 += carry[11]
+       s11 -= carry[11] << 21
+       carry[13] = (s13 + (1 << 20)) >> 21
+       s14 += carry[13]
+       s13 -= carry[13] << 21
+       carry[15] = (s15 + (1 << 20)) >> 21
+       s16 += carry[15]
+       s15 -= carry[15] << 21
+
+       s5 += s17 * 666643
+       s6 += s17 * 470296
+       s7 += s17 * 654183
+       s8 -= s17 * 997805
+       s9 += s17 * 136657
+       s10 -= s17 * 683901
+       s17 = 0
+
+       s4 += s16 * 666643
+       s5 += s16 * 470296
+       s6 += s16 * 654183
+       s7 -= s16 * 997805
+       s8 += s16 * 136657
+       s9 -= s16 * 683901
+       s16 = 0
+
+       s3 += s15 * 666643
+       s4 += s15 * 470296
+       s5 += s15 * 654183
+       s6 -= s15 * 997805
+       s7 += s15 * 136657
+       s8 -= s15 * 683901
+       s15 = 0
+
+       s2 += s14 * 666643
+       s3 += s14 * 470296
+       s4 += s14 * 654183
+       s5 -= s14 * 997805
+       s6 += s14 * 136657
+       s7 -= s14 * 683901
+       s14 = 0
+
+       s1 += s13 * 666643
+       s2 += s13 * 470296
+       s3 += s13 * 654183
+       s4 -= s13 * 997805
+       s5 += s13 * 136657
+       s6 -= s13 * 683901
+       s13 = 0
+
+       s0 += s12 * 666643
+       s1 += s12 * 470296
+       s2 += s12 * 654183
+       s3 -= s12 * 997805
+       s4 += s12 * 136657
+       s5 -= s12 * 683901
+       s12 = 0
+
+       carry[0] = (s0 + (1 << 20)) >> 21
+       s1 += carry[0]
+       s0 -= carry[0] << 21
+       carry[2] = (s2 + (1 << 20)) >> 21
+       s3 += carry[2]
+       s2 -= carry[2] << 21
+       carry[4] = (s4 + (1 << 20)) >> 21
+       s5 += carry[4]
+       s4 -= carry[4] << 21
+       carry[6] = (s6 + (1 << 20)) >> 21
+       s7 += carry[6]
+       s6 -= carry[6] << 21
+       carry[8] = (s8 + (1 << 20)) >> 21
+       s9 += carry[8]
+       s8 -= carry[8] << 21
+       carry[10] = (s10 + (1 << 20)) >> 21
+       s11 += carry[10]
+       s10 -= carry[10] << 21
+
+       carry[1] = (s1 + (1 << 20)) >> 21
+       s2 += carry[1]
+       s1 -= carry[1] << 21
+       carry[3] = (s3 + (1 << 20)) >> 21
+       s4 += carry[3]
+       s3 -= carry[3] << 21
+       carry[5] = (s5 + (1 << 20)) >> 21
+       s6 += carry[5]
+       s5 -= carry[5] << 21
+       carry[7] = (s7 + (1 << 20)) >> 21
+       s8 += carry[7]
+       s7 -= carry[7] << 21
+       carry[9] = (s9 + (1 << 20)) >> 21
+       s10 += carry[9]
+       s9 -= carry[9] << 21
+       carry[11] = (s11 + (1 << 20)) >> 21
+       s12 += carry[11]
+       s11 -= carry[11] << 21
+
+       s0 += s12 * 666643
+       s1 += s12 * 470296
+       s2 += s12 * 654183
+       s3 -= s12 * 997805
+       s4 += s12 * 136657
+       s5 -= s12 * 683901
+       s12 = 0
+
+       carry[0] = s0 >> 21
+       s1 += carry[0]
+       s0 -= carry[0] << 21
+       carry[1] = s1 >> 21
+       s2 += carry[1]
+       s1 -= carry[1] << 21
+       carry[2] = s2 >> 21
+       s3 += carry[2]
+       s2 -= carry[2] << 21
+       carry[3] = s3 >> 21
+       s4 += carry[3]
+       s3 -= carry[3] << 21
+       carry[4] = s4 >> 21
+       s5 += carry[4]
+       s4 -= carry[4] << 21
+       carry[5] = s5 >> 21
+       s6 += carry[5]
+       s5 -= carry[5] << 21
+       carry[6] = s6 >> 21
+       s7 += carry[6]
+       s6 -= carry[6] << 21
+       carry[7] = s7 >> 21
+       s8 += carry[7]
+       s7 -= carry[7] << 21
+       carry[8] = s8 >> 21
+       s9 += carry[8]
+       s8 -= carry[8] << 21
+       carry[9] = s9 >> 21
+       s10 += carry[9]
+       s9 -= carry[9] << 21
+       carry[10] = s10 >> 21
+       s11 += carry[10]
+       s10 -= carry[10] << 21
+       carry[11] = s11 >> 21
+       s12 += carry[11]
+       s11 -= carry[11] << 21
+
+       s0 += s12 * 666643
+       s1 += s12 * 470296
+       s2 += s12 * 654183
+       s3 -= s12 * 997805
+       s4 += s12 * 136657
+       s5 -= s12 * 683901
+       s12 = 0
+
+       carry[0] = s0 >> 21
+       s1 += carry[0]
+       s0 -= carry[0] << 21
+       carry[1] = s1 >> 21
+       s2 += carry[1]
+       s1 -= carry[1] << 21
+       carry[2] = s2 >> 21
+       s3 += carry[2]
+       s2 -= carry[2] << 21
+       carry[3] = s3 >> 21
+       s4 += carry[3]
+       s3 -= carry[3] << 21
+       carry[4] = s4 >> 21
+       s5 += carry[4]
+       s4 -= carry[4] << 21
+       carry[5] = s5 >> 21
+       s6 += carry[5]
+       s5 -= carry[5] << 21
+       carry[6] = s6 >> 21
+       s7 += carry[6]
+       s6 -= carry[6] << 21
+       carry[7] = s7 >> 21
+       s8 += carry[7]
+       s7 -= carry[7] << 21
+       carry[8] = s8 >> 21
+       s9 += carry[8]
+       s8 -= carry[8] << 21
+       carry[9] = s9 >> 21
+       s10 += carry[9]
+       s9 -= carry[9] << 21
+       carry[10] = s10 >> 21
+       s11 += carry[10]
+       s10 -= carry[10] << 21
+
+       out[0] = byte(s0 >> 0)
+       out[1] = byte(s0 >> 8)
+       out[2] = byte((s0 >> 16) | (s1 << 5))
+       out[3] = byte(s1 >> 3)
+       out[4] = byte(s1 >> 11)
+       out[5] = byte((s1 >> 19) | (s2 << 2))
+       out[6] = byte(s2 >> 6)
+       out[7] = byte((s2 >> 14) | (s3 << 7))
+       out[8] = byte(s3 >> 1)
+       out[9] = byte(s3 >> 9)
+       out[10] = byte((s3 >> 17) | (s4 << 4))
+       out[11] = byte(s4 >> 4)
+       out[12] = byte(s4 >> 12)
+       out[13] = byte((s4 >> 20) | (s5 << 1))
+       out[14] = byte(s5 >> 7)
+       out[15] = byte((s5 >> 15) | (s6 << 6))
+       out[16] = byte(s6 >> 2)
+       out[17] = byte(s6 >> 10)
+       out[18] = byte((s6 >> 18) | (s7 << 3))
+       out[19] = byte(s7 >> 5)
+       out[20] = byte(s7 >> 13)
+       out[21] = byte(s8 >> 0)
+       out[22] = byte(s8 >> 8)
+       out[23] = byte((s8 >> 16) | (s9 << 5))
+       out[24] = byte(s9 >> 3)
+       out[25] = byte(s9 >> 11)
+       out[26] = byte((s9 >> 19) | (s10 << 2))
+       out[27] = byte(s10 >> 6)
+       out[28] = byte((s10 >> 14) | (s11 << 7))
+       out[29] = byte(s11 >> 1)
+       out[30] = byte(s11 >> 9)
+       out[31] = byte(s11 >> 17)
+}
+
+// nonAdjacentForm computes a width-w non-adjacent form for this scalar.
+//
+// w must be between 2 and 8, or nonAdjacentForm will panic.
+func (s *Scalar) nonAdjacentForm(w uint) [256]int8 {
+       // This implementation is adapted from the one
+       // in curve25519-dalek and is documented there:
+       // https://github.com/dalek-cryptography/curve25519-dalek/blob/f630041af28e9a405255f98a8a93adca18e4315b/src/scalar.rs#L800-L871
+       if s.s[31] > 127 {
+               panic("scalar has high bit set illegally")
+       }
+       if w < 2 {
+               panic("w must be at least 2 by the definition of NAF")
+       } else if w > 8 {
+               panic("NAF digits must fit in int8")
+       }
+
+       var naf [256]int8
+       var digits [5]uint64
+
+       for i := 0; i < 4; i++ {
+               digits[i] = binary.LittleEndian.Uint64(s.s[i*8:])
+       }
+
+       width := uint64(1 << w)
+       windowMask := uint64(width - 1)
+
+       pos := uint(0)
+       carry := uint64(0)
+       for pos < 256 {
+               indexU64 := pos / 64
+               indexBit := pos % 64
+               var bitBuf uint64
+               if indexBit < 64-w {
+                       // This window's bits are contained in a single u64
+                       bitBuf = digits[indexU64] >> indexBit
+               } else {
+                       // Combine the current 64 bits with bits from the next 64
+                       bitBuf = (digits[indexU64] >> indexBit) | (digits[1+indexU64] << (64 - indexBit))
+               }
+
+               // Add carry into the current window
+               window := carry + (bitBuf & windowMask)
+
+               if window&1 == 0 {
+                       // If the window value is even, preserve the carry and continue.
+                       // Why is the carry preserved?
+                       // If carry == 0 and window & 1 == 0,
+                       //    then the next carry should be 0
+                       // If carry == 1 and window & 1 == 0,
+                       //    then bit_buf & 1 == 1 so the next carry should be 1
+                       pos += 1
+                       continue
+               }
+
+               if window < width/2 {
+                       carry = 0
+                       naf[pos] = int8(window)
+               } else {
+                       carry = 1
+                       naf[pos] = int8(window) - int8(width)
+               }
+
+               pos += w
+       }
+       return naf
+}
+
+func (s *Scalar) signedRadix16() [64]int8 {
+       if s.s[31] > 127 {
+               panic("scalar has high bit set illegally")
+       }
+
+       var digits [64]int8
+
+       // Compute unsigned radix-16 digits:
+       for i := 0; i < 32; i++ {
+               digits[2*i] = int8(s.s[i] & 15)
+               digits[2*i+1] = int8((s.s[i] >> 4) & 15)
+       }
+
+       // Recenter coefficients:
+       for i := 0; i < 63; i++ {
+               carry := (digits[i] + 8) >> 4
+               digits[i] -= carry << 4
+               digits[i+1] += carry
+       }
+
+       return digits
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/scalar_alias_test.go b/src/crypto/ed25519/internal/edwards25519/scalar_alias_test.go
new file mode 100644 (file)
index 0000000..18d800d
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/scalar_alias_test.go
@@ -0,0 +1,93 @@
+// Copyright (c) 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package edwards25519
+
+import (
+       "testing"
+       "testing/quick"
+)
+
+func TestScalarAliasing(t *testing.T) {
+       checkAliasingOneArg := func(f func(v, x *Scalar) *Scalar, v, x Scalar) bool {
+               x1, v1 := x, x
+
+               // Calculate a reference f(x) without aliasing.
+               if out := f(&v, &x); out != &v || !isReduced(out) {
+                       return false
+               }
+
+               // Test aliasing the argument and the receiver.
+               if out := f(&v1, &v1); out != &v1 || v1 != v || !isReduced(out) {
+                       return false
+               }
+
+               // Ensure the arguments was not modified.
+               return x == x1
+       }
+
+       checkAliasingTwoArgs := func(f func(v, x, y *Scalar) *Scalar, v, x, y Scalar) bool {
+               x1, y1, v1 := x, y, Scalar{}
+
+               // Calculate a reference f(x, y) without aliasing.
+               if out := f(&v, &x, &y); out != &v || !isReduced(out) {
+                       return false
+               }
+
+               // Test aliasing the first argument and the receiver.
+               v1 = x
+               if out := f(&v1, &v1, &y); out != &v1 || v1 != v || !isReduced(out) {
+                       return false
+               }
+               // Test aliasing the second argument and the receiver.
+               v1 = y
+               if out := f(&v1, &x, &v1); out != &v1 || v1 != v || !isReduced(out) {
+                       return false
+               }
+
+               // Calculate a reference f(x, x) without aliasing.
+               if out := f(&v, &x, &x); out != &v || !isReduced(out) {
+                       return false
+               }
+
+               // Test aliasing the first argument and the receiver.
+               v1 = x
+               if out := f(&v1, &v1, &x); out != &v1 || v1 != v || !isReduced(out) {
+                       return false
+               }
+               // Test aliasing the second argument and the receiver.
+               v1 = x
+               if out := f(&v1, &x, &v1); out != &v1 || v1 != v || !isReduced(out) {
+                       return false
+               }
+               // Test aliasing both arguments and the receiver.
+               v1 = x
+               if out := f(&v1, &v1, &v1); out != &v1 || v1 != v || !isReduced(out) {
+                       return false
+               }
+
+               // Ensure the arguments were not modified.
+               return x == x1 && y == y1
+       }
+
+       for name, f := range map[string]interface{}{
+               "Negate": func(v, x Scalar) bool {
+                       return checkAliasingOneArg((*Scalar).Negate, v, x)
+               },
+               "Multiply": func(v, x, y Scalar) bool {
+                       return checkAliasingTwoArgs((*Scalar).Multiply, v, x, y)
+               },
+               "Add": func(v, x, y Scalar) bool {
+                       return checkAliasingTwoArgs((*Scalar).Add, v, x, y)
+               },
+               "Subtract": func(v, x, y Scalar) bool {
+                       return checkAliasingTwoArgs((*Scalar).Subtract, v, x, y)
+               },
+       } {
+               err := quick.Check(f, &quick.Config{MaxCountScale: 1 << 5})
+               if err != nil {
+                       t.Errorf("%v: %v", name, err)
+               }
+       }
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/scalar_test.go b/src/crypto/ed25519/internal/edwards25519/scalar_test.go
new file mode 100644 (file)
index 0000000..704caff
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/scalar_test.go
@@ -0,0 +1,233 @@
+// Copyright (c) 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package edwards25519
+
+import (
+       "bytes"
+       "encoding/hex"
+       "math/big"
+       mathrand "math/rand"
+       "reflect"
+       "testing"
+       "testing/quick"
+)
+
+// Generate returns a valid (reduced modulo l) Scalar with a distribution
+// weighted towards high, low, and edge values.
+func (Scalar) Generate(rand *mathrand.Rand, size int) reflect.Value {
+       s := scZero
+       diceRoll := rand.Intn(100)
+       switch {
+       case diceRoll == 0:
+       case diceRoll == 1:
+               s = scOne
+       case diceRoll == 2:
+               s = scMinusOne
+       case diceRoll < 5:
+               // Generate a low scalar in [0, 2^125).
+               rand.Read(s.s[:16])
+               s.s[15] &= (1 << 5) - 1
+       case diceRoll < 10:
+               // Generate a high scalar in [2^252, 2^252 + 2^124).
+               s.s[31] = 1 << 4
+               rand.Read(s.s[:16])
+               s.s[15] &= (1 << 4) - 1
+       default:
+               // Generate a valid scalar in [0, l) by returning [0, 2^252) which has a
+               // negligibly different distribution (the former has a 2^-127.6 chance
+               // of being out of the latter range).
+               rand.Read(s.s[:])
+               s.s[31] &= (1 << 4) - 1
+       }
+       return reflect.ValueOf(s)
+}
+
+// quickCheckConfig1024 will make each quickcheck test run (1024 * -quickchecks)
+// times. The default value of -quickchecks is 100.
+var quickCheckConfig1024 = &quick.Config{MaxCountScale: 1 << 10}
+
+func TestScalarGenerate(t *testing.T) {
+       f := func(sc Scalar) bool {
+               return isReduced(&sc)
+       }
+       if err := quick.Check(f, quickCheckConfig1024); err != nil {
+               t.Errorf("generated unreduced scalar: %v", err)
+       }
+}
+
+func TestScalarSetCanonicalBytes(t *testing.T) {
+       f1 := func(in [32]byte, sc Scalar) bool {
+               // Mask out top 4 bits to guarantee value falls in [0, l).
+               in[len(in)-1] &= (1 << 4) - 1
+               if _, err := sc.SetCanonicalBytes(in[:]); err != nil {
+                       return false
+               }
+               return bytes.Equal(in[:], sc.Bytes()) && isReduced(&sc)
+       }
+       if err := quick.Check(f1, quickCheckConfig1024); err != nil {
+               t.Errorf("failed bytes->scalar->bytes round-trip: %v", err)
+       }
+
+       f2 := func(sc1, sc2 Scalar) bool {
+               if _, err := sc2.SetCanonicalBytes(sc1.Bytes()); err != nil {
+                       return false
+               }
+               return sc1 == sc2
+       }
+       if err := quick.Check(f2, quickCheckConfig1024); err != nil {
+               t.Errorf("failed scalar->bytes->scalar round-trip: %v", err)
+       }
+
+       b := scMinusOne.s
+       b[31] += 1
+       s := scOne
+       if out, err := s.SetCanonicalBytes(b[:]); err == nil {
+               t.Errorf("SetCanonicalBytes worked on a non-canonical value")
+       } else if s != scOne {
+               t.Errorf("SetCanonicalBytes modified its receiver")
+       } else if out != nil {
+               t.Errorf("SetCanonicalBytes did not return nil with an error")
+       }
+}
+
+func TestScalarSetUniformBytes(t *testing.T) {
+       mod, _ := new(big.Int).SetString("27742317777372353535851937790883648493", 10)
+       mod.Add(mod, new(big.Int).Lsh(big.NewInt(1), 252))
+       f := func(in [64]byte, sc Scalar) bool {
+               sc.SetUniformBytes(in[:])
+               if !isReduced(&sc) {
+                       return false
+               }
+               scBig := bigIntFromLittleEndianBytes(sc.s[:])
+               inBig := bigIntFromLittleEndianBytes(in[:])
+               return inBig.Mod(inBig, mod).Cmp(scBig) == 0
+       }
+       if err := quick.Check(f, quickCheckConfig1024); err != nil {
+               t.Error(err)
+       }
+}
+
+func TestScalarSetBytesWithClamping(t *testing.T) {
+       // Generated with libsodium.js 1.0.18 crypto_scalarmult_ed25519_base.
+
+       random := "633d368491364dc9cd4c1bf891b1d59460face1644813240a313e61f2c88216e"
+       s := new(Scalar).SetBytesWithClamping(decodeHex(random))
+       p := new(Point).ScalarBaseMult(s)
+       want := "1d87a9026fd0126a5736fe1628c95dd419172b5b618457e041c9c861b2494a94"
+       if got := hex.EncodeToString(p.Bytes()); got != want {
+               t.Errorf("random: got %q, want %q", got, want)
+       }
+
+       zero := "0000000000000000000000000000000000000000000000000000000000000000"
+       s = new(Scalar).SetBytesWithClamping(decodeHex(zero))
+       p = new(Point).ScalarBaseMult(s)
+       want = "693e47972caf527c7883ad1b39822f026f47db2ab0e1919955b8993aa04411d1"
+       if got := hex.EncodeToString(p.Bytes()); got != want {
+               t.Errorf("zero: got %q, want %q", got, want)
+       }
+
+       one := "ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff"
+       s = new(Scalar).SetBytesWithClamping(decodeHex(one))
+       p = new(Point).ScalarBaseMult(s)
+       want = "12e9a68b73fd5aacdbcaf3e88c46fea6ebedb1aa84eed1842f07f8edab65e3a7"
+       if got := hex.EncodeToString(p.Bytes()); got != want {
+               t.Errorf("one: got %q, want %q", got, want)
+       }
+}
+
+func bigIntFromLittleEndianBytes(b []byte) *big.Int {
+       bb := make([]byte, len(b))
+       for i := range b {
+               bb[i] = b[len(b)-i-1]
+       }
+       return new(big.Int).SetBytes(bb)
+}
+
+func TestScalarMultiplyDistributesOverAdd(t *testing.T) {
+       multiplyDistributesOverAdd := func(x, y, z Scalar) bool {
+               // Compute t1 = (x+y)*z
+               var t1 Scalar
+               t1.Add(&x, &y)
+               t1.Multiply(&t1, &z)
+
+               // Compute t2 = x*z + y*z
+               var t2 Scalar
+               var t3 Scalar
+               t2.Multiply(&x, &z)
+               t3.Multiply(&y, &z)
+               t2.Add(&t2, &t3)
+
+               return t1 == t2 && isReduced(&t1) && isReduced(&t3)
+       }
+
+       if err := quick.Check(multiplyDistributesOverAdd, quickCheckConfig1024); err != nil {
+               t.Error(err)
+       }
+}
+
+func TestScalarAddLikeSubNeg(t *testing.T) {
+       addLikeSubNeg := func(x, y Scalar) bool {
+               // Compute t1 = x - y
+               var t1 Scalar
+               t1.Subtract(&x, &y)
+
+               // Compute t2 = -y + x
+               var t2 Scalar
+               t2.Negate(&y)
+               t2.Add(&t2, &x)
+
+               return t1 == t2 && isReduced(&t1)
+       }
+
+       if err := quick.Check(addLikeSubNeg, quickCheckConfig1024); err != nil {
+               t.Error(err)
+       }
+}
+
+func TestScalarNonAdjacentForm(t *testing.T) {
+       s := Scalar{[32]byte{
+               0x1a, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d,
+               0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8, 0x26, 0x4d,
+               0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1,
+               0x58, 0x9e, 0x7b, 0x7f, 0x23, 0x76, 0xef, 0x09,
+       }}
+       expectedNaf := [256]int8{
+               0, 13, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, -11, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1,
+               0, 0, 0, 0, 9, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 11, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0,
+               -9, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 9, 0,
+               0, 0, 0, -15, 0, 0, 0, 0, -7, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, -3, 0,
+               0, 0, 0, -11, 0, 0, 0, 0, -7, 0, 0, 0, 0, -13, 0, 0, 0, 0, 11, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 1, 0, 0,
+               0, 0, 0, -15, 0, 0, 0, 0, 1, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 13, 0, 0, 0,
+               0, 0, 0, 11, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 7,
+               0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 15, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
+       }
+
+       sNaf := s.nonAdjacentForm(5)
+
+       for i := 0; i < 256; i++ {
+               if expectedNaf[i] != sNaf[i] {
+                       t.Errorf("Wrong digit at position %d, got %d, expected %d", i, sNaf[i], expectedNaf[i])
+               }
+       }
+}
+
+type notZeroScalar Scalar
+
+func (notZeroScalar) Generate(rand *mathrand.Rand, size int) reflect.Value {
+       var s Scalar
+       for s == scZero {
+               s = Scalar{}.Generate(rand, size).Interface().(Scalar)
+       }
+       return reflect.ValueOf(notZeroScalar(s))
+}
+
+func TestScalarEqual(t *testing.T) {
+       if scOne.Equal(&scMinusOne) == 1 {
+               t.Errorf("scOne.Equal(&scMinusOne) is true")
+       }
+       if scMinusOne.Equal(&scMinusOne) == 0 {
+               t.Errorf("scMinusOne.Equal(&scMinusOne) is false")
+       }
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/scalarmult.go b/src/crypto/ed25519/internal/edwards25519/scalarmult.go
new file mode 100644 (file)
index 0000000..f7ca3ce
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/scalarmult.go
@@ -0,0 +1,214 @@
+// Copyright (c) 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package edwards25519
+
+import "sync"
+
+// basepointTable is a set of 32 affineLookupTables, where table i is generated
+// from 256i * basepoint. It is precomputed the first time it's used.
+func basepointTable() *[32]affineLookupTable {
+       basepointTablePrecomp.initOnce.Do(func() {
+               p := NewGeneratorPoint()
+               for i := 0; i < 32; i++ {
+                       basepointTablePrecomp.table[i].FromP3(p)
+                       for j := 0; j < 8; j++ {
+                               p.Add(p, p)
+                       }
+               }
+       })
+       return &basepointTablePrecomp.table
+}
+
+var basepointTablePrecomp struct {
+       table    [32]affineLookupTable
+       initOnce sync.Once
+}
+
+// ScalarBaseMult sets v = x * B, where B is the canonical generator, and
+// returns v.
+//
+// The scalar multiplication is done in constant time.
+func (v *Point) ScalarBaseMult(x *Scalar) *Point {
+       basepointTable := basepointTable()
+
+       // Write x = sum(x_i * 16^i) so  x*B = sum( B*x_i*16^i )
+       // as described in the Ed25519 paper
+       //
+       // Group even and odd coefficients
+       // x*B     = x_0*16^0*B + x_2*16^2*B + ... + x_62*16^62*B
+       //         + x_1*16^1*B + x_3*16^3*B + ... + x_63*16^63*B
+       // x*B     = x_0*16^0*B + x_2*16^2*B + ... + x_62*16^62*B
+       //    + 16*( x_1*16^0*B + x_3*16^2*B + ... + x_63*16^62*B)
+       //
+       // We use a lookup table for each i to get x_i*16^(2*i)*B
+       // and do four doublings to multiply by 16.
+       digits := x.signedRadix16()
+
+       multiple := &affineCached{}
+       tmp1 := &projP1xP1{}
+       tmp2 := &projP2{}
+
+       // Accumulate the odd components first
+       v.Set(NewIdentityPoint())
+       for i := 1; i < 64; i += 2 {
+               basepointTable[i/2].SelectInto(multiple, digits[i])
+               tmp1.AddAffine(v, multiple)
+               v.fromP1xP1(tmp1)
+       }
+
+       // Multiply by 16
+       tmp2.FromP3(v)       // tmp2 =    v in P2 coords
+       tmp1.Double(tmp2)    // tmp1 =  2*v in P1xP1 coords
+       tmp2.FromP1xP1(tmp1) // tmp2 =  2*v in P2 coords
+       tmp1.Double(tmp2)    // tmp1 =  4*v in P1xP1 coords
+       tmp2.FromP1xP1(tmp1) // tmp2 =  4*v in P2 coords
+       tmp1.Double(tmp2)    // tmp1 =  8*v in P1xP1 coords
+       tmp2.FromP1xP1(tmp1) // tmp2 =  8*v in P2 coords
+       tmp1.Double(tmp2)    // tmp1 = 16*v in P1xP1 coords
+       v.fromP1xP1(tmp1)    // now v = 16*(odd components)
+
+       // Accumulate the even components
+       for i := 0; i < 64; i += 2 {
+               basepointTable[i/2].SelectInto(multiple, digits[i])
+               tmp1.AddAffine(v, multiple)
+               v.fromP1xP1(tmp1)
+       }
+
+       return v
+}
+
+// ScalarMult sets v = x * q, and returns v.
+//
+// The scalar multiplication is done in constant time.
+func (v *Point) ScalarMult(x *Scalar, q *Point) *Point {
+       checkInitialized(q)
+
+       var table projLookupTable
+       table.FromP3(q)
+
+       // Write x = sum(x_i * 16^i)
+       // so  x*Q = sum( Q*x_i*16^i )
+       //         = Q*x_0 + 16*(Q*x_1 + 16*( ... + Q*x_63) ... )
+       //           <------compute inside out---------
+       //
+       // We use the lookup table to get the x_i*Q values
+       // and do four doublings to compute 16*Q
+       digits := x.signedRadix16()
+
+       // Unwrap first loop iteration to save computing 16*identity
+       multiple := &projCached{}
+       tmp1 := &projP1xP1{}
+       tmp2 := &projP2{}
+       table.SelectInto(multiple, digits[63])
+
+       v.Set(NewIdentityPoint())
+       tmp1.Add(v, multiple) // tmp1 = x_63*Q in P1xP1 coords
+       for i := 62; i >= 0; i-- {
+               tmp2.FromP1xP1(tmp1) // tmp2 =    (prev) in P2 coords
+               tmp1.Double(tmp2)    // tmp1 =  2*(prev) in P1xP1 coords
+               tmp2.FromP1xP1(tmp1) // tmp2 =  2*(prev) in P2 coords
+               tmp1.Double(tmp2)    // tmp1 =  4*(prev) in P1xP1 coords
+               tmp2.FromP1xP1(tmp1) // tmp2 =  4*(prev) in P2 coords
+               tmp1.Double(tmp2)    // tmp1 =  8*(prev) in P1xP1 coords
+               tmp2.FromP1xP1(tmp1) // tmp2 =  8*(prev) in P2 coords
+               tmp1.Double(tmp2)    // tmp1 = 16*(prev) in P1xP1 coords
+               v.fromP1xP1(tmp1)    //    v = 16*(prev) in P3 coords
+               table.SelectInto(multiple, digits[i])
+               tmp1.Add(v, multiple) // tmp1 = x_i*Q + 16*(prev) in P1xP1 coords
+       }
+       v.fromP1xP1(tmp1)
+       return v
+}
+
+// basepointNafTable is the nafLookupTable8 for the basepoint.
+// It is precomputed the first time it's used.
+func basepointNafTable() *nafLookupTable8 {
+       basepointNafTablePrecomp.initOnce.Do(func() {
+               basepointNafTablePrecomp.table.FromP3(NewGeneratorPoint())
+       })
+       return &basepointNafTablePrecomp.table
+}
+
+var basepointNafTablePrecomp struct {
+       table    nafLookupTable8
+       initOnce sync.Once
+}
+
+// VarTimeDoubleScalarBaseMult sets v = a * A + b * B, where B is the canonical
+// generator, and returns v.
+//
+// Execution time depends on the inputs.
+func (v *Point) VarTimeDoubleScalarBaseMult(a *Scalar, A *Point, b *Scalar) *Point {
+       checkInitialized(A)
+
+       // Similarly to the single variable-base approach, we compute
+       // digits and use them with a lookup table.  However, because
+       // we are allowed to do variable-time operations, we don't
+       // need constant-time lookups or constant-time digit
+       // computations.
+       //
+       // So we use a non-adjacent form of some width w instead of
+       // radix 16.  This is like a binary representation (one digit
+       // for each binary place) but we allow the digits to grow in
+       // magnitude up to 2^{w-1} so that the nonzero digits are as
+       // sparse as possible.  Intuitively, this "condenses" the
+       // "mass" of the scalar onto sparse coefficients (meaning
+       // fewer additions).
+
+       basepointNafTable := basepointNafTable()
+       var aTable nafLookupTable5
+       aTable.FromP3(A)
+       // Because the basepoint is fixed, we can use a wider NAF
+       // corresponding to a bigger table.
+       aNaf := a.nonAdjacentForm(5)
+       bNaf := b.nonAdjacentForm(8)
+
+       // Find the first nonzero coefficient.
+       i := 255
+       for j := i; j >= 0; j-- {
+               if aNaf[j] != 0 || bNaf[j] != 0 {
+                       break
+               }
+       }
+
+       multA := &projCached{}
+       multB := &affineCached{}
+       tmp1 := &projP1xP1{}
+       tmp2 := &projP2{}
+       tmp2.Zero()
+
+       // Move from high to low bits, doubling the accumulator
+       // at each iteration and checking whether there is a nonzero
+       // coefficient to look up a multiple of.
+       for ; i >= 0; i-- {
+               tmp1.Double(tmp2)
+
+               // Only update v if we have a nonzero coeff to add in.
+               if aNaf[i] > 0 {
+                       v.fromP1xP1(tmp1)
+                       aTable.SelectInto(multA, aNaf[i])
+                       tmp1.Add(v, multA)
+               } else if aNaf[i] < 0 {
+                       v.fromP1xP1(tmp1)
+                       aTable.SelectInto(multA, -aNaf[i])
+                       tmp1.Sub(v, multA)
+               }
+
+               if bNaf[i] > 0 {
+                       v.fromP1xP1(tmp1)
+                       basepointNafTable.SelectInto(multB, bNaf[i])
+                       tmp1.AddAffine(v, multB)
+               } else if bNaf[i] < 0 {
+                       v.fromP1xP1(tmp1)
+                       basepointNafTable.SelectInto(multB, -bNaf[i])
+                       tmp1.SubAffine(v, multB)
+               }
+
+               tmp2.FromP1xP1(tmp1)
+       }
+
+       v.fromP2(tmp2)
+       return v
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/scalarmult_test.go b/src/crypto/ed25519/internal/edwards25519/scalarmult_test.go
new file mode 100644 (file)
index 0000000..c2027f5
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/scalarmult_test.go
@@ -0,0 +1,209 @@
+// Copyright (c) 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package edwards25519
+
+import (
+       "testing"
+       "testing/quick"
+)
+
+var (
+       // quickCheckConfig32 will make each quickcheck test run (32 * -quickchecks)
+       // times. The default value of -quickchecks is 100.
+       quickCheckConfig32 = &quick.Config{MaxCountScale: 1 << 5}
+
+       // a random scalar generated using dalek.
+       dalekScalar = Scalar{[32]byte{219, 106, 114, 9, 174, 249, 155, 89, 69, 203, 201, 93, 92, 116, 234, 187, 78, 115, 103, 172, 182, 98, 62, 103, 187, 136, 13, 100, 248, 110, 12, 4}}
+       // the above, times the edwards25519 basepoint.
+       dalekScalarBasepoint, _ = new(Point).SetBytes([]byte{0xf4, 0xef, 0x7c, 0xa, 0x34, 0x55, 0x7b, 0x9f, 0x72, 0x3b, 0xb6, 0x1e, 0xf9, 0x46, 0x9, 0x91, 0x1c, 0xb9, 0xc0, 0x6c, 0x17, 0x28, 0x2d, 0x8b, 0x43, 0x2b, 0x5, 0x18, 0x6a, 0x54, 0x3e, 0x48})
+)
+
+func TestScalarMultSmallScalars(t *testing.T) {
+       var z Scalar
+       var p Point
+       p.ScalarMult(&z, B)
+       if I.Equal(&p) != 1 {
+               t.Error("0*B != 0")
+       }
+       checkOnCurve(t, &p)
+
+       z = Scalar{[32]byte{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}
+       p.ScalarMult(&z, B)
+       if B.Equal(&p) != 1 {
+               t.Error("1*B != 1")
+       }
+       checkOnCurve(t, &p)
+}
+
+func TestScalarMultVsDalek(t *testing.T) {
+       var p Point
+       p.ScalarMult(&dalekScalar, B)
+       if dalekScalarBasepoint.Equal(&p) != 1 {
+               t.Error("Scalar mul does not match dalek")
+       }
+       checkOnCurve(t, &p)
+}
+
+func TestBaseMultVsDalek(t *testing.T) {
+       var p Point
+       p.ScalarBaseMult(&dalekScalar)
+       if dalekScalarBasepoint.Equal(&p) != 1 {
+               t.Error("Scalar mul does not match dalek")
+       }
+       checkOnCurve(t, &p)
+}
+
+func TestVarTimeDoubleBaseMultVsDalek(t *testing.T) {
+       var p Point
+       var z Scalar
+       p.VarTimeDoubleScalarBaseMult(&dalekScalar, B, &z)
+       if dalekScalarBasepoint.Equal(&p) != 1 {
+               t.Error("VarTimeDoubleScalarBaseMult fails with b=0")
+       }
+       checkOnCurve(t, &p)
+       p.VarTimeDoubleScalarBaseMult(&z, B, &dalekScalar)
+       if dalekScalarBasepoint.Equal(&p) != 1 {
+               t.Error("VarTimeDoubleScalarBaseMult fails with a=0")
+       }
+       checkOnCurve(t, &p)
+}
+
+func TestScalarMultDistributesOverAdd(t *testing.T) {
+       scalarMultDistributesOverAdd := func(x, y Scalar) bool {
+               var z Scalar
+               z.Add(&x, &y)
+               var p, q, r, check Point
+               p.ScalarMult(&x, B)
+               q.ScalarMult(&y, B)
+               r.ScalarMult(&z, B)
+               check.Add(&p, &q)
+               checkOnCurve(t, &p, &q, &r, &check)
+               return check.Equal(&r) == 1
+       }
+
+       if err := quick.Check(scalarMultDistributesOverAdd, quickCheckConfig32); err != nil {
+               t.Error(err)
+       }
+}
+
+func TestScalarMultNonIdentityPoint(t *testing.T) {
+       // Check whether p.ScalarMult and q.ScalaBaseMult give the same,
+       // when p and q are originally set to the base point.
+
+       scalarMultNonIdentityPoint := func(x Scalar) bool {
+               var p, q Point
+               p.Set(B)
+               q.Set(B)
+
+               p.ScalarMult(&x, B)
+               q.ScalarBaseMult(&x)
+
+               checkOnCurve(t, &p, &q)
+
+               return p.Equal(&q) == 1
+       }
+
+       if err := quick.Check(scalarMultNonIdentityPoint, quickCheckConfig32); err != nil {
+               t.Error(err)
+       }
+}
+
+func TestBasepointTableGeneration(t *testing.T) {
+       // The basepoint table is 32 affineLookupTables,
+       // corresponding to (16^2i)*B for table i.
+       basepointTable := basepointTable()
+
+       tmp1 := &projP1xP1{}
+       tmp2 := &projP2{}
+       tmp3 := &Point{}
+       tmp3.Set(B)
+       table := make([]affineLookupTable, 32)
+       for i := 0; i < 32; i++ {
+               // Build the table
+               table[i].FromP3(tmp3)
+               // Assert equality with the hardcoded one
+               if table[i] != basepointTable[i] {
+                       t.Errorf("Basepoint table %d does not match", i)
+               }
+
+               // Set p = (16^2)*p = 256*p = 2^8*p
+               tmp2.FromP3(tmp3)
+               for j := 0; j < 7; j++ {
+                       tmp1.Double(tmp2)
+                       tmp2.FromP1xP1(tmp1)
+               }
+               tmp1.Double(tmp2)
+               tmp3.fromP1xP1(tmp1)
+               checkOnCurve(t, tmp3)
+       }
+}
+
+func TestScalarMultMatchesBaseMult(t *testing.T) {
+       scalarMultMatchesBaseMult := func(x Scalar) bool {
+               var p, q Point
+               p.ScalarMult(&x, B)
+               q.ScalarBaseMult(&x)
+               checkOnCurve(t, &p, &q)
+               return p.Equal(&q) == 1
+       }
+
+       if err := quick.Check(scalarMultMatchesBaseMult, quickCheckConfig32); err != nil {
+               t.Error(err)
+       }
+}
+
+func TestBasepointNafTableGeneration(t *testing.T) {
+       var table nafLookupTable8
+       table.FromP3(B)
+
+       if table != *basepointNafTable() {
+               t.Error("BasepointNafTable does not match")
+       }
+}
+
+func TestVarTimeDoubleBaseMultMatchesBaseMult(t *testing.T) {
+       varTimeDoubleBaseMultMatchesBaseMult := func(x, y Scalar) bool {
+               var p, q1, q2, check Point
+
+               p.VarTimeDoubleScalarBaseMult(&x, B, &y)
+
+               q1.ScalarBaseMult(&x)
+               q2.ScalarBaseMult(&y)
+               check.Add(&q1, &q2)
+
+               checkOnCurve(t, &p, &check, &q1, &q2)
+               return p.Equal(&check) == 1
+       }
+
+       if err := quick.Check(varTimeDoubleBaseMultMatchesBaseMult, quickCheckConfig32); err != nil {
+               t.Error(err)
+       }
+}
+
+// Benchmarks.
+
+func BenchmarkScalarBaseMult(t *testing.B) {
+       var p Point
+
+       for i := 0; i < t.N; i++ {
+               p.ScalarBaseMult(&dalekScalar)
+       }
+}
+
+func BenchmarkScalarMult(t *testing.B) {
+       var p Point
+
+       for i := 0; i < t.N; i++ {
+               p.ScalarMult(&dalekScalar, B)
+       }
+}
+
+func BenchmarkVarTimeDoubleScalarBaseMult(t *testing.B) {
+       var p Point
+
+       for i := 0; i < t.N; i++ {
+               p.VarTimeDoubleScalarBaseMult(&dalekScalar, B, &dalekScalar)
+       }
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/tables.go b/src/crypto/ed25519/internal/edwards25519/tables.go
new file mode 100644 (file)
index 0000000..beec956
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/tables.go
@@ -0,0 +1,129 @@
+// Copyright (c) 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package edwards25519
+
+import (
+       "crypto/subtle"
+)
+
+// A dynamic lookup table for variable-base, constant-time scalar muls.
+type projLookupTable struct {
+       points [8]projCached
+}
+
+// A precomputed lookup table for fixed-base, constant-time scalar muls.
+type affineLookupTable struct {
+       points [8]affineCached
+}
+
+// A dynamic lookup table for variable-base, variable-time scalar muls.
+type nafLookupTable5 struct {
+       points [8]projCached
+}
+
+// A precomputed lookup table for fixed-base, variable-time scalar muls.
+type nafLookupTable8 struct {
+       points [64]affineCached
+}
+
+// Constructors.
+
+// Builds a lookup table at runtime. Fast.
+func (v *projLookupTable) FromP3(q *Point) {
+       // Goal: v.points[i] = (i+1)*Q, i.e., Q, 2Q, ..., 8Q
+       // This allows lookup of -8Q, ..., -Q, 0, Q, ..., 8Q
+       v.points[0].FromP3(q)
+       tmpP3 := Point{}
+       tmpP1xP1 := projP1xP1{}
+       for i := 0; i < 7; i++ {
+               // Compute (i+1)*Q as Q + i*Q and convert to a ProjCached
+               // This is needlessly complicated because the API has explicit
+               // recievers instead of creating stack objects and relying on RVO
+               v.points[i+1].FromP3(tmpP3.fromP1xP1(tmpP1xP1.Add(q, &v.points[i])))
+       }
+}
+
+// This is not optimised for speed; fixed-base tables should be precomputed.
+func (v *affineLookupTable) FromP3(q *Point) {
+       // Goal: v.points[i] = (i+1)*Q, i.e., Q, 2Q, ..., 8Q
+       // This allows lookup of -8Q, ..., -Q, 0, Q, ..., 8Q
+       v.points[0].FromP3(q)
+       tmpP3 := Point{}
+       tmpP1xP1 := projP1xP1{}
+       for i := 0; i < 7; i++ {
+               // Compute (i+1)*Q as Q + i*Q and convert to AffineCached
+               v.points[i+1].FromP3(tmpP3.fromP1xP1(tmpP1xP1.AddAffine(q, &v.points[i])))
+       }
+}
+
+// Builds a lookup table at runtime. Fast.
+func (v *nafLookupTable5) FromP3(q *Point) {
+       // Goal: v.points[i] = (2*i+1)*Q, i.e., Q, 3Q, 5Q, ..., 15Q
+       // This allows lookup of -15Q, ..., -3Q, -Q, 0, Q, 3Q, ..., 15Q
+       v.points[0].FromP3(q)
+       q2 := Point{}
+       q2.Add(q, q)
+       tmpP3 := Point{}
+       tmpP1xP1 := projP1xP1{}
+       for i := 0; i < 7; i++ {
+               v.points[i+1].FromP3(tmpP3.fromP1xP1(tmpP1xP1.Add(&q2, &v.points[i])))
+       }
+}
+
+// This is not optimised for speed; fixed-base tables should be precomputed.
+func (v *nafLookupTable8) FromP3(q *Point) {
+       v.points[0].FromP3(q)
+       q2 := Point{}
+       q2.Add(q, q)
+       tmpP3 := Point{}
+       tmpP1xP1 := projP1xP1{}
+       for i := 0; i < 63; i++ {
+               v.points[i+1].FromP3(tmpP3.fromP1xP1(tmpP1xP1.AddAffine(&q2, &v.points[i])))
+       }
+}
+
+// Selectors.
+
+// Set dest to x*Q, where -8 <= x <= 8, in constant time.
+func (v *projLookupTable) SelectInto(dest *projCached, x int8) {
+       // Compute xabs = |x|
+       xmask := x >> 7
+       xabs := uint8((x + xmask) ^ xmask)
+
+       dest.Zero()
+       for j := 1; j <= 8; j++ {
+               // Set dest = j*Q if |x| = j
+               cond := subtle.ConstantTimeByteEq(xabs, uint8(j))
+               dest.Select(&v.points[j-1], dest, cond)
+       }
+       // Now dest = |x|*Q, conditionally negate to get x*Q
+       dest.CondNeg(int(xmask & 1))
+}
+
+// Set dest to x*Q, where -8 <= x <= 8, in constant time.
+func (v *affineLookupTable) SelectInto(dest *affineCached, x int8) {
+       // Compute xabs = |x|
+       xmask := x >> 7
+       xabs := uint8((x + xmask) ^ xmask)
+
+       dest.Zero()
+       for j := 1; j <= 8; j++ {
+               // Set dest = j*Q if |x| = j
+               cond := subtle.ConstantTimeByteEq(xabs, uint8(j))
+               dest.Select(&v.points[j-1], dest, cond)
+       }
+       // Now dest = |x|*Q, conditionally negate to get x*Q
+       dest.CondNeg(int(xmask & 1))
+}
+
+// Given odd x with 0 < x < 2^4, return x*Q (in variable time).
+func (v *nafLookupTable5) SelectInto(dest *projCached, x int8) {
+       *dest = v.points[x/2]
+}
+
+// Given odd x with 0 < x < 2^7, return x*Q (in variable time).
+func (v *nafLookupTable8) SelectInto(dest *affineCached, x int8) {
+       *dest = v.points[x/2]
+}

diff --git a/src/crypto/ed25519/internal/edwards25519/tables_test.go b/src/crypto/ed25519/internal/edwards25519/tables_test.go
new file mode 100644 (file)
index 0000000..b5d161a
--- /dev/null
+++ b/src/crypto/ed25519/internal/edwards25519/tables_test.go
@@ -0,0 +1,119 @@
+// Copyright (c) 2019 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package edwards25519
+
+import (
+       "testing"
+)
+
+func TestProjLookupTable(t *testing.T) {
+       var table projLookupTable
+       table.FromP3(B)
+
+       var tmp1, tmp2, tmp3 projCached
+       table.SelectInto(&tmp1, 6)
+       table.SelectInto(&tmp2, -2)
+       table.SelectInto(&tmp3, -4)
+       // Expect T1 + T2 + T3 = identity
+
+       var accP1xP1 projP1xP1
+       accP3 := NewIdentityPoint()
+
+       accP1xP1.Add(accP3, &tmp1)
+       accP3.fromP1xP1(&accP1xP1)
+       accP1xP1.Add(accP3, &tmp2)
+       accP3.fromP1xP1(&accP1xP1)
+       accP1xP1.Add(accP3, &tmp3)
+       accP3.fromP1xP1(&accP1xP1)
+
+       if accP3.Equal(I) != 1 {
+               t.Errorf("Consistency check on ProjLookupTable.SelectInto failed!  %x %x %x", tmp1, tmp2, tmp3)
+       }
+}
+
+func TestAffineLookupTable(t *testing.T) {
+       var table affineLookupTable
+       table.FromP3(B)
+
+       var tmp1, tmp2, tmp3 affineCached
+       table.SelectInto(&tmp1, 3)
+       table.SelectInto(&tmp2, -7)
+       table.SelectInto(&tmp3, 4)
+       // Expect T1 + T2 + T3 = identity
+
+       var accP1xP1 projP1xP1
+       accP3 := NewIdentityPoint()
+
+       accP1xP1.AddAffine(accP3, &tmp1)
+       accP3.fromP1xP1(&accP1xP1)
+       accP1xP1.AddAffine(accP3, &tmp2)
+       accP3.fromP1xP1(&accP1xP1)
+       accP1xP1.AddAffine(accP3, &tmp3)
+       accP3.fromP1xP1(&accP1xP1)
+
+       if accP3.Equal(I) != 1 {
+               t.Errorf("Consistency check on ProjLookupTable.SelectInto failed!  %x %x %x", tmp1, tmp2, tmp3)
+       }
+}
+
+func TestNafLookupTable5(t *testing.T) {
+       var table nafLookupTable5
+       table.FromP3(B)
+
+       var tmp1, tmp2, tmp3, tmp4 projCached
+       table.SelectInto(&tmp1, 9)
+       table.SelectInto(&tmp2, 11)
+       table.SelectInto(&tmp3, 7)
+       table.SelectInto(&tmp4, 13)
+       // Expect T1 + T2 = T3 + T4
+
+       var accP1xP1 projP1xP1
+       lhs := NewIdentityPoint()
+       rhs := NewIdentityPoint()
+
+       accP1xP1.Add(lhs, &tmp1)
+       lhs.fromP1xP1(&accP1xP1)
+       accP1xP1.Add(lhs, &tmp2)
+       lhs.fromP1xP1(&accP1xP1)
+
+       accP1xP1.Add(rhs, &tmp3)
+       rhs.fromP1xP1(&accP1xP1)
+       accP1xP1.Add(rhs, &tmp4)
+       rhs.fromP1xP1(&accP1xP1)
+
+       if lhs.Equal(rhs) != 1 {
+               t.Errorf("Consistency check on nafLookupTable5 failed")
+       }
+}
+
+func TestNafLookupTable8(t *testing.T) {
+       var table nafLookupTable8
+       table.FromP3(B)
+
+       var tmp1, tmp2, tmp3, tmp4 affineCached
+       table.SelectInto(&tmp1, 49)
+       table.SelectInto(&tmp2, 11)
+       table.SelectInto(&tmp3, 35)
+       table.SelectInto(&tmp4, 25)
+       // Expect T1 + T2 = T3 + T4
+
+       var accP1xP1 projP1xP1
+       lhs := NewIdentityPoint()
+       rhs := NewIdentityPoint()
+
+       accP1xP1.AddAffine(lhs, &tmp1)
+       lhs.fromP1xP1(&accP1xP1)
+       accP1xP1.AddAffine(lhs, &tmp2)
+       lhs.fromP1xP1(&accP1xP1)
+
+       accP1xP1.AddAffine(rhs, &tmp3)
+       rhs.fromP1xP1(&accP1xP1)
+       accP1xP1.AddAffine(rhs, &tmp4)
+       rhs.fromP1xP1(&accP1xP1)
+
+       if lhs.Equal(rhs) != 1 {
+               t.Errorf("Consistency check on nafLookupTable8 failed")
+       }
+}

diff --git a/src/go/build/deps_test.go b/src/go/build/deps_test.go
index 9cdfb898ba961b8a7b973f94935bca8a2907519e..bb6d92f89acacf9986d0dbffe47d32c327d119b3 100644 (file)
--- a/src/go/build/deps_test.go
+++ b/src/go/build/deps_test.go
@@ -392,6 +392,8 @@ var depsRules = `
        < crypto
        < crypto/subtle
        < crypto/internal/subtle
+       < crypto/ed25519/internal/edwards25519/field
+       < crypto/ed25519/internal/edwards25519
        < crypto/cipher
        < crypto/aes, crypto/des, crypto/hmac, crypto/md5, crypto/rc4,
          crypto/sha1, crypto/sha256, crypto/sha512
@@ -403,7 +405,6 @@ var depsRules = `
        CRYPTO, FMT, math/big
        < crypto/rand
        < crypto/internal/randutil
-       < crypto/ed25519/internal/edwards25519
        < crypto/ed25519
        < encoding/asn1
        < golang.org/x/crypto/cryptobyte/asn1

diff --git a/src/go/types/stdlib_test.go b/src/go/types/stdlib_test.go
index 29f71137df8394256a7770186f40ece33692ae7b..b565956e086e0dee61ee50217764de81d6507089 100644 (file)
--- a/src/go/types/stdlib_test.go
+++ b/src/go/types/stdlib_test.go
@@ -306,6 +306,13 @@ func (w *walker) walk(dir string) {
                return
        }
 
+       // skip submodules, which might not be vendored
+       for _, f := range files {
+               if f.Name() == "go.mod" {
+                       return
+               }
+       }
+
        // apply pkgh to the files in directory dir
        // but ignore files directly under $GOROOT/src (might be temporary test files).
        if dir != filepath.Join(runtime.GOROOT(), "src") {