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// Copyright 2012 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package bn256
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func lineFunctionAdd(r, p *twistPoint, q *curvePoint, r2 *gfP2, pool *bnPool) (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 := newGFp2(pool).Mul(p.x, r.t, pool)
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D := newGFp2(pool).Add(p.y, r.z)
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D.Square(D, pool)
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D.Sub(D, r2)
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D.Sub(D, r.t)
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D.Mul(D, r.t, pool)
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H := newGFp2(pool).Sub(B, r.x)
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I := newGFp2(pool).Square(H, pool)
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E := newGFp2(pool).Add(I, I)
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E.Add(E, E)
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J := newGFp2(pool).Mul(H, E, pool)
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L1 := newGFp2(pool).Sub(D, r.y)
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L1.Sub(L1, r.y)
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V := newGFp2(pool).Mul(r.x, E, pool)
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rOut = newTwistPoint(pool)
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rOut.x.Square(L1, pool)
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rOut.x.Sub(rOut.x, J)
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rOut.x.Sub(rOut.x, V)
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rOut.x.Sub(rOut.x, V)
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rOut.z.Add(r.z, H)
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rOut.z.Square(rOut.z, pool)
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rOut.z.Sub(rOut.z, r.t)
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rOut.z.Sub(rOut.z, I)
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t := newGFp2(pool).Sub(V, rOut.x)
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t.Mul(t, L1, pool)
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t2 := newGFp2(pool).Mul(r.y, J, pool)
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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, pool)
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t.Add(p.y, rOut.z)
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t.Square(t, pool)
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t.Sub(t, r2)
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t.Sub(t, rOut.t)
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t2.Mul(L1, p.x, pool)
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t2.Add(t2, t2)
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a = newGFp2(pool)
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a.Sub(t2, t)
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c = newGFp2(pool)
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c.MulScalar(rOut.z, q.y)
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c.Add(c, c)
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b = newGFp2(pool)
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b.SetZero()
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b.Sub(b, L1)
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b.MulScalar(b, q.x)
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b.Add(b, b)
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B.Put(pool)
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D.Put(pool)
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H.Put(pool)
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I.Put(pool)
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E.Put(pool)
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J.Put(pool)
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L1.Put(pool)
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V.Put(pool)
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t.Put(pool)
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t2.Put(pool)
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return
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}
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func lineFunctionDouble(r *twistPoint, q *curvePoint, pool *bnPool) (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 := newGFp2(pool).Square(r.x, pool)
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B := newGFp2(pool).Square(r.y, pool)
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C := newGFp2(pool).Square(B, pool)
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D := newGFp2(pool).Add(r.x, B)
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D.Square(D, pool)
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D.Sub(D, A)
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D.Sub(D, C)
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D.Add(D, D)
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E := newGFp2(pool).Add(A, A)
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E.Add(E, A)
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G := newGFp2(pool).Square(E, pool)
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rOut = newTwistPoint(pool)
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rOut.x.Sub(G, D)
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rOut.x.Sub(rOut.x, D)
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rOut.z.Add(r.y, r.z)
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rOut.z.Square(rOut.z, pool)
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rOut.z.Sub(rOut.z, B)
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rOut.z.Sub(rOut.z, r.t)
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rOut.y.Sub(D, rOut.x)
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rOut.y.Mul(rOut.y, E, pool)
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t := newGFp2(pool).Add(C, C)
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t.Add(t, t)
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t.Add(t, t)
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rOut.y.Sub(rOut.y, t)
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rOut.t.Square(rOut.z, pool)
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t.Mul(E, r.t, pool)
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t.Add(t, t)
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b = newGFp2(pool)
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b.SetZero()
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b.Sub(b, t)
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b.MulScalar(b, q.x)
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a = newGFp2(pool)
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a.Add(r.x, E)
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a.Square(a, pool)
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a.Sub(a, A)
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a.Sub(a, G)
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t.Add(B, B)
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t.Add(t, t)
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a.Sub(a, t)
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c = newGFp2(pool)
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c.Mul(rOut.z, r.t, pool)
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c.Add(c, c)
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c.MulScalar(c, q.y)
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A.Put(pool)
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B.Put(pool)
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C.Put(pool)
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D.Put(pool)
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E.Put(pool)
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G.Put(pool)
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t.Put(pool)
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return
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}
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func mulLine(ret *gfP12, a, b, c *gfP2, pool *bnPool) {
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a2 := newGFp6(pool)
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a2.x.SetZero()
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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, pool)
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t3 := newGFp6(pool).MulScalar(ret.y, c, pool)
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t := newGFp2(pool)
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t.Add(b, c)
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t2 := newGFp6(pool)
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t2.x.SetZero()
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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, pool)
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ret.x.Sub(ret.x, a2)
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ret.x.Sub(ret.x, ret.y)
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a2.MulTau(a2, pool)
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ret.y.Add(ret.y, a2)
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a2.Put(pool)
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t3.Put(pool)
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t2.Put(pool)
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t.Put(pool)
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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, pool *bnPool) *gfP12 {
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ret := newGFp12(pool)
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ret.SetOne()
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aAffine := newTwistPoint(pool)
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aAffine.Set(q)
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aAffine.MakeAffine(pool)
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bAffine := newCurvePoint(pool)
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bAffine.Set(p)
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bAffine.MakeAffine(pool)
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minusA := newTwistPoint(pool)
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minusA.Negative(aAffine, pool)
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r := newTwistPoint(pool)
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r.Set(aAffine)
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r2 := newGFp2(pool)
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r2.Square(aAffine.y, pool)
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for i := len(sixuPlus2NAF) - 1; i > 0; i-- {
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a, b, c, newR := lineFunctionDouble(r, bAffine, pool)
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if i != len(sixuPlus2NAF)-1 {
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ret.Square(ret, pool)
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}
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mulLine(ret, a, b, c, pool)
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a.Put(pool)
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b.Put(pool)
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c.Put(pool)
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r.Put(pool)
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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, pool)
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case -1:
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a, b, c, newR = lineFunctionAdd(r, minusA, bAffine, r2, pool)
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default:
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continue
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}
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mulLine(ret, a, b, c, pool)
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a.Put(pool)
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b.Put(pool)
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c.Put(pool)
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r.Put(pool)
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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 := newTwistPoint(pool)
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q1.x.Conjugate(aAffine.x)
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q1.x.Mul(q1.x, xiToPMinus1Over3, pool)
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q1.y.Conjugate(aAffine.y)
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q1.y.Mul(q1.y, xiToPMinus1Over2, pool)
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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 := newTwistPoint(pool)
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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, pool)
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a, b, c, newR := lineFunctionAdd(r, q1, bAffine, r2, pool)
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mulLine(ret, a, b, c, pool)
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a.Put(pool)
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b.Put(pool)
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c.Put(pool)
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r.Put(pool)
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r = newR
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r2.Square(minusQ2.y, pool)
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a, b, c, newR = lineFunctionAdd(r, minusQ2, bAffine, r2, pool)
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mulLine(ret, a, b, c, pool)
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a.Put(pool)
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b.Put(pool)
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c.Put(pool)
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r.Put(pool)
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r = newR
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aAffine.Put(pool)
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bAffine.Put(pool)
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minusA.Put(pool)
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r.Put(pool)
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r2.Put(pool)
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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, pool *bnPool) *gfP12 {
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t1 := newGFp12(pool)
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// This is the p^6-Frobenius
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t1.x.Negative(in.x)
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t1.y.Set(in.y)
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inv := newGFp12(pool)
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inv.Invert(in, pool)
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t1.Mul(t1, inv, pool)
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t2 := newGFp12(pool).FrobeniusP2(t1, pool)
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t1.Mul(t1, t2, pool)
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fp := newGFp12(pool).Frobenius(t1, pool)
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fp2 := newGFp12(pool).FrobeniusP2(t1, pool)
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fp3 := newGFp12(pool).Frobenius(fp2, pool)
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fu, fu2, fu3 := newGFp12(pool), newGFp12(pool), newGFp12(pool)
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fu.Exp(t1, u, pool)
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fu2.Exp(fu, u, pool)
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fu3.Exp(fu2, u, pool)
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y3 := newGFp12(pool).Frobenius(fu, pool)
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fu2p := newGFp12(pool).Frobenius(fu2, pool)
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fu3p := newGFp12(pool).Frobenius(fu3, pool)
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y2 := newGFp12(pool).FrobeniusP2(fu2, pool)
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y0 := newGFp12(pool)
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y0.Mul(fp, fp2, pool)
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y0.Mul(y0, fp3, pool)
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y1, y4, y5 := newGFp12(pool), newGFp12(pool), newGFp12(pool)
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y1.Conjugate(t1)
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y5.Conjugate(fu2)
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y3.Conjugate(y3)
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y4.Mul(fu, fu2p, pool)
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y4.Conjugate(y4)
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y6 := newGFp12(pool)
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y6.Mul(fu3, fu3p, pool)
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y6.Conjugate(y6)
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t0 := newGFp12(pool)
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t0.Square(y6, pool)
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t0.Mul(t0, y4, pool)
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t0.Mul(t0, y5, pool)
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t1.Mul(y3, y5, pool)
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t1.Mul(t1, t0, pool)
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t0.Mul(t0, y2, pool)
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t1.Square(t1, pool)
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t1.Mul(t1, t0, pool)
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t1.Square(t1, pool)
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t0.Mul(t1, y1, pool)
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t1.Mul(t1, y0, pool)
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t0.Square(t0, pool)
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t0.Mul(t0, t1, pool)
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inv.Put(pool)
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t1.Put(pool)
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t2.Put(pool)
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fp.Put(pool)
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fp2.Put(pool)
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fp3.Put(pool)
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fu.Put(pool)
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fu2.Put(pool)
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fu3.Put(pool)
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fu2p.Put(pool)
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fu3p.Put(pool)
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y0.Put(pool)
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y1.Put(pool)
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y2.Put(pool)
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y3.Put(pool)
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y4.Put(pool)
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y5.Put(pool)
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y6.Put(pool)
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return t0
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}
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func optimalAte(a *twistPoint, b *curvePoint, pool *bnPool) *gfP12 {
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e := miller(a, b, pool)
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ret := finalExponentiation(e, pool)
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e.Put(pool)
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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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