mirror of
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465 lines
9.6 KiB
Go
465 lines
9.6 KiB
Go
package bn256
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import "math/big"
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// gfP12b6 implements the field of size p¹² as a quadratic extension of gfP6
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// where t²=s.
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type gfP12b6 struct {
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x, y gfP6 // value is xt + y
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}
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func gfP12b6Decode(in *gfP12b6) *gfP12b6 {
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out := &gfP12b6{}
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out.x = *gfP6Decode(&in.x)
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out.y = *gfP6Decode(&in.y)
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return out
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}
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var gfP12b6Gen *gfP12b6 = &gfP12b6{
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x: gfP6{
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x: gfP2{
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x: *fromBigInt(bigFromHex("256943fbdb2bf87ab91ae7fbeaff14e146cf7e2279b9d155d13461e09b22f523")),
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y: *fromBigInt(bigFromHex("0167b0280051495c6af1ec23ba2cd2ff1cdcdeca461a5ab0b5449e9091308310")),
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},
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y: gfP2{
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x: *fromBigInt(bigFromHex("8ffe1c0e9de45fd0fed790ac26be91f6b3f0a49c084fe29a3fb6ed288ad7994d")),
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y: *fromBigInt(bigFromHex("1664a1366beb3196f0443e15f5f9042a947354a5678430d45ba031cff06db927")),
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},
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z: gfP2{
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x: *fromBigInt(bigFromHex("7fc6eb2aa771d99c9234fddd31752edfd60723e05a4ebfdeb5c33fbd47e0cf06")),
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y: *fromBigInt(bigFromHex("6fa6b6fa6dd6b6d3b19a959a110e748154eef796dc0fc2dd766ea414de786968")),
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},
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},
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y: gfP6{
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x: gfP2{
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x: *fromBigInt(bigFromHex("082cde173022da8cd09b28a2d80a8cee53894436a52007f978dc37f36116d39b")),
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y: *fromBigInt(bigFromHex("3fa7ed741eaed99a58f53e3df82df7ccd3407bcc7b1d44a9441920ced5fb824f")),
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},
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y: gfP2{
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x: *fromBigInt(bigFromHex("5e7addaddf7fbfe16291b4e89af50b8217ddc47ba3cba833c6e77c3fb027685e")),
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y: *fromBigInt(bigFromHex("79d0c8337072c93fef482bb055f44d6247ccac8e8e12525854b3566236337ebe")),
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},
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z: gfP2{
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x: *fromBigInt(bigFromHex("7f7c6d52b475e6aaa827fdc5b4175ac6929320f782d998f86b6b57cda42a0426")),
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y: *fromBigInt(bigFromHex("36a699de7c136f78eee2dbac4ca9727bff0cee02ee920f5822e65ea170aa9669")),
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},
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},
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}
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func (e *gfP12b6) String() string {
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return "(" + e.x.String() + "," + e.y.String() + ")"
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}
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func (e *gfP12b6) ToGfP12() *gfP12 {
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ret := &gfP12{}
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ret.z.y.Set(&e.y.z)
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ret.x.y.Set(&e.y.y)
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ret.y.x.Set(&e.y.x)
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ret.y.y.Set(&e.x.z)
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ret.z.x.Set(&e.x.y)
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ret.x.x.Set(&e.x.x)
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return ret
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}
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func (e *gfP12b6) SetGfP12(a *gfP12) *gfP12b6 {
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e.y.z.Set(&a.z.y) //a
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e.y.y.Set(&a.x.y) //b
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e.y.x.Set(&a.y.x)
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e.x.z.Set(&a.y.y)
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e.x.y.Set(&a.z.x) //c
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e.x.x.Set(&a.x.x)
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return e
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}
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func (e *gfP12b6) Set(a *gfP12b6) *gfP12b6 {
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e.x.Set(&a.x)
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e.y.Set(&a.y)
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return e
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}
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func (e *gfP12b6) SetZero() *gfP12b6 {
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e.x.SetZero()
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e.y.SetZero()
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return e
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}
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func (e *gfP12b6) SetOne() *gfP12b6 {
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e.x.SetZero()
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e.y.SetOne()
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return e
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}
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func (e *gfP12b6) IsZero() bool {
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return e.x.IsZero() && e.y.IsZero()
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}
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func (e *gfP12b6) IsOne() bool {
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return e.x.IsZero() && e.y.IsOne()
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}
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func (e *gfP12b6) Neg(a *gfP12b6) *gfP12b6 {
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e.x.Neg(&a.x)
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e.y.Neg(&a.y)
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return e
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}
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func (e *gfP12b6) Conjugate(a *gfP12b6) *gfP12b6 {
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e.x.Neg(&a.x)
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e.y.Set(&a.y)
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return e
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}
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func (e *gfP12b6) Add(a, b *gfP12b6) *gfP12b6 {
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e.x.Add(&a.x, &b.x)
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e.y.Add(&a.y, &b.y)
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return e
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}
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func (e *gfP12b6) Sub(a, b *gfP12b6) *gfP12b6 {
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e.x.Sub(&a.x, &b.x)
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e.y.Sub(&a.y, &b.y)
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return e
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}
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func (e *gfP12b6) Mul(a, b *gfP12b6) *gfP12b6 {
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tmp := &gfP12b6{}
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tmp.MulNC(a, b)
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e.x.Set(&tmp.x)
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e.y.Set(&tmp.y)
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return e
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}
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// Mul without value copy, will use e directly, so e can't be same as a and b.
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func (e *gfP12b6) MulNC(a, b *gfP12b6) *gfP12b6 {
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// "Multiplication and Squaring on Pairing-Friendly Fields"
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// Section 4, Karatsuba method.
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// http://eprint.iacr.org/2006/471.pdf
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//(a0+a1*t)(b0+b1*t)=c0+c1*t, where
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//c0 = a0*b0 +a1*b1*s
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//c1 = (a0 + a1)(b0 + b1) - a0*b0 - a1*b1 = a0*b1 + a1*b0
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tx := &e.x
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ty := &e.y
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v0, v1 := &gfP6{}, &gfP6{}
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v0.MulNC(&a.y, &b.y)
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v1.MulNC(&a.x, &b.x)
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tx.Add(&a.x, &a.y)
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ty.Add(&b.x, &b.y)
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tx.Mul(tx, ty)
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tx.Sub(tx, v0)
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tx.Sub(tx, v1)
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ty.MulS(v1)
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ty.Add(ty, v0)
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return e
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}
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func (e *gfP12b6) MulScalar(a *gfP12b6, b *gfP6) *gfP12b6 {
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e.x.Mul(&a.x, b)
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e.y.Mul(&a.y, b)
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return e
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}
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func (e *gfP12b6) MulGfP(a *gfP12b6, b *gfP) *gfP12b6 {
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e.x.MulGfP(&a.x, b)
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e.y.MulGfP(&a.y, b)
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return e
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}
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func (e *gfP12b6) MulGfP2(a *gfP12b6, b *gfP2) *gfP12b6 {
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e.x.MulScalar(&a.x, b)
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e.y.MulScalar(&a.y, b)
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return e
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}
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func (e *gfP12b6) Square(a *gfP12b6) *gfP12b6 {
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tmp := &gfP12b6{}
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tmp.SquareNC(a)
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e.x.Set(&tmp.x)
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e.y.Set(&tmp.y)
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return e
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}
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// Square without value copy, will use e directly, so e can't be same as a.
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func (e *gfP12b6) SquareNC(a *gfP12b6) *gfP12b6 {
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// Complex squaring algorithm
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// (xt+y)² = (x^2*s + y^2) + 2*x*y*t
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tx := &e.x
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ty := &e.y
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tx.SquareNC(&a.x).MulS(tx)
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ty.SquareNC(&a.y)
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ty.Add(tx, ty)
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tx.Mul(&a.x, &a.y)
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tx.Add(tx, tx)
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return e
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}
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// Cyclo6Square is used in final exponentiation after easy part(a ^ ((p^2 + 1)(p^6-1))).
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// Note that after the easy part of the final exponentiation,
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// the resulting element lies in cyclotomic subgroup.
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// "New software speed records for cryptographic pairings"
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// Section 3.3, Final exponentiation
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// https://cryptojedi.org/papers/dclxvi-20100714.pdf
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// The fomula reference:
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// Granger/Scott (PKC2010).
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// Section 3.2
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// https://eprint.iacr.org/2009/565.pdf
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func (e *gfP12b6) Cyclo6Square(a *gfP12b6) *gfP12b6 {
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tmp := &gfP12b6{}
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tmp.Cyclo6SquareNC(a)
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e.x.Set(&tmp.x)
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e.y.Set(&tmp.y)
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return e
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}
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// Special Square without value copy, will use e directly, so e can't be same as a.
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func (e *gfP12b6) Cyclo6SquareNC(a *gfP12b6) *gfP12b6 {
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f02 := &e.y.x
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f01 := &e.y.y
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f00 := &e.y.z
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f12 := &e.x.x
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f11 := &e.x.y
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f10 := &e.x.z
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t00, t01, t02, t10, t11, t12 := &gfP2{}, &gfP2{}, &gfP2{}, &gfP2{}, &gfP2{}, &gfP2{}
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gfP4Square(t11, t00, &a.x.y, &a.y.z)
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gfP4Square(t12, t01, &a.y.x, &a.x.z)
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gfP4Square(t02, t10, &a.x.x, &a.y.y)
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f00.MulU1(t02)
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t02.Set(t10)
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t10.Set(f00)
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f00.Add(t00, t00)
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t00.Add(f00, t00)
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f00.Add(t01, t01)
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t01.Add(f00, t01)
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f00.Add(t02, t02)
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t02.Add(f00, t02)
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f00.Add(t10, t10)
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t10.Add(f00, t10)
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f00.Add(t11, t11)
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t11.Add(f00, t11)
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f00.Add(t12, t12)
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t12.Add(f00, t12)
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f00.Add(&a.y.z, &a.y.z)
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f00.Neg(f00)
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f01.Add(&a.y.y, &a.y.y)
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f01.Neg(f01)
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f02.Add(&a.y.x, &a.y.x)
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f02.Neg(f02)
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f10.Add(&a.x.z, &a.x.z)
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f11.Add(&a.x.y, &a.x.y)
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f12.Add(&a.x.x, &a.x.x)
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f00.Add(f00, t00)
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f01.Add(f01, t01)
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f02.Add(f02, t02)
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f10.Add(f10, t10)
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f11.Add(f11, t11)
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f12.Add(f12, t12)
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return e
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}
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func (e *gfP12b6) Cyclo6Squares(a *gfP12b6, n int) *gfP12b6 {
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// Square first round
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in := &gfP12b6{}
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f02 := &in.y.x
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f01 := &in.y.y
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f00 := &in.y.z
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f12 := &in.x.x
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f11 := &in.x.y
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f10 := &in.x.z
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t00, t01, t02, t10, t11, t12 := &gfP2{}, &gfP2{}, &gfP2{}, &gfP2{}, &gfP2{}, &gfP2{}
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gfP4Square(t11, t00, &a.x.y, &a.y.z)
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gfP4Square(t12, t01, &a.y.x, &a.x.z)
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gfP4Square(t02, t10, &a.x.x, &a.y.y)
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f00.MulU1(t02)
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t02.Set(t10)
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t10.Set(f00)
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f00.Add(t00, t00)
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t00.Add(f00, t00)
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f00.Add(t01, t01)
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t01.Add(f00, t01)
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f00.Add(t02, t02)
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t02.Add(f00, t02)
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f00.Add(t10, t10)
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t10.Add(f00, t10)
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f00.Add(t11, t11)
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t11.Add(f00, t11)
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f00.Add(t12, t12)
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t12.Add(f00, t12)
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f00.Add(&a.y.z, &a.y.z)
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f00.Neg(f00)
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f01.Add(&a.y.y, &a.y.y)
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f01.Neg(f01)
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f02.Add(&a.y.x, &a.y.x)
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f02.Neg(f02)
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f10.Add(&a.x.z, &a.x.z)
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f11.Add(&a.x.y, &a.x.y)
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f12.Add(&a.x.x, &a.x.x)
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f00.Add(f00, t00)
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f01.Add(f01, t01)
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f02.Add(f02, t02)
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f10.Add(f10, t10)
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f11.Add(f11, t11)
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f12.Add(f12, t12)
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tmp := &gfP12b6{}
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var tmp2 *gfP12b6
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for i := 1; i < n; i++ {
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f02 = &tmp.y.x
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f01 = &tmp.y.y
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f00 = &tmp.y.z
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f12 = &tmp.x.x
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f11 = &tmp.x.y
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f10 = &tmp.x.z
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gfP4Square(t11, t00, &in.x.y, &in.y.z)
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gfP4Square(t12, t01, &in.y.x, &in.x.z)
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gfP4Square(t02, t10, &in.x.x, &in.y.y)
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f00.MulU1(t02)
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t02.Set(t10)
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t10.Set(f00)
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f00.Add(t00, t00)
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t00.Add(f00, t00)
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f00.Add(t01, t01)
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t01.Add(f00, t01)
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f00.Add(t02, t02)
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t02.Add(f00, t02)
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f00.Add(t10, t10)
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t10.Add(f00, t10)
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f00.Add(t11, t11)
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t11.Add(f00, t11)
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f00.Add(t12, t12)
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t12.Add(f00, t12)
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f00.Add(&in.y.z, &in.y.z)
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f00.Neg(f00)
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f01.Add(&in.y.y, &in.y.y)
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f01.Neg(f01)
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f02.Add(&in.y.x, &in.y.x)
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f02.Neg(f02)
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f10.Add(&in.x.z, &in.x.z)
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f11.Add(&in.x.y, &in.x.y)
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f12.Add(&in.x.x, &in.x.x)
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f00.Add(f00, t00)
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f01.Add(f01, t01)
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f02.Add(f02, t02)
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f10.Add(f10, t10)
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f11.Add(f11, t11)
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f12.Add(f12, t12)
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// Switch references
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tmp2 = in
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in = tmp
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tmp = tmp2
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}
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e.x.Set(&in.x)
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e.y.Set(&in.y)
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return e
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}
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func gfP4Square(retX, retY, x, y *gfP2) {
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retX.SquareU(x)
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retY.Square(y)
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retY.Add(retX, retY)
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retX.Mul(x, y)
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retX.Add(retX, retX)
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}
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func (c *gfP12b6) Exp(a *gfP12b6, power *big.Int) *gfP12b6 {
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sum := (&gfP12b6{}).SetOne()
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t := &gfP12b6{}
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for i := power.BitLen() - 1; i >= 0; i-- {
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t.Square(sum)
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if power.Bit(i) != 0 {
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sum.Mul(t, a)
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} else {
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sum.Set(t)
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}
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}
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c.Set(sum)
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return c
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}
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func (e *gfP12b6) Invert(a *gfP12b6) *gfP12b6 {
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// See "Implementing cryptographic pairings", M. Scott, section 3.2.
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// ftp://136.206.11.249/pub/crypto/pairings.pdf
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t0, t1 := &gfP6{}, &gfP6{}
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t0.MulNC(&a.y, &a.y)
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t1.MulNC(&a.x, &a.x).MulS(t1)
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t0.Sub(t0, t1)
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t0.Invert(t0)
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e.x.Neg(&a.x)
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e.y.Set(&a.y)
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e.MulScalar(e, t0)
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return e
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}
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// Frobenius computes (xt+y)^p
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// = x^p t^p + y^p
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// = x^p t^(p-1) t + y^p
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// = x^p s^((p-1)/2) t + y^p
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// sToPMinus1Over2
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func (e *gfP12b6) Frobenius(a *gfP12b6) *gfP12b6 {
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e.x.Frobenius(&a.x)
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e.y.Frobenius(&a.y)
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e.x.MulGfP(&e.x, sToPMinus1Over2)
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return e
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}
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// FrobeniusP2 computes (xt+y)^p² = x^p² t ·s^((p²-1)/2) + y^p²
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func (e *gfP12b6) FrobeniusP2(a *gfP12b6) *gfP12b6 {
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e.x.FrobeniusP2(&a.x)
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e.y.FrobeniusP2(&a.y)
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e.x.MulGfP(&e.x, sToPSquaredMinus1Over2)
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return e
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}
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func (e *gfP12b6) FrobeniusP4(a *gfP12b6) *gfP12b6 {
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e.x.FrobeniusP4(&a.x)
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e.y.FrobeniusP4(&a.y)
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e.x.MulGfP(&e.x, sToPSquaredMinus1)
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return e
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}
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func (e *gfP12b6) FrobeniusP6(a *gfP12b6) *gfP12b6 {
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e.x.Neg(&a.x)
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e.y.Set(&a.y)
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return e
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}
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// Select sets q to p1 if cond == 1, and to p2 if cond == 0.
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func (q *gfP12b6) Select(p1, p2 *gfP12b6, cond int) *gfP12b6 {
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q.x.Select(&p1.x, &p2.x, cond)
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q.y.Select(&p1.y, &p2.y, cond)
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|
return q
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|
}
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