272 lines
6.4 KiB
Go
272 lines
6.4 KiB
Go
package bn256
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func lineFunctionAdd(r, p *twistPoint, q *curvePoint, r2 *gfP2) (a, b, c *gfP2, rOut *twistPoint) {
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// See the mixed addition algorithm from "Faster Computation of the
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// Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf
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B := (&gfP2{}).Mul(&p.x, &r.t)
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D := (&gfP2{}).Add(&p.y, &r.z)
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D.Square(D).Sub(D, r2).Sub(D, &r.t).Mul(D, &r.t)
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H := (&gfP2{}).Sub(B, &r.x)
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I := (&gfP2{}).Square(H)
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E := (&gfP2{}).Add(I, I)
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E.Add(E, E)
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J := (&gfP2{}).Mul(H, E)
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L1 := (&gfP2{}).Sub(D, &r.y)
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L1.Sub(L1, &r.y)
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V := (&gfP2{}).Mul(&r.x, E)
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rOut = &twistPoint{}
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rOut.x.Square(L1).Sub(&rOut.x, J).Sub(&rOut.x, V).Sub(&rOut.x, V)
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rOut.z.Add(&r.z, H).Square(&rOut.z).Sub(&rOut.z, &r.t).Sub(&rOut.z, I)
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t := (&gfP2{}).Sub(V, &rOut.x)
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t.Mul(t, L1)
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t2 := (&gfP2{}).Mul(&r.y, J)
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t2.Add(t2, t2)
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rOut.y.Sub(t, t2)
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rOut.t.Square(&rOut.z)
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t.Add(&p.y, &rOut.z).Square(t).Sub(t, r2).Sub(t, &rOut.t)
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t2.Mul(L1, &p.x)
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t2.Add(t2, t2)
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a = (&gfP2{}).Sub(t2, t)
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c = (&gfP2{}).MulScalar(&rOut.z, &q.y)
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c.Add(c, c)
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b = (&gfP2{}).Neg(L1)
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b.MulScalar(b, &q.x).Add(b, b)
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return
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}
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func lineFunctionDouble(r *twistPoint, q *curvePoint) (a, b, c *gfP2, rOut *twistPoint) {
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// See the doubling algorithm for a=0 from "Faster Computation of the
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// Tate Pairing", http://arxiv.org/pdf/0904.0854v3.pdf
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A := (&gfP2{}).Square(&r.x)
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B := (&gfP2{}).Square(&r.y)
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C := (&gfP2{}).Square(B)
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D := (&gfP2{}).Add(&r.x, B)
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D.Square(D).Sub(D, A).Sub(D, C).Add(D, D)
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E := (&gfP2{}).Add(A, A)
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E.Add(E, A)
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G := (&gfP2{}).Square(E)
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rOut = &twistPoint{}
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rOut.x.Sub(G, D).Sub(&rOut.x, D)
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rOut.z.Add(&r.y, &r.z).Square(&rOut.z).Sub(&rOut.z, B).Sub(&rOut.z, &r.t)
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rOut.y.Sub(D, &rOut.x).Mul(&rOut.y, E)
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t := (&gfP2{}).Add(C, C)
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t.Add(t, t).Add(t, t)
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rOut.y.Sub(&rOut.y, t)
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rOut.t.Square(&rOut.z)
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t.Mul(E, &r.t).Add(t, t)
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b = (&gfP2{}).Neg(t)
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b.MulScalar(b, &q.x)
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a = (&gfP2{}).Add(&r.x, E)
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a.Square(a).Sub(a, A).Sub(a, G)
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t.Add(B, B).Add(t, t)
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a.Sub(a, t)
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c = (&gfP2{}).Mul(&rOut.z, &r.t)
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c.Add(c, c).MulScalar(c, &q.y)
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return
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}
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func mulLine(ret *gfP12, a, b, c *gfP2) {
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a2 := &gfP6{}
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a2.y.Set(a)
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a2.z.Set(b)
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a2.Mul(a2, &ret.x)
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t3 := (&gfP6{}).MulScalar(&ret.y, c)
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t := (&gfP2{}).Add(b, c)
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t2 := &gfP6{}
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t2.y.Set(a)
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t2.z.Set(t)
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ret.x.Add(&ret.x, &ret.y)
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ret.y.Set(t3)
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ret.x.Mul(&ret.x, t2).Sub(&ret.x, a2).Sub(&ret.x, &ret.y)
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a2.MulTau(a2)
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ret.y.Add(&ret.y, a2)
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}
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// sixuPlus2NAF is 6u+2 in non-adjacent form.
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var sixuPlus2NAF = []int8{0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0,
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0, 1, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 1,
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1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1,
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1, 0, 0, -1, 0, 0, 0, 1, 1, 0, -1, 0, 0, 1, 0, 1, 1}
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// miller implements the Miller loop for calculating the Optimal Ate pairing.
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// See algorithm 1 from http://cryptojedi.org/papers/dclxvi-20100714.pdf
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func miller(q *twistPoint, p *curvePoint) *gfP12 {
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ret := (&gfP12{}).SetOne()
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aAffine := &twistPoint{}
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aAffine.Set(q)
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aAffine.MakeAffine()
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bAffine := &curvePoint{}
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bAffine.Set(p)
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bAffine.MakeAffine()
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minusA := &twistPoint{}
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minusA.Neg(aAffine)
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r := &twistPoint{}
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r.Set(aAffine)
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r2 := (&gfP2{}).Square(&aAffine.y)
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for i := len(sixuPlus2NAF) - 1; i > 0; i-- {
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a, b, c, newR := lineFunctionDouble(r, bAffine)
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if i != len(sixuPlus2NAF)-1 {
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ret.Square(ret)
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}
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mulLine(ret, a, b, c)
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r = newR
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switch sixuPlus2NAF[i-1] {
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case 1:
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a, b, c, newR = lineFunctionAdd(r, aAffine, bAffine, r2)
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case -1:
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a, b, c, newR = lineFunctionAdd(r, minusA, bAffine, r2)
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default:
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continue
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}
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mulLine(ret, a, b, c)
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r = newR
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}
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// In order to calculate Q1 we have to convert q from the sextic twist
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// to the full GF(p^12) group, apply the Frobenius there, and convert
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// back.
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//
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// The twist isomorphism is (x', y') -> (xω², yω³). If we consider just
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// x for a moment, then after applying the Frobenius, we have x̄ω^(2p)
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// where x̄ is the conjugate of x. If we are going to apply the inverse
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// isomorphism we need a value with a single coefficient of ω² so we
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// rewrite this as x̄ω^(2p-2)ω². ξ⁶ = ω and, due to the construction of
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// p, 2p-2 is a multiple of six. Therefore we can rewrite as
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// x̄ξ^((p-1)/3)ω² and applying the inverse isomorphism eliminates the
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// ω².
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//
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// A similar argument can be made for the y value.
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q1 := &twistPoint{}
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q1.x.Conjugate(&aAffine.x).Mul(&q1.x, xiToPMinus1Over3)
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q1.y.Conjugate(&aAffine.y).Mul(&q1.y, xiToPMinus1Over2)
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q1.z.SetOne()
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q1.t.SetOne()
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// For Q2 we are applying the p² Frobenius. The two conjugations cancel
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// out and we are left only with the factors from the isomorphism. In
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// the case of x, we end up with a pure number which is why
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// xiToPSquaredMinus1Over3 is ∈ GF(p). With y we get a factor of -1. We
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// ignore this to end up with -Q2.
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minusQ2 := &twistPoint{}
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minusQ2.x.MulScalar(&aAffine.x, xiToPSquaredMinus1Over3)
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minusQ2.y.Set(&aAffine.y)
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minusQ2.z.SetOne()
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minusQ2.t.SetOne()
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r2.Square(&q1.y)
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a, b, c, newR := lineFunctionAdd(r, q1, bAffine, r2)
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mulLine(ret, a, b, c)
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r = newR
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r2.Square(&minusQ2.y)
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a, b, c, newR = lineFunctionAdd(r, minusQ2, bAffine, r2)
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mulLine(ret, a, b, c)
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r = newR
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return ret
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}
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// finalExponentiation computes the (p¹²-1)/Order-th power of an element of
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// GF(p¹²) to obtain an element of GT (steps 13-15 of algorithm 1 from
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// http://cryptojedi.org/papers/dclxvi-20100714.pdf)
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func finalExponentiation(in *gfP12) *gfP12 {
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t1 := &gfP12{}
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// This is the p^6-Frobenius
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t1.x.Neg(&in.x)
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t1.y.Set(&in.y)
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inv := &gfP12{}
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inv.Invert(in)
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t1.Mul(t1, inv)
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t2 := (&gfP12{}).FrobeniusP2(t1)
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t1.Mul(t1, t2)
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fp := (&gfP12{}).Frobenius(t1)
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fp2 := (&gfP12{}).FrobeniusP2(t1)
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fp3 := (&gfP12{}).Frobenius(fp2)
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fu := (&gfP12{}).Exp(t1, u)
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fu2 := (&gfP12{}).Exp(fu, u)
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fu3 := (&gfP12{}).Exp(fu2, u)
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y3 := (&gfP12{}).Frobenius(fu)
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fu2p := (&gfP12{}).Frobenius(fu2)
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fu3p := (&gfP12{}).Frobenius(fu3)
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y2 := (&gfP12{}).FrobeniusP2(fu2)
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y0 := &gfP12{}
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y0.Mul(fp, fp2).Mul(y0, fp3)
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y1 := (&gfP12{}).Conjugate(t1)
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y5 := (&gfP12{}).Conjugate(fu2)
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y3.Conjugate(y3)
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y4 := (&gfP12{}).Mul(fu, fu2p)
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y4.Conjugate(y4)
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y6 := (&gfP12{}).Mul(fu3, fu3p)
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y6.Conjugate(y6)
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t0 := (&gfP12{}).Square(y6)
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t0.Mul(t0, y4).Mul(t0, y5)
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t1.Mul(y3, y5).Mul(t1, t0)
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t0.Mul(t0, y2)
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t1.Square(t1).Mul(t1, t0).Square(t1)
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t0.Mul(t1, y1)
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t1.Mul(t1, y0)
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t0.Square(t0).Mul(t0, t1)
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return t0
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}
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func optimalAte(a *twistPoint, b *curvePoint) *gfP12 {
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e := miller(a, b)
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ret := finalExponentiation(e)
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if a.IsInfinity() || b.IsInfinity() {
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ret.SetOne()
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}
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return ret
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}
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