From 96d1e4ab5938d263457a9c18fdf6fdf0581c6ec6 Mon Sep 17 00:00:00 2001
From: Keith Randall <khr@golang.org>
Date: Tue, 14 Oct 2014 14:09:56 -0700
Subject: [PATCH] math/big: Allow non-prime modulus for ModInverse

The inverse is defined whenever the element and the
modulus are relatively prime.  The code already handles
this situation, but the spec does not.

Test that it does indeed work.

Fixes #8875

LGTM=agl
R=agl
CC=golang-codereviews
https://golang.org/cl/155010043
---
 src/math/big/int.go      | 15 ++++++++-------
 src/math/big/int_test.go | 36 ++++++++++++++++++++++++++----------
 2 files changed, 34 insertions(+), 17 deletions(-)

diff --git a/src/math/big/int.go b/src/math/big/int.go
index fc53719d71..d22e39e7c9 100644
--- a/src/math/big/int.go
+++ b/src/math/big/int.go
@@ -752,15 +752,16 @@ func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
 	return z
 }
 
-// ModInverse sets z to the multiplicative inverse of g in the group ℤ/pℤ (where
-// p is a prime) and returns z.
-func (z *Int) ModInverse(g, p *Int) *Int {
+// ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
+// and returns z. If g and n are not relatively prime, the result is undefined.
+func (z *Int) ModInverse(g, n *Int) *Int {
 	var d Int
-	d.GCD(z, nil, g, p)
-	// x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking
-	// that modulo p results in g*x = 1, therefore x is the inverse element.
+	d.GCD(z, nil, g, n)
+	// x and y are such that g*x + n*y = d. Since g and n are
+	// relatively prime, d = 1. Taking that modulo n results in
+	// g*x = 1, therefore x is the inverse element.
 	if z.neg {
-		z.Add(z, p)
+		z.Add(z, n)
 	}
 	return z
 }
diff --git a/src/math/big/int_test.go b/src/math/big/int_test.go
index ec05fbb1c0..6070cf325d 100644
--- a/src/math/big/int_test.go
+++ b/src/math/big/int_test.go
@@ -1448,24 +1448,40 @@ func TestNot(t *testing.T) {
 
 var modInverseTests = []struct {
 	element string
-	prime   string
+	modulus string
 }{
-	{"1", "7"},
-	{"1", "13"},
+	{"1234567", "458948883992"},
 	{"239487239847", "2410312426921032588552076022197566074856950548502459942654116941958108831682612228890093858261341614673227141477904012196503648957050582631942730706805009223062734745341073406696246014589361659774041027169249453200378729434170325843778659198143763193776859869524088940195577346119843545301547043747207749969763750084308926339295559968882457872412993810129130294592999947926365264059284647209730384947211681434464714438488520940127459844288859336526896320919633919"},
 }
 
 func TestModInverse(t *testing.T) {
-	var element, prime Int
+	var element, modulus, gcd, inverse Int
 	one := NewInt(1)
 	for i, test := range modInverseTests {
 		(&element).SetString(test.element, 10)
-		(&prime).SetString(test.prime, 10)
-		inverse := new(Int).ModInverse(&element, &prime)
-		inverse.Mul(inverse, &element)
-		inverse.Mod(inverse, &prime)
-		if inverse.Cmp(one) != 0 {
-			t.Errorf("#%d: failed (e·e^(-1)=%s)", i, inverse)
+		(&modulus).SetString(test.modulus, 10)
+		(&inverse).ModInverse(&element, &modulus)
+		(&inverse).Mul(&inverse, &element)
+		(&inverse).Mod(&inverse, &modulus)
+		if (&inverse).Cmp(one) != 0 {
+			t.Errorf("#%d: failed (e·e^(-1)=%s)", i, &inverse)
+		}
+	}
+	// exhaustive test for small values
+	for n := 2; n < 100; n++ {
+		(&modulus).SetInt64(int64(n))
+		for x := 1; x < n; x++ {
+			(&element).SetInt64(int64(x))
+			(&gcd).GCD(nil, nil, &element, &modulus)
+			if (&gcd).Cmp(one) != 0 {
+				continue
+			}
+			(&inverse).ModInverse(&element, &modulus)
+			(&inverse).Mul(&inverse, &element)
+			(&inverse).Mod(&inverse, &modulus)
+			if (&inverse).Cmp(one) != 0 {
+				t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
+			}
 		}
 	}
 }
-- 
2.50.0