1 files changed, 398 insertions, 0 deletions
diff --git a/crypto/bn256/optate.go b/crypto/bn256/optate.go
new file mode 100644
index 000000000..24e0bbc13
--- /dev/null
+++ b/crypto/bn256/optate.go
@@ -0,0 +1,398 @@
+// 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, 1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0,
+	0, 1, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 1,
+	1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1,
+	1, 0, 0, -1, 0, 0, 0, 1, 1, 0, -1, 0, 0, 1, 0, 1, 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
+}