return p256ToAffine(&x1, &y1, &z1)
}
-// Field elements are represented as nine, unsigned 32-bit words.
-//
-// The value of a field element is:
-// x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228)
-//
-// That is, each limb is alternately 29 or 28-bits wide in little-endian
-// order.
-//
-// This means that a field element hits 2**257, rather than 2**256 as we would
-// like. A 28, 29, ... pattern would cause us to hit 2**256, but that causes
-// problems when multiplying as terms end up one bit short of a limb which
-// would require much bit-shifting to correct.
-//
-// Finally, the values stored in a field element are in Montgomery form. So the
-// value |y| is stored as (y*R) mod p, where p is the P-256 prime and R is
-// 2**257.
-
-const (
- p256Limbs = 9
- bottom29Bits = 0x1fffffff
-)
-
-var (
- // p256One is the number 1 as a field element.
- p256One = [p256Limbs]uint32{2, 0, 0, 0xffff800, 0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff, 0}
- p256Zero = [p256Limbs]uint32{0, 0, 0, 0, 0, 0, 0, 0, 0}
- // p256P is the prime modulus as a field element.
- p256P = [p256Limbs]uint32{0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, 0, 0, 0x200000, 0xf000000, 0xfffffff}
- // p2562P is the twice prime modulus as a field element.
- p2562P = [p256Limbs]uint32{0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, 0, 0, 0x400000, 0xe000000, 0x1fffffff}
-)
-
// p256Precomputed contains precomputed values to aid the calculation of scalar
// multiples of the base point, G. It's actually two, equal length, tables
// concatenated.
0xbe73d6f, 0xaa88141, 0xd976c81, 0x7e7a9cc, 0x18beb771, 0xd773cbd, 0x13f51951, 0x9d0c177, 0x1c49a78,
}
-// Field element operations:
-
-const bottom28Bits = 0xfffffff
-
-// nonZeroToAllOnes returns:
-//
-// 0xffffffff for 0 < x <= 2**31
-// 0 for x == 0 or x > 2**31.
-func nonZeroToAllOnes(x uint32) uint32 {
- return ((x - 1) >> 31) - 1
-}
-
-// p256ReduceCarry adds a multiple of p in order to cancel |carry|,
-// which is a term at 2**257.
-//
-// On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28.
-// On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29.
-func p256ReduceCarry(inout *[p256Limbs]uint32, carry uint32) {
- carry_mask := nonZeroToAllOnes(carry)
-
- inout[0] += carry << 1
- inout[3] += 0x10000000 & carry_mask
- // carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the
- // previous line therefore this doesn't underflow.
- inout[3] -= carry << 11
- inout[4] += (0x20000000 - 1) & carry_mask
- inout[5] += (0x10000000 - 1) & carry_mask
- inout[6] += (0x20000000 - 1) & carry_mask
- inout[6] -= carry << 22
- // This may underflow if carry is non-zero but, if so, we'll fix it in the
- // next line.
- inout[7] -= 1 & carry_mask
- inout[7] += carry << 25
-}
-
-// p256Sum sets out = in+in2.
-//
-// On entry: in[i]+in2[i] must not overflow a 32-bit word.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Sum(out, in, in2 *[p256Limbs]uint32) {
- carry := uint32(0)
- for i := 0; ; i++ {
- out[i] = in[i] + in2[i]
- out[i] += carry
- carry = out[i] >> 29
- out[i] &= bottom29Bits
-
- i++
- if i == p256Limbs {
- break
- }
-
- out[i] = in[i] + in2[i]
- out[i] += carry
- carry = out[i] >> 28
- out[i] &= bottom28Bits
- }
-
- p256ReduceCarry(out, carry)
-}
-
-const (
- two30m2 = 1<<30 - 1<<2
- two30p13m2 = 1<<30 + 1<<13 - 1<<2
- two31m2 = 1<<31 - 1<<2
- two31m3 = 1<<31 - 1<<3
- two31p24m2 = 1<<31 + 1<<24 - 1<<2
- two30m27m2 = 1<<30 - 1<<27 - 1<<2
-)
-
-// p256Zero31 is 0 mod p.
-var p256Zero31 = [p256Limbs]uint32{two31m3, two30m2, two31m2, two30p13m2, two31m2, two30m2, two31p24m2, two30m27m2, two31m2}
-
-// p256Diff sets out = in-in2.
-//
-// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
-// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Diff(out, in, in2 *[p256Limbs]uint32) {
- var carry uint32
-
- for i := 0; ; i++ {
- out[i] = in[i] - in2[i]
- out[i] += p256Zero31[i]
- out[i] += carry
- carry = out[i] >> 29
- out[i] &= bottom29Bits
-
- i++
- if i == p256Limbs {
- break
- }
-
- out[i] = in[i] - in2[i]
- out[i] += p256Zero31[i]
- out[i] += carry
- carry = out[i] >> 28
- out[i] &= bottom28Bits
- }
-
- p256ReduceCarry(out, carry)
-}
-
-// p256ReduceDegree sets out = tmp/R mod p where tmp contains 64-bit words with
-// the same 29,28,... bit positions as a field element.
-//
-// The values in field elements are in Montgomery form: x*R mod p where R =
-// 2**257. Since we just multiplied two Montgomery values together, the result
-// is x*y*R*R mod p. We wish to divide by R in order for the result also to be
-// in Montgomery form.
-//
-// On entry: tmp[i] < 2**64.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256ReduceDegree(out *[p256Limbs]uint32, tmp [17]uint64) {
- // The following table may be helpful when reading this code:
- //
- // Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10...
- // Width (bits): 29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29
- // Start bit: 0 | 29| 57| 86|114|143|171|200|228|257|285
- // (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285
- var tmp2 [18]uint32
- var carry, x, xMask uint32
-
- // tmp contains 64-bit words with the same 29,28,29-bit positions as a
- // field element. So the top of an element of tmp might overlap with
- // another element two positions down. The following loop eliminates
- // this overlap.
- tmp2[0] = uint32(tmp[0]) & bottom29Bits
-
- tmp2[1] = uint32(tmp[0]) >> 29
- tmp2[1] |= (uint32(tmp[0]>>32) << 3) & bottom28Bits
- tmp2[1] += uint32(tmp[1]) & bottom28Bits
- carry = tmp2[1] >> 28
- tmp2[1] &= bottom28Bits
-
- for i := 2; i < 17; i++ {
- tmp2[i] = (uint32(tmp[i-2] >> 32)) >> 25
- tmp2[i] += (uint32(tmp[i-1])) >> 28
- tmp2[i] += (uint32(tmp[i-1]>>32) << 4) & bottom29Bits
- tmp2[i] += uint32(tmp[i]) & bottom29Bits
- tmp2[i] += carry
- carry = tmp2[i] >> 29
- tmp2[i] &= bottom29Bits
-
- i++
- if i == 17 {
- break
- }
- tmp2[i] = uint32(tmp[i-2]>>32) >> 25
- tmp2[i] += uint32(tmp[i-1]) >> 29
- tmp2[i] += ((uint32(tmp[i-1] >> 32)) << 3) & bottom28Bits
- tmp2[i] += uint32(tmp[i]) & bottom28Bits
- tmp2[i] += carry
- carry = tmp2[i] >> 28
- tmp2[i] &= bottom28Bits
- }
-
- tmp2[17] = uint32(tmp[15]>>32) >> 25
- tmp2[17] += uint32(tmp[16]) >> 29
- tmp2[17] += uint32(tmp[16]>>32) << 3
- tmp2[17] += carry
-
- // Montgomery elimination of terms:
- //
- // Since R is 2**257, we can divide by R with a bitwise shift if we can
- // ensure that the right-most 257 bits are all zero. We can make that true
- // by adding multiplies of p without affecting the value.
- //
- // So we eliminate limbs from right to left. Since the bottom 29 bits of p
- // are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0.
- // We can do that for 8 further limbs and then right shift to eliminate the
- // extra factor of R.
- for i := 0; ; i += 2 {
- tmp2[i+1] += tmp2[i] >> 29
- x = tmp2[i] & bottom29Bits
- xMask = nonZeroToAllOnes(x)
- tmp2[i] = 0
-
- // The bounds calculations for this loop are tricky. Each iteration of
- // the loop eliminates two words by adding values to words to their
- // right.
- //
- // The following table contains the amounts added to each word (as an
- // offset from the value of i at the top of the loop). The amounts are
- // accounted for from the first and second half of the loop separately
- // and are written as, for example, 28 to mean a value <2**28.
- //
- // Word: 3 4 5 6 7 8 9 10
- // Added in top half: 28 11 29 21 29 28
- // 28 29
- // 29
- // Added in bottom half: 29 10 28 21 28 28
- // 29
- //
- // The value that is currently offset 7 will be offset 5 for the next
- // iteration and then offset 3 for the iteration after that. Therefore
- // the total value added will be the values added at 7, 5 and 3.
- //
- // The following table accumulates these values. The sums at the bottom
- // are written as, for example, 29+28, to mean a value < 2**29+2**28.
- //
- // Word: 3 4 5 6 7 8 9 10 11 12 13
- // 28 11 10 29 21 29 28 28 28 28 28
- // 29 28 11 28 29 28 29 28 29 28
- // 29 28 21 21 29 21 29 21
- // 10 29 28 21 28 21 28
- // 28 29 28 29 28 29 28
- // 11 10 29 10 29 10
- // 29 28 11 28 11
- // 29 29
- // --------------------------------------------
- // 30+ 31+ 30+ 31+ 30+
- // 28+ 29+ 28+ 29+ 21+
- // 21+ 28+ 21+ 28+ 10
- // 10 21+ 10 21+
- // 11 11
- //
- // So the greatest amount is added to tmp2[10] and tmp2[12]. If
- // tmp2[10/12] has an initial value of <2**29, then the maximum value
- // will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32,
- // as required.
- tmp2[i+3] += (x << 10) & bottom28Bits
- tmp2[i+4] += (x >> 18)
-
- tmp2[i+6] += (x << 21) & bottom29Bits
- tmp2[i+7] += x >> 8
-
- // At position 200, which is the starting bit position for word 7, we
- // have a factor of 0xf000000 = 2**28 - 2**24.
- tmp2[i+7] += 0x10000000 & xMask
- tmp2[i+8] += (x - 1) & xMask
- tmp2[i+7] -= (x << 24) & bottom28Bits
- tmp2[i+8] -= x >> 4
-
- tmp2[i+8] += 0x20000000 & xMask
- tmp2[i+8] -= x
- tmp2[i+8] += (x << 28) & bottom29Bits
- tmp2[i+9] += ((x >> 1) - 1) & xMask
-
- if i+1 == p256Limbs {
- break
- }
- tmp2[i+2] += tmp2[i+1] >> 28
- x = tmp2[i+1] & bottom28Bits
- xMask = nonZeroToAllOnes(x)
- tmp2[i+1] = 0
-
- tmp2[i+4] += (x << 11) & bottom29Bits
- tmp2[i+5] += (x >> 18)
-
- tmp2[i+7] += (x << 21) & bottom28Bits
- tmp2[i+8] += x >> 7
-
- // At position 199, which is the starting bit of the 8th word when
- // dealing with a context starting on an odd word, we have a factor of
- // 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th
- // word from i+1 is i+8.
- tmp2[i+8] += 0x20000000 & xMask
- tmp2[i+9] += (x - 1) & xMask
- tmp2[i+8] -= (x << 25) & bottom29Bits
- tmp2[i+9] -= x >> 4
-
- tmp2[i+9] += 0x10000000 & xMask
- tmp2[i+9] -= x
- tmp2[i+10] += (x - 1) & xMask
- }
-
- // We merge the right shift with a carry chain. The words above 2**257 have
- // widths of 28,29,... which we need to correct when copying them down.
- carry = 0
- for i := 0; i < 8; i++ {
- // The maximum value of tmp2[i + 9] occurs on the first iteration and
- // is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is
- // therefore safe.
- out[i] = tmp2[i+9]
- out[i] += carry
- out[i] += (tmp2[i+10] << 28) & bottom29Bits
- carry = out[i] >> 29
- out[i] &= bottom29Bits
-
- i++
- out[i] = tmp2[i+9] >> 1
- out[i] += carry
- carry = out[i] >> 28
- out[i] &= bottom28Bits
- }
-
- out[8] = tmp2[17]
- out[8] += carry
- carry = out[8] >> 29
- out[8] &= bottom29Bits
-
- p256ReduceCarry(out, carry)
-}
-
-// p256Square sets out=in*in.
-//
-// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Square(out, in *[p256Limbs]uint32) {
- var tmp [17]uint64
-
- tmp[0] = uint64(in[0]) * uint64(in[0])
- tmp[1] = uint64(in[0]) * (uint64(in[1]) << 1)
- tmp[2] = uint64(in[0])*(uint64(in[2])<<1) +
- uint64(in[1])*(uint64(in[1])<<1)
- tmp[3] = uint64(in[0])*(uint64(in[3])<<1) +
- uint64(in[1])*(uint64(in[2])<<1)
- tmp[4] = uint64(in[0])*(uint64(in[4])<<1) +
- uint64(in[1])*(uint64(in[3])<<2) +
- uint64(in[2])*uint64(in[2])
- tmp[5] = uint64(in[0])*(uint64(in[5])<<1) +
- uint64(in[1])*(uint64(in[4])<<1) +
- uint64(in[2])*(uint64(in[3])<<1)
- tmp[6] = uint64(in[0])*(uint64(in[6])<<1) +
- uint64(in[1])*(uint64(in[5])<<2) +
- uint64(in[2])*(uint64(in[4])<<1) +
- uint64(in[3])*(uint64(in[3])<<1)
- tmp[7] = uint64(in[0])*(uint64(in[7])<<1) +
- uint64(in[1])*(uint64(in[6])<<1) +
- uint64(in[2])*(uint64(in[5])<<1) +
- uint64(in[3])*(uint64(in[4])<<1)
- // tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60,
- // which is < 2**64 as required.
- tmp[8] = uint64(in[0])*(uint64(in[8])<<1) +
- uint64(in[1])*(uint64(in[7])<<2) +
- uint64(in[2])*(uint64(in[6])<<1) +
- uint64(in[3])*(uint64(in[5])<<2) +
- uint64(in[4])*uint64(in[4])
- tmp[9] = uint64(in[1])*(uint64(in[8])<<1) +
- uint64(in[2])*(uint64(in[7])<<1) +
- uint64(in[3])*(uint64(in[6])<<1) +
- uint64(in[4])*(uint64(in[5])<<1)
- tmp[10] = uint64(in[2])*(uint64(in[8])<<1) +
- uint64(in[3])*(uint64(in[7])<<2) +
- uint64(in[4])*(uint64(in[6])<<1) +
- uint64(in[5])*(uint64(in[5])<<1)
- tmp[11] = uint64(in[3])*(uint64(in[8])<<1) +
- uint64(in[4])*(uint64(in[7])<<1) +
- uint64(in[5])*(uint64(in[6])<<1)
- tmp[12] = uint64(in[4])*(uint64(in[8])<<1) +
- uint64(in[5])*(uint64(in[7])<<2) +
- uint64(in[6])*uint64(in[6])
- tmp[13] = uint64(in[5])*(uint64(in[8])<<1) +
- uint64(in[6])*(uint64(in[7])<<1)
- tmp[14] = uint64(in[6])*(uint64(in[8])<<1) +
- uint64(in[7])*(uint64(in[7])<<1)
- tmp[15] = uint64(in[7]) * (uint64(in[8]) << 1)
- tmp[16] = uint64(in[8]) * uint64(in[8])
-
- p256ReduceDegree(out, tmp)
-}
-
-// p256Mul sets out=in*in2.
-//
-// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
-// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Mul(out, in, in2 *[p256Limbs]uint32) {
- var tmp [17]uint64
-
- tmp[0] = uint64(in[0]) * uint64(in2[0])
- tmp[1] = uint64(in[0])*(uint64(in2[1])<<0) +
- uint64(in[1])*(uint64(in2[0])<<0)
- tmp[2] = uint64(in[0])*(uint64(in2[2])<<0) +
- uint64(in[1])*(uint64(in2[1])<<1) +
- uint64(in[2])*(uint64(in2[0])<<0)
- tmp[3] = uint64(in[0])*(uint64(in2[3])<<0) +
- uint64(in[1])*(uint64(in2[2])<<0) +
- uint64(in[2])*(uint64(in2[1])<<0) +
- uint64(in[3])*(uint64(in2[0])<<0)
- tmp[4] = uint64(in[0])*(uint64(in2[4])<<0) +
- uint64(in[1])*(uint64(in2[3])<<1) +
- uint64(in[2])*(uint64(in2[2])<<0) +
- uint64(in[3])*(uint64(in2[1])<<1) +
- uint64(in[4])*(uint64(in2[0])<<0)
- tmp[5] = uint64(in[0])*(uint64(in2[5])<<0) +
- uint64(in[1])*(uint64(in2[4])<<0) +
- uint64(in[2])*(uint64(in2[3])<<0) +
- uint64(in[3])*(uint64(in2[2])<<0) +
- uint64(in[4])*(uint64(in2[1])<<0) +
- uint64(in[5])*(uint64(in2[0])<<0)
- tmp[6] = uint64(in[0])*(uint64(in2[6])<<0) +
- uint64(in[1])*(uint64(in2[5])<<1) +
- uint64(in[2])*(uint64(in2[4])<<0) +
- uint64(in[3])*(uint64(in2[3])<<1) +
- uint64(in[4])*(uint64(in2[2])<<0) +
- uint64(in[5])*(uint64(in2[1])<<1) +
- uint64(in[6])*(uint64(in2[0])<<0)
- tmp[7] = uint64(in[0])*(uint64(in2[7])<<0) +
- uint64(in[1])*(uint64(in2[6])<<0) +
- uint64(in[2])*(uint64(in2[5])<<0) +
- uint64(in[3])*(uint64(in2[4])<<0) +
- uint64(in[4])*(uint64(in2[3])<<0) +
- uint64(in[5])*(uint64(in2[2])<<0) +
- uint64(in[6])*(uint64(in2[1])<<0) +
- uint64(in[7])*(uint64(in2[0])<<0)
- // tmp[8] has the greatest value but doesn't overflow. See logic in
- // p256Square.
- tmp[8] = uint64(in[0])*(uint64(in2[8])<<0) +
- uint64(in[1])*(uint64(in2[7])<<1) +
- uint64(in[2])*(uint64(in2[6])<<0) +
- uint64(in[3])*(uint64(in2[5])<<1) +
- uint64(in[4])*(uint64(in2[4])<<0) +
- uint64(in[5])*(uint64(in2[3])<<1) +
- uint64(in[6])*(uint64(in2[2])<<0) +
- uint64(in[7])*(uint64(in2[1])<<1) +
- uint64(in[8])*(uint64(in2[0])<<0)
- tmp[9] = uint64(in[1])*(uint64(in2[8])<<0) +
- uint64(in[2])*(uint64(in2[7])<<0) +
- uint64(in[3])*(uint64(in2[6])<<0) +
- uint64(in[4])*(uint64(in2[5])<<0) +
- uint64(in[5])*(uint64(in2[4])<<0) +
- uint64(in[6])*(uint64(in2[3])<<0) +
- uint64(in[7])*(uint64(in2[2])<<0) +
- uint64(in[8])*(uint64(in2[1])<<0)
- tmp[10] = uint64(in[2])*(uint64(in2[8])<<0) +
- uint64(in[3])*(uint64(in2[7])<<1) +
- uint64(in[4])*(uint64(in2[6])<<0) +
- uint64(in[5])*(uint64(in2[5])<<1) +
- uint64(in[6])*(uint64(in2[4])<<0) +
- uint64(in[7])*(uint64(in2[3])<<1) +
- uint64(in[8])*(uint64(in2[2])<<0)
- tmp[11] = uint64(in[3])*(uint64(in2[8])<<0) +
- uint64(in[4])*(uint64(in2[7])<<0) +
- uint64(in[5])*(uint64(in2[6])<<0) +
- uint64(in[6])*(uint64(in2[5])<<0) +
- uint64(in[7])*(uint64(in2[4])<<0) +
- uint64(in[8])*(uint64(in2[3])<<0)
- tmp[12] = uint64(in[4])*(uint64(in2[8])<<0) +
- uint64(in[5])*(uint64(in2[7])<<1) +
- uint64(in[6])*(uint64(in2[6])<<0) +
- uint64(in[7])*(uint64(in2[5])<<1) +
- uint64(in[8])*(uint64(in2[4])<<0)
- tmp[13] = uint64(in[5])*(uint64(in2[8])<<0) +
- uint64(in[6])*(uint64(in2[7])<<0) +
- uint64(in[7])*(uint64(in2[6])<<0) +
- uint64(in[8])*(uint64(in2[5])<<0)
- tmp[14] = uint64(in[6])*(uint64(in2[8])<<0) +
- uint64(in[7])*(uint64(in2[7])<<1) +
- uint64(in[8])*(uint64(in2[6])<<0)
- tmp[15] = uint64(in[7])*(uint64(in2[8])<<0) +
- uint64(in[8])*(uint64(in2[7])<<0)
- tmp[16] = uint64(in[8]) * (uint64(in2[8]) << 0)
-
- p256ReduceDegree(out, tmp)
-}
-
-func p256Assign(out, in *[p256Limbs]uint32) {
- *out = *in
-}
-
-// p256Invert calculates |out| = |in|^{-1}
-//
-// Based on Fermat's Little Theorem:
-//
-// a^p = a (mod p)
-// a^{p-1} = 1 (mod p)
-// a^{p-2} = a^{-1} (mod p)
-func p256Invert(out, in *[p256Limbs]uint32) {
- var ftmp, ftmp2 [p256Limbs]uint32
-
- // each e_I will hold |in|^{2^I - 1}
- var e2, e4, e8, e16, e32, e64 [p256Limbs]uint32
-
- p256Square(&ftmp, in) // 2^1
- p256Mul(&ftmp, in, &ftmp) // 2^2 - 2^0
- p256Assign(&e2, &ftmp)
- p256Square(&ftmp, &ftmp) // 2^3 - 2^1
- p256Square(&ftmp, &ftmp) // 2^4 - 2^2
- p256Mul(&ftmp, &ftmp, &e2) // 2^4 - 2^0
- p256Assign(&e4, &ftmp)
- p256Square(&ftmp, &ftmp) // 2^5 - 2^1
- p256Square(&ftmp, &ftmp) // 2^6 - 2^2
- p256Square(&ftmp, &ftmp) // 2^7 - 2^3
- p256Square(&ftmp, &ftmp) // 2^8 - 2^4
- p256Mul(&ftmp, &ftmp, &e4) // 2^8 - 2^0
- p256Assign(&e8, &ftmp)
- for i := 0; i < 8; i++ {
- p256Square(&ftmp, &ftmp)
- } // 2^16 - 2^8
- p256Mul(&ftmp, &ftmp, &e8) // 2^16 - 2^0
- p256Assign(&e16, &ftmp)
- for i := 0; i < 16; i++ {
- p256Square(&ftmp, &ftmp)
- } // 2^32 - 2^16
- p256Mul(&ftmp, &ftmp, &e16) // 2^32 - 2^0
- p256Assign(&e32, &ftmp)
- for i := 0; i < 32; i++ {
- p256Square(&ftmp, &ftmp)
- } // 2^64 - 2^32
- p256Assign(&e64, &ftmp)
- p256Mul(&ftmp, &ftmp, in) // 2^64 - 2^32 + 2^0
- for i := 0; i < 192; i++ {
- p256Square(&ftmp, &ftmp)
- } // 2^256 - 2^224 + 2^192
-
- p256Mul(&ftmp2, &e64, &e32) // 2^64 - 2^0
- for i := 0; i < 16; i++ {
- p256Square(&ftmp2, &ftmp2)
- } // 2^80 - 2^16
- p256Mul(&ftmp2, &ftmp2, &e16) // 2^80 - 2^0
- for i := 0; i < 8; i++ {
- p256Square(&ftmp2, &ftmp2)
- } // 2^88 - 2^8
- p256Mul(&ftmp2, &ftmp2, &e8) // 2^88 - 2^0
- for i := 0; i < 4; i++ {
- p256Square(&ftmp2, &ftmp2)
- } // 2^92 - 2^4
- p256Mul(&ftmp2, &ftmp2, &e4) // 2^92 - 2^0
- p256Square(&ftmp2, &ftmp2) // 2^93 - 2^1
- p256Square(&ftmp2, &ftmp2) // 2^94 - 2^2
- p256Mul(&ftmp2, &ftmp2, &e2) // 2^94 - 2^0
- p256Square(&ftmp2, &ftmp2) // 2^95 - 2^1
- p256Square(&ftmp2, &ftmp2) // 2^96 - 2^2
- p256Mul(&ftmp2, &ftmp2, in) // 2^96 - 3
-
- p256Mul(out, &ftmp2, &ftmp) // 2^256 - 2^224 + 2^192 + 2^96 - 3
-}
-
-// p256Scalar3 sets out=3*out.
-//
-// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Scalar3(out *[p256Limbs]uint32) {
- var carry uint32
-
- for i := 0; ; i++ {
- out[i] *= 3
- out[i] += carry
- carry = out[i] >> 29
- out[i] &= bottom29Bits
-
- i++
- if i == p256Limbs {
- break
- }
-
- out[i] *= 3
- out[i] += carry
- carry = out[i] >> 28
- out[i] &= bottom28Bits
- }
-
- p256ReduceCarry(out, carry)
-}
-
-// p256Scalar4 sets out=4*out.
-//
-// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Scalar4(out *[p256Limbs]uint32) {
- var carry, nextCarry uint32
-
- for i := 0; ; i++ {
- nextCarry = out[i] >> 27
- out[i] <<= 2
- out[i] &= bottom29Bits
- out[i] += carry
- carry = nextCarry + (out[i] >> 29)
- out[i] &= bottom29Bits
-
- i++
- if i == p256Limbs {
- break
- }
- nextCarry = out[i] >> 26
- out[i] <<= 2
- out[i] &= bottom28Bits
- out[i] += carry
- carry = nextCarry + (out[i] >> 28)
- out[i] &= bottom28Bits
- }
-
- p256ReduceCarry(out, carry)
-}
-
-// p256Scalar8 sets out=8*out.
-//
-// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Scalar8(out *[p256Limbs]uint32) {
- var carry, nextCarry uint32
-
- for i := 0; ; i++ {
- nextCarry = out[i] >> 26
- out[i] <<= 3
- out[i] &= bottom29Bits
- out[i] += carry
- carry = nextCarry + (out[i] >> 29)
- out[i] &= bottom29Bits
-
- i++
- if i == p256Limbs {
- break
- }
- nextCarry = out[i] >> 25
- out[i] <<= 3
- out[i] &= bottom28Bits
- out[i] += carry
- carry = nextCarry + (out[i] >> 28)
- out[i] &= bottom28Bits
- }
-
- p256ReduceCarry(out, carry)
-}
-
// Group operations:
//
// Elements of the elliptic curve group are represented in Jacobian
p256Diff(yOut, yOut, &tmp)
}
-// p256CopyConditional sets out=in if mask = 0xffffffff in constant time.
-//
-// On entry: mask is either 0 or 0xffffffff.
-func p256CopyConditional(out, in *[p256Limbs]uint32, mask uint32) {
- for i := 0; i < p256Limbs; i++ {
- tmp := mask & (in[i] ^ out[i])
- out[i] ^= tmp
- }
-}
-
// p256SelectAffinePoint sets {out_x,out_y} to the index'th entry of table.
//
// On entry: index < 16, table[0] must be zero.
nIsInfinityMask &^= pIsNoninfiniteMask
}
}
-
-// p256FromBig sets out = R*in.
-func p256FromBig(out *[p256Limbs]uint32, in *big.Int) {
- tmp := new(big.Int).Lsh(in, 257)
- tmp.Mod(tmp, p256Params.P)
-
- for i := 0; i < p256Limbs; i++ {
- if bits := tmp.Bits(); len(bits) > 0 {
- out[i] = uint32(bits[0]) & bottom29Bits
- } else {
- out[i] = 0
- }
- tmp.Rsh(tmp, 29)
-
- i++
- if i == p256Limbs {
- break
- }
-
- if bits := tmp.Bits(); len(bits) > 0 {
- out[i] = uint32(bits[0]) & bottom28Bits
- } else {
- out[i] = 0
- }
- tmp.Rsh(tmp, 28)
- }
-}
-
-// p256ToBig returns a *big.Int containing the value of in.
-func p256ToBig(in *[p256Limbs]uint32) *big.Int {
- result, tmp := new(big.Int), new(big.Int)
-
- result.SetInt64(int64(in[p256Limbs-1]))
- for i := p256Limbs - 2; i >= 0; i-- {
- if (i & 1) == 0 {
- result.Lsh(result, 29)
- } else {
- result.Lsh(result, 28)
- }
- tmp.SetInt64(int64(in[i]))
- result.Add(result, tmp)
- }
-
- result.Mul(result, p256RInverse)
- result.Mod(result, p256Params.P)
- return result
-}
--- /dev/null
+// Copyright 2013 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 && !arm64
+
+package elliptic
+
+import "math/big"
+
+// Field elements are represented as nine, unsigned 32-bit words.
+//
+// The value of a field element is:
+// x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228)
+//
+// That is, each limb is alternately 29 or 28-bits wide in little-endian
+// order.
+//
+// This means that a field element hits 2**257, rather than 2**256 as we would
+// like. A 28, 29, ... pattern would cause us to hit 2**256, but that causes
+// problems when multiplying as terms end up one bit short of a limb which
+// would require much bit-shifting to correct.
+//
+// Finally, the values stored in a field element are in Montgomery form. So the
+// value |y| is stored as (y*R) mod p, where p is the P-256 prime and R is
+// 2**257.
+
+const (
+ p256Limbs = 9
+ bottom29Bits = 0x1fffffff
+)
+
+var (
+ // p256One is the number 1 as a field element.
+ p256One = [p256Limbs]uint32{2, 0, 0, 0xffff800, 0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff, 0}
+ p256Zero = [p256Limbs]uint32{0, 0, 0, 0, 0, 0, 0, 0, 0}
+ // p256P is the prime modulus as a field element.
+ p256P = [p256Limbs]uint32{0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, 0, 0, 0x200000, 0xf000000, 0xfffffff}
+ // p2562P is the twice prime modulus as a field element.
+ p2562P = [p256Limbs]uint32{0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, 0, 0, 0x400000, 0xe000000, 0x1fffffff}
+)
+
+// Field element operations:
+
+const bottom28Bits = 0xfffffff
+
+// nonZeroToAllOnes returns:
+//
+// 0xffffffff for 0 < x <= 2**31
+// 0 for x == 0 or x > 2**31.
+func nonZeroToAllOnes(x uint32) uint32 {
+ return ((x - 1) >> 31) - 1
+}
+
+// p256ReduceCarry adds a multiple of p in order to cancel |carry|,
+// which is a term at 2**257.
+//
+// On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28.
+// On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29.
+func p256ReduceCarry(inout *[p256Limbs]uint32, carry uint32) {
+ carry_mask := nonZeroToAllOnes(carry)
+
+ inout[0] += carry << 1
+ inout[3] += 0x10000000 & carry_mask
+ // carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the
+ // previous line therefore this doesn't underflow.
+ inout[3] -= carry << 11
+ inout[4] += (0x20000000 - 1) & carry_mask
+ inout[5] += (0x10000000 - 1) & carry_mask
+ inout[6] += (0x20000000 - 1) & carry_mask
+ inout[6] -= carry << 22
+ // This may underflow if carry is non-zero but, if so, we'll fix it in the
+ // next line.
+ inout[7] -= 1 & carry_mask
+ inout[7] += carry << 25
+}
+
+// p256Sum sets out = in+in2.
+//
+// On entry: in[i]+in2[i] must not overflow a 32-bit word.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Sum(out, in, in2 *[p256Limbs]uint32) {
+ carry := uint32(0)
+ for i := 0; ; i++ {
+ out[i] = in[i] + in2[i]
+ out[i] += carry
+ carry = out[i] >> 29
+ out[i] &= bottom29Bits
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+
+ out[i] = in[i] + in2[i]
+ out[i] += carry
+ carry = out[i] >> 28
+ out[i] &= bottom28Bits
+ }
+
+ p256ReduceCarry(out, carry)
+}
+
+const (
+ two30m2 = 1<<30 - 1<<2
+ two30p13m2 = 1<<30 + 1<<13 - 1<<2
+ two31m2 = 1<<31 - 1<<2
+ two31m3 = 1<<31 - 1<<3
+ two31p24m2 = 1<<31 + 1<<24 - 1<<2
+ two30m27m2 = 1<<30 - 1<<27 - 1<<2
+)
+
+// p256Zero31 is 0 mod p.
+var p256Zero31 = [p256Limbs]uint32{two31m3, two30m2, two31m2, two30p13m2, two31m2, two30m2, two31p24m2, two30m27m2, two31m2}
+
+// p256Diff sets out = in-in2.
+//
+// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
+// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Diff(out, in, in2 *[p256Limbs]uint32) {
+ var carry uint32
+
+ for i := 0; ; i++ {
+ out[i] = in[i] - in2[i]
+ out[i] += p256Zero31[i]
+ out[i] += carry
+ carry = out[i] >> 29
+ out[i] &= bottom29Bits
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+
+ out[i] = in[i] - in2[i]
+ out[i] += p256Zero31[i]
+ out[i] += carry
+ carry = out[i] >> 28
+ out[i] &= bottom28Bits
+ }
+
+ p256ReduceCarry(out, carry)
+}
+
+// p256ReduceDegree sets out = tmp/R mod p where tmp contains 64-bit words with
+// the same 29,28,... bit positions as a field element.
+//
+// The values in field elements are in Montgomery form: x*R mod p where R =
+// 2**257. Since we just multiplied two Montgomery values together, the result
+// is x*y*R*R mod p. We wish to divide by R in order for the result also to be
+// in Montgomery form.
+//
+// On entry: tmp[i] < 2**64.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256ReduceDegree(out *[p256Limbs]uint32, tmp [17]uint64) {
+ // The following table may be helpful when reading this code:
+ //
+ // Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10...
+ // Width (bits): 29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29
+ // Start bit: 0 | 29| 57| 86|114|143|171|200|228|257|285
+ // (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285
+ var tmp2 [18]uint32
+ var carry, x, xMask uint32
+
+ // tmp contains 64-bit words with the same 29,28,29-bit positions as a
+ // field element. So the top of an element of tmp might overlap with
+ // another element two positions down. The following loop eliminates
+ // this overlap.
+ tmp2[0] = uint32(tmp[0]) & bottom29Bits
+
+ tmp2[1] = uint32(tmp[0]) >> 29
+ tmp2[1] |= (uint32(tmp[0]>>32) << 3) & bottom28Bits
+ tmp2[1] += uint32(tmp[1]) & bottom28Bits
+ carry = tmp2[1] >> 28
+ tmp2[1] &= bottom28Bits
+
+ for i := 2; i < 17; i++ {
+ tmp2[i] = (uint32(tmp[i-2] >> 32)) >> 25
+ tmp2[i] += (uint32(tmp[i-1])) >> 28
+ tmp2[i] += (uint32(tmp[i-1]>>32) << 4) & bottom29Bits
+ tmp2[i] += uint32(tmp[i]) & bottom29Bits
+ tmp2[i] += carry
+ carry = tmp2[i] >> 29
+ tmp2[i] &= bottom29Bits
+
+ i++
+ if i == 17 {
+ break
+ }
+ tmp2[i] = uint32(tmp[i-2]>>32) >> 25
+ tmp2[i] += uint32(tmp[i-1]) >> 29
+ tmp2[i] += ((uint32(tmp[i-1] >> 32)) << 3) & bottom28Bits
+ tmp2[i] += uint32(tmp[i]) & bottom28Bits
+ tmp2[i] += carry
+ carry = tmp2[i] >> 28
+ tmp2[i] &= bottom28Bits
+ }
+
+ tmp2[17] = uint32(tmp[15]>>32) >> 25
+ tmp2[17] += uint32(tmp[16]) >> 29
+ tmp2[17] += uint32(tmp[16]>>32) << 3
+ tmp2[17] += carry
+
+ // Montgomery elimination of terms:
+ //
+ // Since R is 2**257, we can divide by R with a bitwise shift if we can
+ // ensure that the right-most 257 bits are all zero. We can make that true
+ // by adding multiplies of p without affecting the value.
+ //
+ // So we eliminate limbs from right to left. Since the bottom 29 bits of p
+ // are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0.
+ // We can do that for 8 further limbs and then right shift to eliminate the
+ // extra factor of R.
+ for i := 0; ; i += 2 {
+ tmp2[i+1] += tmp2[i] >> 29
+ x = tmp2[i] & bottom29Bits
+ xMask = nonZeroToAllOnes(x)
+ tmp2[i] = 0
+
+ // The bounds calculations for this loop are tricky. Each iteration of
+ // the loop eliminates two words by adding values to words to their
+ // right.
+ //
+ // The following table contains the amounts added to each word (as an
+ // offset from the value of i at the top of the loop). The amounts are
+ // accounted for from the first and second half of the loop separately
+ // and are written as, for example, 28 to mean a value <2**28.
+ //
+ // Word: 3 4 5 6 7 8 9 10
+ // Added in top half: 28 11 29 21 29 28
+ // 28 29
+ // 29
+ // Added in bottom half: 29 10 28 21 28 28
+ // 29
+ //
+ // The value that is currently offset 7 will be offset 5 for the next
+ // iteration and then offset 3 for the iteration after that. Therefore
+ // the total value added will be the values added at 7, 5 and 3.
+ //
+ // The following table accumulates these values. The sums at the bottom
+ // are written as, for example, 29+28, to mean a value < 2**29+2**28.
+ //
+ // Word: 3 4 5 6 7 8 9 10 11 12 13
+ // 28 11 10 29 21 29 28 28 28 28 28
+ // 29 28 11 28 29 28 29 28 29 28
+ // 29 28 21 21 29 21 29 21
+ // 10 29 28 21 28 21 28
+ // 28 29 28 29 28 29 28
+ // 11 10 29 10 29 10
+ // 29 28 11 28 11
+ // 29 29
+ // --------------------------------------------
+ // 30+ 31+ 30+ 31+ 30+
+ // 28+ 29+ 28+ 29+ 21+
+ // 21+ 28+ 21+ 28+ 10
+ // 10 21+ 10 21+
+ // 11 11
+ //
+ // So the greatest amount is added to tmp2[10] and tmp2[12]. If
+ // tmp2[10/12] has an initial value of <2**29, then the maximum value
+ // will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32,
+ // as required.
+ tmp2[i+3] += (x << 10) & bottom28Bits
+ tmp2[i+4] += (x >> 18)
+
+ tmp2[i+6] += (x << 21) & bottom29Bits
+ tmp2[i+7] += x >> 8
+
+ // At position 200, which is the starting bit position for word 7, we
+ // have a factor of 0xf000000 = 2**28 - 2**24.
+ tmp2[i+7] += 0x10000000 & xMask
+ tmp2[i+8] += (x - 1) & xMask
+ tmp2[i+7] -= (x << 24) & bottom28Bits
+ tmp2[i+8] -= x >> 4
+
+ tmp2[i+8] += 0x20000000 & xMask
+ tmp2[i+8] -= x
+ tmp2[i+8] += (x << 28) & bottom29Bits
+ tmp2[i+9] += ((x >> 1) - 1) & xMask
+
+ if i+1 == p256Limbs {
+ break
+ }
+ tmp2[i+2] += tmp2[i+1] >> 28
+ x = tmp2[i+1] & bottom28Bits
+ xMask = nonZeroToAllOnes(x)
+ tmp2[i+1] = 0
+
+ tmp2[i+4] += (x << 11) & bottom29Bits
+ tmp2[i+5] += (x >> 18)
+
+ tmp2[i+7] += (x << 21) & bottom28Bits
+ tmp2[i+8] += x >> 7
+
+ // At position 199, which is the starting bit of the 8th word when
+ // dealing with a context starting on an odd word, we have a factor of
+ // 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th
+ // word from i+1 is i+8.
+ tmp2[i+8] += 0x20000000 & xMask
+ tmp2[i+9] += (x - 1) & xMask
+ tmp2[i+8] -= (x << 25) & bottom29Bits
+ tmp2[i+9] -= x >> 4
+
+ tmp2[i+9] += 0x10000000 & xMask
+ tmp2[i+9] -= x
+ tmp2[i+10] += (x - 1) & xMask
+ }
+
+ // We merge the right shift with a carry chain. The words above 2**257 have
+ // widths of 28,29,... which we need to correct when copying them down.
+ carry = 0
+ for i := 0; i < 8; i++ {
+ // The maximum value of tmp2[i + 9] occurs on the first iteration and
+ // is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is
+ // therefore safe.
+ out[i] = tmp2[i+9]
+ out[i] += carry
+ out[i] += (tmp2[i+10] << 28) & bottom29Bits
+ carry = out[i] >> 29
+ out[i] &= bottom29Bits
+
+ i++
+ out[i] = tmp2[i+9] >> 1
+ out[i] += carry
+ carry = out[i] >> 28
+ out[i] &= bottom28Bits
+ }
+
+ out[8] = tmp2[17]
+ out[8] += carry
+ carry = out[8] >> 29
+ out[8] &= bottom29Bits
+
+ p256ReduceCarry(out, carry)
+}
+
+// p256Square sets out=in*in.
+//
+// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Square(out, in *[p256Limbs]uint32) {
+ var tmp [17]uint64
+
+ tmp[0] = uint64(in[0]) * uint64(in[0])
+ tmp[1] = uint64(in[0]) * (uint64(in[1]) << 1)
+ tmp[2] = uint64(in[0])*(uint64(in[2])<<1) +
+ uint64(in[1])*(uint64(in[1])<<1)
+ tmp[3] = uint64(in[0])*(uint64(in[3])<<1) +
+ uint64(in[1])*(uint64(in[2])<<1)
+ tmp[4] = uint64(in[0])*(uint64(in[4])<<1) +
+ uint64(in[1])*(uint64(in[3])<<2) +
+ uint64(in[2])*uint64(in[2])
+ tmp[5] = uint64(in[0])*(uint64(in[5])<<1) +
+ uint64(in[1])*(uint64(in[4])<<1) +
+ uint64(in[2])*(uint64(in[3])<<1)
+ tmp[6] = uint64(in[0])*(uint64(in[6])<<1) +
+ uint64(in[1])*(uint64(in[5])<<2) +
+ uint64(in[2])*(uint64(in[4])<<1) +
+ uint64(in[3])*(uint64(in[3])<<1)
+ tmp[7] = uint64(in[0])*(uint64(in[7])<<1) +
+ uint64(in[1])*(uint64(in[6])<<1) +
+ uint64(in[2])*(uint64(in[5])<<1) +
+ uint64(in[3])*(uint64(in[4])<<1)
+ // tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60,
+ // which is < 2**64 as required.
+ tmp[8] = uint64(in[0])*(uint64(in[8])<<1) +
+ uint64(in[1])*(uint64(in[7])<<2) +
+ uint64(in[2])*(uint64(in[6])<<1) +
+ uint64(in[3])*(uint64(in[5])<<2) +
+ uint64(in[4])*uint64(in[4])
+ tmp[9] = uint64(in[1])*(uint64(in[8])<<1) +
+ uint64(in[2])*(uint64(in[7])<<1) +
+ uint64(in[3])*(uint64(in[6])<<1) +
+ uint64(in[4])*(uint64(in[5])<<1)
+ tmp[10] = uint64(in[2])*(uint64(in[8])<<1) +
+ uint64(in[3])*(uint64(in[7])<<2) +
+ uint64(in[4])*(uint64(in[6])<<1) +
+ uint64(in[5])*(uint64(in[5])<<1)
+ tmp[11] = uint64(in[3])*(uint64(in[8])<<1) +
+ uint64(in[4])*(uint64(in[7])<<1) +
+ uint64(in[5])*(uint64(in[6])<<1)
+ tmp[12] = uint64(in[4])*(uint64(in[8])<<1) +
+ uint64(in[5])*(uint64(in[7])<<2) +
+ uint64(in[6])*uint64(in[6])
+ tmp[13] = uint64(in[5])*(uint64(in[8])<<1) +
+ uint64(in[6])*(uint64(in[7])<<1)
+ tmp[14] = uint64(in[6])*(uint64(in[8])<<1) +
+ uint64(in[7])*(uint64(in[7])<<1)
+ tmp[15] = uint64(in[7]) * (uint64(in[8]) << 1)
+ tmp[16] = uint64(in[8]) * uint64(in[8])
+
+ p256ReduceDegree(out, tmp)
+}
+
+// p256Mul sets out=in*in2.
+//
+// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
+// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Mul(out, in, in2 *[p256Limbs]uint32) {
+ var tmp [17]uint64
+
+ tmp[0] = uint64(in[0]) * uint64(in2[0])
+ tmp[1] = uint64(in[0])*(uint64(in2[1])<<0) +
+ uint64(in[1])*(uint64(in2[0])<<0)
+ tmp[2] = uint64(in[0])*(uint64(in2[2])<<0) +
+ uint64(in[1])*(uint64(in2[1])<<1) +
+ uint64(in[2])*(uint64(in2[0])<<0)
+ tmp[3] = uint64(in[0])*(uint64(in2[3])<<0) +
+ uint64(in[1])*(uint64(in2[2])<<0) +
+ uint64(in[2])*(uint64(in2[1])<<0) +
+ uint64(in[3])*(uint64(in2[0])<<0)
+ tmp[4] = uint64(in[0])*(uint64(in2[4])<<0) +
+ uint64(in[1])*(uint64(in2[3])<<1) +
+ uint64(in[2])*(uint64(in2[2])<<0) +
+ uint64(in[3])*(uint64(in2[1])<<1) +
+ uint64(in[4])*(uint64(in2[0])<<0)
+ tmp[5] = uint64(in[0])*(uint64(in2[5])<<0) +
+ uint64(in[1])*(uint64(in2[4])<<0) +
+ uint64(in[2])*(uint64(in2[3])<<0) +
+ uint64(in[3])*(uint64(in2[2])<<0) +
+ uint64(in[4])*(uint64(in2[1])<<0) +
+ uint64(in[5])*(uint64(in2[0])<<0)
+ tmp[6] = uint64(in[0])*(uint64(in2[6])<<0) +
+ uint64(in[1])*(uint64(in2[5])<<1) +
+ uint64(in[2])*(uint64(in2[4])<<0) +
+ uint64(in[3])*(uint64(in2[3])<<1) +
+ uint64(in[4])*(uint64(in2[2])<<0) +
+ uint64(in[5])*(uint64(in2[1])<<1) +
+ uint64(in[6])*(uint64(in2[0])<<0)
+ tmp[7] = uint64(in[0])*(uint64(in2[7])<<0) +
+ uint64(in[1])*(uint64(in2[6])<<0) +
+ uint64(in[2])*(uint64(in2[5])<<0) +
+ uint64(in[3])*(uint64(in2[4])<<0) +
+ uint64(in[4])*(uint64(in2[3])<<0) +
+ uint64(in[5])*(uint64(in2[2])<<0) +
+ uint64(in[6])*(uint64(in2[1])<<0) +
+ uint64(in[7])*(uint64(in2[0])<<0)
+ // tmp[8] has the greatest value but doesn't overflow. See logic in
+ // p256Square.
+ tmp[8] = uint64(in[0])*(uint64(in2[8])<<0) +
+ uint64(in[1])*(uint64(in2[7])<<1) +
+ uint64(in[2])*(uint64(in2[6])<<0) +
+ uint64(in[3])*(uint64(in2[5])<<1) +
+ uint64(in[4])*(uint64(in2[4])<<0) +
+ uint64(in[5])*(uint64(in2[3])<<1) +
+ uint64(in[6])*(uint64(in2[2])<<0) +
+ uint64(in[7])*(uint64(in2[1])<<1) +
+ uint64(in[8])*(uint64(in2[0])<<0)
+ tmp[9] = uint64(in[1])*(uint64(in2[8])<<0) +
+ uint64(in[2])*(uint64(in2[7])<<0) +
+ uint64(in[3])*(uint64(in2[6])<<0) +
+ uint64(in[4])*(uint64(in2[5])<<0) +
+ uint64(in[5])*(uint64(in2[4])<<0) +
+ uint64(in[6])*(uint64(in2[3])<<0) +
+ uint64(in[7])*(uint64(in2[2])<<0) +
+ uint64(in[8])*(uint64(in2[1])<<0)
+ tmp[10] = uint64(in[2])*(uint64(in2[8])<<0) +
+ uint64(in[3])*(uint64(in2[7])<<1) +
+ uint64(in[4])*(uint64(in2[6])<<0) +
+ uint64(in[5])*(uint64(in2[5])<<1) +
+ uint64(in[6])*(uint64(in2[4])<<0) +
+ uint64(in[7])*(uint64(in2[3])<<1) +
+ uint64(in[8])*(uint64(in2[2])<<0)
+ tmp[11] = uint64(in[3])*(uint64(in2[8])<<0) +
+ uint64(in[4])*(uint64(in2[7])<<0) +
+ uint64(in[5])*(uint64(in2[6])<<0) +
+ uint64(in[6])*(uint64(in2[5])<<0) +
+ uint64(in[7])*(uint64(in2[4])<<0) +
+ uint64(in[8])*(uint64(in2[3])<<0)
+ tmp[12] = uint64(in[4])*(uint64(in2[8])<<0) +
+ uint64(in[5])*(uint64(in2[7])<<1) +
+ uint64(in[6])*(uint64(in2[6])<<0) +
+ uint64(in[7])*(uint64(in2[5])<<1) +
+ uint64(in[8])*(uint64(in2[4])<<0)
+ tmp[13] = uint64(in[5])*(uint64(in2[8])<<0) +
+ uint64(in[6])*(uint64(in2[7])<<0) +
+ uint64(in[7])*(uint64(in2[6])<<0) +
+ uint64(in[8])*(uint64(in2[5])<<0)
+ tmp[14] = uint64(in[6])*(uint64(in2[8])<<0) +
+ uint64(in[7])*(uint64(in2[7])<<1) +
+ uint64(in[8])*(uint64(in2[6])<<0)
+ tmp[15] = uint64(in[7])*(uint64(in2[8])<<0) +
+ uint64(in[8])*(uint64(in2[7])<<0)
+ tmp[16] = uint64(in[8]) * (uint64(in2[8]) << 0)
+
+ p256ReduceDegree(out, tmp)
+}
+
+func p256Assign(out, in *[p256Limbs]uint32) {
+ *out = *in
+}
+
+// p256Invert calculates |out| = |in|^{-1}
+//
+// Based on Fermat's Little Theorem:
+//
+// a^p = a (mod p)
+// a^{p-1} = 1 (mod p)
+// a^{p-2} = a^{-1} (mod p)
+func p256Invert(out, in *[p256Limbs]uint32) {
+ var ftmp, ftmp2 [p256Limbs]uint32
+
+ // each e_I will hold |in|^{2^I - 1}
+ var e2, e4, e8, e16, e32, e64 [p256Limbs]uint32
+
+ p256Square(&ftmp, in) // 2^1
+ p256Mul(&ftmp, in, &ftmp) // 2^2 - 2^0
+ p256Assign(&e2, &ftmp)
+ p256Square(&ftmp, &ftmp) // 2^3 - 2^1
+ p256Square(&ftmp, &ftmp) // 2^4 - 2^2
+ p256Mul(&ftmp, &ftmp, &e2) // 2^4 - 2^0
+ p256Assign(&e4, &ftmp)
+ p256Square(&ftmp, &ftmp) // 2^5 - 2^1
+ p256Square(&ftmp, &ftmp) // 2^6 - 2^2
+ p256Square(&ftmp, &ftmp) // 2^7 - 2^3
+ p256Square(&ftmp, &ftmp) // 2^8 - 2^4
+ p256Mul(&ftmp, &ftmp, &e4) // 2^8 - 2^0
+ p256Assign(&e8, &ftmp)
+ for i := 0; i < 8; i++ {
+ p256Square(&ftmp, &ftmp)
+ } // 2^16 - 2^8
+ p256Mul(&ftmp, &ftmp, &e8) // 2^16 - 2^0
+ p256Assign(&e16, &ftmp)
+ for i := 0; i < 16; i++ {
+ p256Square(&ftmp, &ftmp)
+ } // 2^32 - 2^16
+ p256Mul(&ftmp, &ftmp, &e16) // 2^32 - 2^0
+ p256Assign(&e32, &ftmp)
+ for i := 0; i < 32; i++ {
+ p256Square(&ftmp, &ftmp)
+ } // 2^64 - 2^32
+ p256Assign(&e64, &ftmp)
+ p256Mul(&ftmp, &ftmp, in) // 2^64 - 2^32 + 2^0
+ for i := 0; i < 192; i++ {
+ p256Square(&ftmp, &ftmp)
+ } // 2^256 - 2^224 + 2^192
+
+ p256Mul(&ftmp2, &e64, &e32) // 2^64 - 2^0
+ for i := 0; i < 16; i++ {
+ p256Square(&ftmp2, &ftmp2)
+ } // 2^80 - 2^16
+ p256Mul(&ftmp2, &ftmp2, &e16) // 2^80 - 2^0
+ for i := 0; i < 8; i++ {
+ p256Square(&ftmp2, &ftmp2)
+ } // 2^88 - 2^8
+ p256Mul(&ftmp2, &ftmp2, &e8) // 2^88 - 2^0
+ for i := 0; i < 4; i++ {
+ p256Square(&ftmp2, &ftmp2)
+ } // 2^92 - 2^4
+ p256Mul(&ftmp2, &ftmp2, &e4) // 2^92 - 2^0
+ p256Square(&ftmp2, &ftmp2) // 2^93 - 2^1
+ p256Square(&ftmp2, &ftmp2) // 2^94 - 2^2
+ p256Mul(&ftmp2, &ftmp2, &e2) // 2^94 - 2^0
+ p256Square(&ftmp2, &ftmp2) // 2^95 - 2^1
+ p256Square(&ftmp2, &ftmp2) // 2^96 - 2^2
+ p256Mul(&ftmp2, &ftmp2, in) // 2^96 - 3
+
+ p256Mul(out, &ftmp2, &ftmp) // 2^256 - 2^224 + 2^192 + 2^96 - 3
+}
+
+// p256Scalar3 sets out=3*out.
+//
+// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Scalar3(out *[p256Limbs]uint32) {
+ var carry uint32
+
+ for i := 0; ; i++ {
+ out[i] *= 3
+ out[i] += carry
+ carry = out[i] >> 29
+ out[i] &= bottom29Bits
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+
+ out[i] *= 3
+ out[i] += carry
+ carry = out[i] >> 28
+ out[i] &= bottom28Bits
+ }
+
+ p256ReduceCarry(out, carry)
+}
+
+// p256Scalar4 sets out=4*out.
+//
+// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Scalar4(out *[p256Limbs]uint32) {
+ var carry, nextCarry uint32
+
+ for i := 0; ; i++ {
+ nextCarry = out[i] >> 27
+ out[i] <<= 2
+ out[i] &= bottom29Bits
+ out[i] += carry
+ carry = nextCarry + (out[i] >> 29)
+ out[i] &= bottom29Bits
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+ nextCarry = out[i] >> 26
+ out[i] <<= 2
+ out[i] &= bottom28Bits
+ out[i] += carry
+ carry = nextCarry + (out[i] >> 28)
+ out[i] &= bottom28Bits
+ }
+
+ p256ReduceCarry(out, carry)
+}
+
+// p256Scalar8 sets out=8*out.
+//
+// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Scalar8(out *[p256Limbs]uint32) {
+ var carry, nextCarry uint32
+
+ for i := 0; ; i++ {
+ nextCarry = out[i] >> 26
+ out[i] <<= 3
+ out[i] &= bottom29Bits
+ out[i] += carry
+ carry = nextCarry + (out[i] >> 29)
+ out[i] &= bottom29Bits
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+ nextCarry = out[i] >> 25
+ out[i] <<= 3
+ out[i] &= bottom28Bits
+ out[i] += carry
+ carry = nextCarry + (out[i] >> 28)
+ out[i] &= bottom28Bits
+ }
+
+ p256ReduceCarry(out, carry)
+}
+
+// p256CopyConditional sets out=in if mask = 0xffffffff in constant time.
+//
+// On entry: mask is either 0 or 0xffffffff.
+func p256CopyConditional(out, in *[p256Limbs]uint32, mask uint32) {
+ for i := 0; i < p256Limbs; i++ {
+ tmp := mask & (in[i] ^ out[i])
+ out[i] ^= tmp
+ }
+}
+
+// p256FromBig sets out = R*in.
+func p256FromBig(out *[p256Limbs]uint32, in *big.Int) {
+ tmp := new(big.Int).Lsh(in, 257)
+ tmp.Mod(tmp, p256Params.P)
+
+ for i := 0; i < p256Limbs; i++ {
+ if bits := tmp.Bits(); len(bits) > 0 {
+ out[i] = uint32(bits[0]) & bottom29Bits
+ } else {
+ out[i] = 0
+ }
+ tmp.Rsh(tmp, 29)
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+
+ if bits := tmp.Bits(); len(bits) > 0 {
+ out[i] = uint32(bits[0]) & bottom28Bits
+ } else {
+ out[i] = 0
+ }
+ tmp.Rsh(tmp, 28)
+ }
+}
+
+// p256ToBig returns a *big.Int containing the value of in.
+func p256ToBig(in *[p256Limbs]uint32) *big.Int {
+ result, tmp := new(big.Int), new(big.Int)
+
+ result.SetInt64(int64(in[p256Limbs-1]))
+ for i := p256Limbs - 2; i >= 0; i-- {
+ if (i & 1) == 0 {
+ result.Lsh(result, 29)
+ } else {
+ result.Lsh(result, 28)
+ }
+ tmp.SetInt64(int64(in[i]))
+ result.Add(result, tmp)
+ }
+
+ result.Mul(result, p256RInverse)
+ result.Mod(result, p256Params.P)
+ return result
+}