--- /dev/null
+// 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
+}
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