From a454c5f5eccbb78178644d5413486c32d0904191 Mon Sep 17 00:00:00 2001 From: Sun Yimin Date: Fri, 28 Apr 2023 11:25:09 +0800 Subject: [PATCH] sm9: reduce gfp2 mul --- sm9/bn256/gfp12.go | 7 ++ sm9/bn256/gfp12_test.go | 251 ++++++++++------------------------------ sm9/bn256/gfp2.go | 71 ++++++------ sm9/bn256/gfp2_test.go | 46 ++++++++ sm9/bn256/gfp4.go | 4 +- 5 files changed, 155 insertions(+), 224 deletions(-) diff --git a/sm9/bn256/gfp12.go b/sm9/bn256/gfp12.go index 6b00595..ddd9cba 100644 --- a/sm9/bn256/gfp12.go +++ b/sm9/bn256/gfp12.go @@ -251,6 +251,13 @@ func (e *gfP12) Invert(a *gfP12) *gfP12 { return e } +func (e *gfP12) Neg(a *gfP12) *gfP12 { + e.x.Neg(&a.x) + e.y.Neg(&a.y) + e.z.Neg(&a.z) + return e +} + // (z + y*w + x*w^2)^p //= z^p + y^p*w*w^(p-1)+x^p*w^2*(w^2)^(p-1) // w2ToP2Minus1 = vToPMinus1 * wToPMinus1 diff --git a/sm9/bn256/gfp12_test.go b/sm9/bn256/gfp12_test.go index 689475f..ef4efe7 100644 --- a/sm9/bn256/gfp12_test.go +++ b/sm9/bn256/gfp12_test.go @@ -5,28 +5,21 @@ import ( "testing" ) +var testdataP4 = gfP4{ + gfP2{ + *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), + *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), + }, + gfP2{ + *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), + *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), + }, +} + func Test_gfP12Square(t *testing.T) { x := &gfP12{ - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, + testdataP4, + testdataP4, *(&gfP4{}).SetOne(), } xmulx := &gfP12{} @@ -42,30 +35,7 @@ func Test_gfP12Square(t *testing.T) { } } -func Test_gfP12Invert(t *testing.T) { - x := &gfP12{ - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - *(&gfP4{}).SetOne(), - } +func testGfP12Invert(t *testing.T, x *gfP12) { xInv := &gfP12{} xInv.Invert(x) @@ -74,35 +44,27 @@ func Test_gfP12Invert(t *testing.T) { if !y.IsOne() { t.Fail() } +} + +func Test_gfP12Invert(t *testing.T) { + x := &gfP12{ + testdataP4, + testdataP4, + *(&gfP4{}).SetOne(), + } + testGfP12Invert(t, x) x = &gfP12{ - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, + testdataP4, + testdataP4, *(&gfP4{}).SetZero(), } - xInv.Invert(x) - - y.Mul(x, xInv) - if !y.IsOne() { - t.Fail() + testGfP12Invert(t, x) + x = &gfP12{ + testdataP4, + testdataP4, + testdataP4, } + testGfP12Invert(t, x) } // Generate wToPMinus1 @@ -227,36 +189,9 @@ func Test_gfP12FrobeniusP3_Case2(t *testing.T) { func Test_gfP12Frobenius(t *testing.T) { x := &gfP12{ - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, + testdataP4, + testdataP4, + testdataP4, } expected := &gfP12{} expected.Exp(x, p) @@ -269,36 +204,9 @@ func Test_gfP12Frobenius(t *testing.T) { func Test_gfP12FrobeniusP2(t *testing.T) { x := &gfP12{ - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, + testdataP4, + testdataP4, + testdataP4, } expected := &gfP12{} p2 := new(big.Int).Mul(p, p) @@ -312,36 +220,9 @@ func Test_gfP12FrobeniusP2(t *testing.T) { func Test_gfP12FrobeniusP3(t *testing.T) { x := &gfP12{ - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, + testdataP4, + testdataP4, + testdataP4, } expected := &gfP12{} p3 := new(big.Int).Mul(p, p) @@ -356,36 +237,9 @@ func Test_gfP12FrobeniusP3(t *testing.T) { func Test_gfP12FrobeniusP6(t *testing.T) { x := &gfP12{ - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, - gfP4{ - gfP2{ - *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), - *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), - }, - gfP2{ - *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), - *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), - }, - }, + testdataP4, + testdataP4, + testdataP4, } expected := &gfP12{} p6 := new(big.Int).Mul(p, p) @@ -412,3 +266,22 @@ func Test_W3(t *testing.T) { t.Errorf("not expected") } } + +func BenchmarkGfP12Frobenius(b *testing.B) { + x := &gfP12{ + testdataP4, + testdataP4, + testdataP4, + } + expected := &gfP12{} + expected.Exp(x, p) + got := &gfP12{} + b.ReportAllocs() + b.ResetTimer() + for i := 0; i < b.N; i++ { + got.Frobenius(x) + if *expected != *got { + b.Errorf("got %v, expected %v", got, expected) + } + } +} diff --git a/sm9/bn256/gfp2.go b/sm9/bn256/gfp2.go index b0359e2..872c0e7 100644 --- a/sm9/bn256/gfp2.go +++ b/sm9/bn256/gfp2.go @@ -9,9 +9,9 @@ import ( // http://eprint.iacr.org/2006/471.pdf. // gfP2 implements a field of size p² as a quadratic extension of the base field -// where i²=-2. +// where u²=-2, beta=-2. type gfP2 struct { - x, y gfP // value is xi+y. + x, y gfP // value is xu+y. } func gfP2Decode(in *gfP2) *gfP2 { @@ -105,45 +105,50 @@ func (e *gfP2) Triple(a *gfP2) *gfP2 { // See "Multiplication and Squaring in Pairing-Friendly Fields", // http://eprint.iacr.org/2006/471.pdf // The Karatsuba method -//(a0+a1*i)(b0+b1*i)=c0+c1*i, where +//(a0+a1*u)(b0+b1*u)=c0+c1*u, where //c0 = a0*b0 - 2a1*b1 //c1 = (a0 + a1)(b0 + b1) - a0*b0 - a1*b1 = a0*b1 + a1*b0 func (e *gfP2) Mul(a, b *gfP2) *gfP2 { - tx, t := &gfP{}, &gfP{} - gfpMul(tx, &a.x, &b.y) - gfpMul(t, &b.x, &a.y) - gfpAdd(tx, tx, t) + tx, ty, v0, v1 := &gfP{}, &gfP{}, &gfP{}, &gfP{} - ty := &gfP{} - gfpMul(ty, &a.y, &b.y) - gfpMul(t, &a.x, &b.x) - gfpMul(t, t, two) - gfpSub(ty, ty, t) + gfpMul(v0, &a.y, &b.y) + gfpMul(v1, &a.x, &b.x) + + gfpAdd(tx, &a.x, &a.y) + gfpAdd(ty, &b.x, &b.y) + gfpMul(tx, tx, ty) + gfpSub(tx, tx, v0) + gfpSub(tx, tx, v1) + + gfpSub(ty, v0, v1) + gfpSub(ty, ty, v1) e.x.Set(tx) e.y.Set(ty) return e } -// MulU: a * b * i -//(a0+a1*i)(b0+b1*i)*i=c0+c1*i, where -//c1 = (a0*b0 - 2a1*b1)i +// MulU: a * b * u +//(a0+a1*u)(b0+b1*u)*u=c0+c1*u, where +//c1 = (a0*b0 - 2a1*b1)u //c0 = -2 * ((a0 + a1)(b0 + b1) - a0*b0 - a1*b1) = -2 * (a0*b1 + a1*b0) func (e *gfP2) MulU(a, b *gfP2) *gfP2 { - // ty = -2 * (a0 * b1 + a1 * b0) - ty, t := &gfP{}, &gfP{} - gfpMul(ty, &a.x, &b.y) - gfpMul(t, &b.x, &a.y) - gfpAdd(ty, ty, t) + tx, ty, v0, v1 := &gfP{}, &gfP{}, &gfP{}, &gfP{} + + gfpMul(v0, &a.y, &b.y) + gfpMul(v1, &a.x, &b.x) + + gfpAdd(tx, &a.x, &a.y) + gfpAdd(ty, &b.x, &b.y) + + gfpMul(ty, tx, ty) + gfpSub(ty, ty, v0) + gfpSub(ty, ty, v1) gfpAdd(ty, ty, ty) gfpNeg(ty, ty) - - // tx = a0 * b0 - 2 * a1 * b1 - tx := &gfP{} - gfpMul(tx, &a.y, &b.y) - gfpMul(t, &a.x, &b.x) - gfpMul(t, t, two) - gfpSub(tx, tx, t) + + gfpSub(tx, v0, v1) + gfpSub(tx, tx, v1) e.x.Set(tx) e.y.Set(ty) @@ -152,7 +157,7 @@ func (e *gfP2) MulU(a, b *gfP2) *gfP2 { func (e *gfP2) Square(a *gfP2) *gfP2 { // Complex squaring algorithm: - // (xi+y)² = y^2-2*x^2 + 2*i*x*y + // (xu+y)² = y^2-2*x^2 + 2*u*x*y tx, ty := &gfP{}, &gfP{} gfpMul(tx, &a.x, &a.x) gfpMul(ty, &a.y, &a.y) @@ -169,7 +174,7 @@ func (e *gfP2) Square(a *gfP2) *gfP2 { func (e *gfP2) SquareU(a *gfP2) *gfP2 { // Complex squaring algorithm: - // (xi+y)²*i = (y^2-2*x^2)i - 4*x*y + // (xu+y)²*u = (y^2-2*x^2)u - 4*x*y tx, ty := &gfP{}, &gfP{} // tx = a0^2 - 2 * a1^2 @@ -231,10 +236,10 @@ func (e *gfP2) Exp(f *gfP2, power *big.Int) *gfP2 { return e } -// （xi+y)^p = x * i^p + y -// = x * i * i^(p-1) + y -// = (-x)*i + y -// here i^(p-1) = -1 +// （xu+y)^p = x * u^p + y +// = x * u * u^(p-1) + y +// = (-x)*u + y +// here u^(p-1) = -1 func (e *gfP2) Frobenius(a *gfP2) *gfP2 { e.Conjugate(a) return e diff --git a/sm9/bn256/gfp2_test.go b/sm9/bn256/gfp2_test.go index 8639343..b6fd6d7 100644 --- a/sm9/bn256/gfp2_test.go +++ b/sm9/bn256/gfp2_test.go @@ -126,6 +126,52 @@ func Test_gfP2Sqrt(t *testing.T) { } } +func BenchmarkGfP2Mul(b *testing.B) { + x := &gfP2{ + *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), + *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), + } + y := &gfP2{ + *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), + *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), + } + b.ReportAllocs() + b.ResetTimer() + for i := 0; i < b.N; i++ { + t := &gfP2{} + t.Mul(x, y) + } +} + +func BenchmarkGfP2MulU(b *testing.B) { + x := &gfP2{ + *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), + *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), + } + y := &gfP2{ + *fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")), + *fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")), + } + b.ReportAllocs() + b.ResetTimer() + for i := 0; i < b.N; i++ { + t := &gfP2{} + t.MulU(x, y) + } +} + +func BenchmarkGfP2Square(b *testing.B) { + x := &gfP2{ + *fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")), + *fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")), + } + b.ReportAllocs() + b.ResetTimer() + for i := 0; i < b.N; i++ { + x.Square(x) + } +} + /* func Test_gfP2QuadraticResidue(t *testing.T) { x := &gfP2{ diff --git a/sm9/bn256/gfp4.go b/sm9/bn256/gfp4.go index 772a8c5..763159b 100644 --- a/sm9/bn256/gfp4.go +++ b/sm9/bn256/gfp4.go @@ -8,9 +8,9 @@ import "math/big" // // gfP4 implements the field of size p^4 as a quadratic extension of gfP2 -// where u²=i. +// where v²=ξ and ξ=u. type gfP4 struct { - x, y gfP2 // value is xi+y. + x, y gfP2 // value is xv+y. } func gfP4Decode(in *gfP4) *gfP4 {