mirror of
https://github.com/octoleo/restic.git
synced 2024-12-25 12:09:59 +00:00
396 lines
8.5 KiB
Go
396 lines
8.5 KiB
Go
// Copyright 2012 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 bn256
|
|
|
|
func lineFunctionAdd(r, p *twistPoint, q *curvePoint, r2 *gfP2, pool *bnPool) (a, b, c *gfP2, rOut *twistPoint) {
|
|
// See the mixed addition algorithm from "Faster Computation of the
|
|
// Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf
|
|
|
|
B := newGFp2(pool).Mul(p.x, r.t, pool)
|
|
|
|
D := newGFp2(pool).Add(p.y, r.z)
|
|
D.Square(D, pool)
|
|
D.Sub(D, r2)
|
|
D.Sub(D, r.t)
|
|
D.Mul(D, r.t, pool)
|
|
|
|
H := newGFp2(pool).Sub(B, r.x)
|
|
I := newGFp2(pool).Square(H, pool)
|
|
|
|
E := newGFp2(pool).Add(I, I)
|
|
E.Add(E, E)
|
|
|
|
J := newGFp2(pool).Mul(H, E, pool)
|
|
|
|
L1 := newGFp2(pool).Sub(D, r.y)
|
|
L1.Sub(L1, r.y)
|
|
|
|
V := newGFp2(pool).Mul(r.x, E, pool)
|
|
|
|
rOut = newTwistPoint(pool)
|
|
rOut.x.Square(L1, pool)
|
|
rOut.x.Sub(rOut.x, J)
|
|
rOut.x.Sub(rOut.x, V)
|
|
rOut.x.Sub(rOut.x, V)
|
|
|
|
rOut.z.Add(r.z, H)
|
|
rOut.z.Square(rOut.z, pool)
|
|
rOut.z.Sub(rOut.z, r.t)
|
|
rOut.z.Sub(rOut.z, I)
|
|
|
|
t := newGFp2(pool).Sub(V, rOut.x)
|
|
t.Mul(t, L1, pool)
|
|
t2 := newGFp2(pool).Mul(r.y, J, pool)
|
|
t2.Add(t2, t2)
|
|
rOut.y.Sub(t, t2)
|
|
|
|
rOut.t.Square(rOut.z, pool)
|
|
|
|
t.Add(p.y, rOut.z)
|
|
t.Square(t, pool)
|
|
t.Sub(t, r2)
|
|
t.Sub(t, rOut.t)
|
|
|
|
t2.Mul(L1, p.x, pool)
|
|
t2.Add(t2, t2)
|
|
a = newGFp2(pool)
|
|
a.Sub(t2, t)
|
|
|
|
c = newGFp2(pool)
|
|
c.MulScalar(rOut.z, q.y)
|
|
c.Add(c, c)
|
|
|
|
b = newGFp2(pool)
|
|
b.SetZero()
|
|
b.Sub(b, L1)
|
|
b.MulScalar(b, q.x)
|
|
b.Add(b, b)
|
|
|
|
B.Put(pool)
|
|
D.Put(pool)
|
|
H.Put(pool)
|
|
I.Put(pool)
|
|
E.Put(pool)
|
|
J.Put(pool)
|
|
L1.Put(pool)
|
|
V.Put(pool)
|
|
t.Put(pool)
|
|
t2.Put(pool)
|
|
|
|
return
|
|
}
|
|
|
|
func lineFunctionDouble(r *twistPoint, q *curvePoint, pool *bnPool) (a, b, c *gfP2, rOut *twistPoint) {
|
|
// See the doubling algorithm for a=0 from "Faster Computation of the
|
|
// Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf
|
|
|
|
A := newGFp2(pool).Square(r.x, pool)
|
|
B := newGFp2(pool).Square(r.y, pool)
|
|
C := newGFp2(pool).Square(B, pool)
|
|
|
|
D := newGFp2(pool).Add(r.x, B)
|
|
D.Square(D, pool)
|
|
D.Sub(D, A)
|
|
D.Sub(D, C)
|
|
D.Add(D, D)
|
|
|
|
E := newGFp2(pool).Add(A, A)
|
|
E.Add(E, A)
|
|
|
|
G := newGFp2(pool).Square(E, pool)
|
|
|
|
rOut = newTwistPoint(pool)
|
|
rOut.x.Sub(G, D)
|
|
rOut.x.Sub(rOut.x, D)
|
|
|
|
rOut.z.Add(r.y, r.z)
|
|
rOut.z.Square(rOut.z, pool)
|
|
rOut.z.Sub(rOut.z, B)
|
|
rOut.z.Sub(rOut.z, r.t)
|
|
|
|
rOut.y.Sub(D, rOut.x)
|
|
rOut.y.Mul(rOut.y, E, pool)
|
|
t := newGFp2(pool).Add(C, C)
|
|
t.Add(t, t)
|
|
t.Add(t, t)
|
|
rOut.y.Sub(rOut.y, t)
|
|
|
|
rOut.t.Square(rOut.z, pool)
|
|
|
|
t.Mul(E, r.t, pool)
|
|
t.Add(t, t)
|
|
b = newGFp2(pool)
|
|
b.SetZero()
|
|
b.Sub(b, t)
|
|
b.MulScalar(b, q.x)
|
|
|
|
a = newGFp2(pool)
|
|
a.Add(r.x, E)
|
|
a.Square(a, pool)
|
|
a.Sub(a, A)
|
|
a.Sub(a, G)
|
|
t.Add(B, B)
|
|
t.Add(t, t)
|
|
a.Sub(a, t)
|
|
|
|
c = newGFp2(pool)
|
|
c.Mul(rOut.z, r.t, pool)
|
|
c.Add(c, c)
|
|
c.MulScalar(c, q.y)
|
|
|
|
A.Put(pool)
|
|
B.Put(pool)
|
|
C.Put(pool)
|
|
D.Put(pool)
|
|
E.Put(pool)
|
|
G.Put(pool)
|
|
t.Put(pool)
|
|
|
|
return
|
|
}
|
|
|
|
func mulLine(ret *gfP12, a, b, c *gfP2, pool *bnPool) {
|
|
a2 := newGFp6(pool)
|
|
a2.x.SetZero()
|
|
a2.y.Set(a)
|
|
a2.z.Set(b)
|
|
a2.Mul(a2, ret.x, pool)
|
|
t3 := newGFp6(pool).MulScalar(ret.y, c, pool)
|
|
|
|
t := newGFp2(pool)
|
|
t.Add(b, c)
|
|
t2 := newGFp6(pool)
|
|
t2.x.SetZero()
|
|
t2.y.Set(a)
|
|
t2.z.Set(t)
|
|
ret.x.Add(ret.x, ret.y)
|
|
|
|
ret.y.Set(t3)
|
|
|
|
ret.x.Mul(ret.x, t2, pool)
|
|
ret.x.Sub(ret.x, a2)
|
|
ret.x.Sub(ret.x, ret.y)
|
|
a2.MulTau(a2, pool)
|
|
ret.y.Add(ret.y, a2)
|
|
|
|
a2.Put(pool)
|
|
t3.Put(pool)
|
|
t2.Put(pool)
|
|
t.Put(pool)
|
|
}
|
|
|
|
// sixuPlus2NAF is 6u+2 in non-adjacent form.
|
|
var sixuPlus2NAF = []int8{0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, -1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 1}
|
|
|
|
// miller implements the Miller loop for calculating the Optimal Ate pairing.
|
|
// See algorithm 1 from http://cryptojedi.org/papers/dclxvi-20100714.pdf
|
|
func miller(q *twistPoint, p *curvePoint, pool *bnPool) *gfP12 {
|
|
ret := newGFp12(pool)
|
|
ret.SetOne()
|
|
|
|
aAffine := newTwistPoint(pool)
|
|
aAffine.Set(q)
|
|
aAffine.MakeAffine(pool)
|
|
|
|
bAffine := newCurvePoint(pool)
|
|
bAffine.Set(p)
|
|
bAffine.MakeAffine(pool)
|
|
|
|
minusA := newTwistPoint(pool)
|
|
minusA.Negative(aAffine, pool)
|
|
|
|
r := newTwistPoint(pool)
|
|
r.Set(aAffine)
|
|
|
|
r2 := newGFp2(pool)
|
|
r2.Square(aAffine.y, pool)
|
|
|
|
for i := len(sixuPlus2NAF) - 1; i > 0; i-- {
|
|
a, b, c, newR := lineFunctionDouble(r, bAffine, pool)
|
|
if i != len(sixuPlus2NAF)-1 {
|
|
ret.Square(ret, pool)
|
|
}
|
|
|
|
mulLine(ret, a, b, c, pool)
|
|
a.Put(pool)
|
|
b.Put(pool)
|
|
c.Put(pool)
|
|
r.Put(pool)
|
|
r = newR
|
|
|
|
switch sixuPlus2NAF[i-1] {
|
|
case 1:
|
|
a, b, c, newR = lineFunctionAdd(r, aAffine, bAffine, r2, pool)
|
|
case -1:
|
|
a, b, c, newR = lineFunctionAdd(r, minusA, bAffine, r2, pool)
|
|
default:
|
|
continue
|
|
}
|
|
|
|
mulLine(ret, a, b, c, pool)
|
|
a.Put(pool)
|
|
b.Put(pool)
|
|
c.Put(pool)
|
|
r.Put(pool)
|
|
r = newR
|
|
}
|
|
|
|
// In order to calculate Q1 we have to convert q from the sextic twist
|
|
// to the full GF(p^12) group, apply the Frobenius there, and convert
|
|
// back.
|
|
//
|
|
// The twist isomorphism is (x', y') -> (xω², yω³). If we consider just
|
|
// x for a moment, then after applying the Frobenius, we have x̄ω^(2p)
|
|
// where x̄ is the conjugate of x. If we are going to apply the inverse
|
|
// isomorphism we need a value with a single coefficient of ω² so we
|
|
// rewrite this as x̄ω^(2p-2)ω². ξ⁶ = ω and, due to the construction of
|
|
// p, 2p-2 is a multiple of six. Therefore we can rewrite as
|
|
// x̄ξ^((p-1)/3)ω² and applying the inverse isomorphism eliminates the
|
|
// ω².
|
|
//
|
|
// A similar argument can be made for the y value.
|
|
|
|
q1 := newTwistPoint(pool)
|
|
q1.x.Conjugate(aAffine.x)
|
|
q1.x.Mul(q1.x, xiToPMinus1Over3, pool)
|
|
q1.y.Conjugate(aAffine.y)
|
|
q1.y.Mul(q1.y, xiToPMinus1Over2, pool)
|
|
q1.z.SetOne()
|
|
q1.t.SetOne()
|
|
|
|
// For Q2 we are applying the p² Frobenius. The two conjugations cancel
|
|
// out and we are left only with the factors from the isomorphism. In
|
|
// the case of x, we end up with a pure number which is why
|
|
// xiToPSquaredMinus1Over3 is ∈ GF(p). With y we get a factor of -1. We
|
|
// ignore this to end up with -Q2.
|
|
|
|
minusQ2 := newTwistPoint(pool)
|
|
minusQ2.x.MulScalar(aAffine.x, xiToPSquaredMinus1Over3)
|
|
minusQ2.y.Set(aAffine.y)
|
|
minusQ2.z.SetOne()
|
|
minusQ2.t.SetOne()
|
|
|
|
r2.Square(q1.y, pool)
|
|
a, b, c, newR := lineFunctionAdd(r, q1, bAffine, r2, pool)
|
|
mulLine(ret, a, b, c, pool)
|
|
a.Put(pool)
|
|
b.Put(pool)
|
|
c.Put(pool)
|
|
r.Put(pool)
|
|
r = newR
|
|
|
|
r2.Square(minusQ2.y, pool)
|
|
a, b, c, newR = lineFunctionAdd(r, minusQ2, bAffine, r2, pool)
|
|
mulLine(ret, a, b, c, pool)
|
|
a.Put(pool)
|
|
b.Put(pool)
|
|
c.Put(pool)
|
|
r.Put(pool)
|
|
r = newR
|
|
|
|
aAffine.Put(pool)
|
|
bAffine.Put(pool)
|
|
minusA.Put(pool)
|
|
r.Put(pool)
|
|
r2.Put(pool)
|
|
|
|
return ret
|
|
}
|
|
|
|
// finalExponentiation computes the (p¹²-1)/Order-th power of an element of
|
|
// GF(p¹²) to obtain an element of GT (steps 13-15 of algorithm 1 from
|
|
// http://cryptojedi.org/papers/dclxvi-20100714.pdf)
|
|
func finalExponentiation(in *gfP12, pool *bnPool) *gfP12 {
|
|
t1 := newGFp12(pool)
|
|
|
|
// This is the p^6-Frobenius
|
|
t1.x.Negative(in.x)
|
|
t1.y.Set(in.y)
|
|
|
|
inv := newGFp12(pool)
|
|
inv.Invert(in, pool)
|
|
t1.Mul(t1, inv, pool)
|
|
|
|
t2 := newGFp12(pool).FrobeniusP2(t1, pool)
|
|
t1.Mul(t1, t2, pool)
|
|
|
|
fp := newGFp12(pool).Frobenius(t1, pool)
|
|
fp2 := newGFp12(pool).FrobeniusP2(t1, pool)
|
|
fp3 := newGFp12(pool).Frobenius(fp2, pool)
|
|
|
|
fu, fu2, fu3 := newGFp12(pool), newGFp12(pool), newGFp12(pool)
|
|
fu.Exp(t1, u, pool)
|
|
fu2.Exp(fu, u, pool)
|
|
fu3.Exp(fu2, u, pool)
|
|
|
|
y3 := newGFp12(pool).Frobenius(fu, pool)
|
|
fu2p := newGFp12(pool).Frobenius(fu2, pool)
|
|
fu3p := newGFp12(pool).Frobenius(fu3, pool)
|
|
y2 := newGFp12(pool).FrobeniusP2(fu2, pool)
|
|
|
|
y0 := newGFp12(pool)
|
|
y0.Mul(fp, fp2, pool)
|
|
y0.Mul(y0, fp3, pool)
|
|
|
|
y1, y4, y5 := newGFp12(pool), newGFp12(pool), newGFp12(pool)
|
|
y1.Conjugate(t1)
|
|
y5.Conjugate(fu2)
|
|
y3.Conjugate(y3)
|
|
y4.Mul(fu, fu2p, pool)
|
|
y4.Conjugate(y4)
|
|
|
|
y6 := newGFp12(pool)
|
|
y6.Mul(fu3, fu3p, pool)
|
|
y6.Conjugate(y6)
|
|
|
|
t0 := newGFp12(pool)
|
|
t0.Square(y6, pool)
|
|
t0.Mul(t0, y4, pool)
|
|
t0.Mul(t0, y5, pool)
|
|
t1.Mul(y3, y5, pool)
|
|
t1.Mul(t1, t0, pool)
|
|
t0.Mul(t0, y2, pool)
|
|
t1.Square(t1, pool)
|
|
t1.Mul(t1, t0, pool)
|
|
t1.Square(t1, pool)
|
|
t0.Mul(t1, y1, pool)
|
|
t1.Mul(t1, y0, pool)
|
|
t0.Square(t0, pool)
|
|
t0.Mul(t0, t1, pool)
|
|
|
|
inv.Put(pool)
|
|
t1.Put(pool)
|
|
t2.Put(pool)
|
|
fp.Put(pool)
|
|
fp2.Put(pool)
|
|
fp3.Put(pool)
|
|
fu.Put(pool)
|
|
fu2.Put(pool)
|
|
fu3.Put(pool)
|
|
fu2p.Put(pool)
|
|
fu3p.Put(pool)
|
|
y0.Put(pool)
|
|
y1.Put(pool)
|
|
y2.Put(pool)
|
|
y3.Put(pool)
|
|
y4.Put(pool)
|
|
y5.Put(pool)
|
|
y6.Put(pool)
|
|
|
|
return t0
|
|
}
|
|
|
|
func optimalAte(a *twistPoint, b *curvePoint, pool *bnPool) *gfP12 {
|
|
e := miller(a, b, pool)
|
|
ret := finalExponentiation(e, pool)
|
|
e.Put(pool)
|
|
|
|
if a.IsInfinity() || b.IsInfinity() {
|
|
ret.SetOne()
|
|
}
|
|
|
|
return ret
|
|
}
|