191 lines
7.9 KiB
Go
191 lines
7.9 KiB
Go
package p751toolbox
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// Represents a 3-isogeny phi, holding the data necessary to evaluate phi.
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type ThreeIsogeny struct {
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x ExtensionFieldElement
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z ExtensionFieldElement
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}
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// Given a three-torsion point x3 = x(P_3) on the curve E_(A:C), construct the
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// three-isogeny phi : E_(A:C) -> E_(A:C)/<P_3> = E_(A':C').
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//
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// Returns a tuple (codomain, isogeny) = (E_(A':C'), phi).
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func ComputeThreeIsogeny(x3 *ProjectivePoint) (ProjectiveCurveParameters, ThreeIsogeny) {
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var isogeny ThreeIsogeny
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isogeny.x = x3.X
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isogeny.z = x3.Z
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// We want to compute
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// (A':C') = (Z^4 + 18X^2Z^2 - 27X^4 : 4XZ^3)
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// To do this, use the identity 18X^2Z^2 - 27X^4 = 9X^2(2Z^2 - 3X^2)
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var codomain ProjectiveCurveParameters
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var v0, v1, v2, v3 ExtensionFieldElement
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v1.Square(&x3.X) // = X^2
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v0.Add(&v1, &v1).Add(&v1, &v0) // = 3X^2
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v1.Add(&v0, &v0).Add(&v1, &v0) // = 9X^2
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v2.Square(&x3.Z) // = Z^2
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v3.Square(&v2) // = Z^4
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v2.Add(&v2, &v2) // = 2Z^2
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v0.Sub(&v2, &v0) // = 2Z^2 - 3X^2
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v1.Mul(&v1, &v0) // = 9X^2(2Z^2 - 3X^2)
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v0.Mul(&x3.X, &x3.Z) // = XZ
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v0.Add(&v0, &v0) // = 2XZ
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codomain.A.Add(&v3, &v1) // = Z^4 + 9X^2(2Z^2 - 3X^2)
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codomain.C.Mul(&v0, &v2) // = 4XZ^3
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return codomain, isogeny
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}
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// Given a 3-isogeny phi and a point xP = x(P), compute x(Q), the x-coordinate
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// of the image Q = phi(P) of P under phi : E_(A:C) -> E_(A':C').
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//
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// The output xQ = x(Q) is then a point on the curve E_(A':C'); the curve
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// parameters are returned by the Compute3Isogeny function used to construct
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// phi.
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func (phi *ThreeIsogeny) Eval(xP *ProjectivePoint) ProjectivePoint {
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var xQ ProjectivePoint
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var t0, t1, t2 ExtensionFieldElement
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t0.Mul(&phi.x, &xP.X) // = X3*XP
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t1.Mul(&phi.z, &xP.Z) // = Z3*XP
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t2.Sub(&t0, &t1) // = X3*XP - Z3*ZP
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t0.Mul(&phi.z, &xP.X) // = Z3*XP
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t1.Mul(&phi.x, &xP.Z) // = X3*ZP
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t0.Sub(&t0, &t1) // = Z3*XP - X3*ZP
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t2.Square(&t2) // = (X3*XP - Z3*ZP)^2
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t0.Square(&t0) // = (Z3*XP - X3*ZP)^2
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xQ.X.Mul(&t2, &xP.X) // = XP*(X3*XP - Z3*ZP)^2
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xQ.Z.Mul(&t0, &xP.Z) // = ZP*(Z3*XP - X3*ZP)^2
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return xQ
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}
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// Represents a 4-isogeny phi, holding the data necessary to evaluate phi.
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//
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// See ComputeFourIsogeny for more details.
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type FourIsogeny struct {
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Xsq_plus_Zsq ExtensionFieldElement
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Xsq_minus_Zsq ExtensionFieldElement
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XZ2 ExtensionFieldElement
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Xpow4 ExtensionFieldElement
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Zpow4 ExtensionFieldElement
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}
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// Given a four-torsion point x4 = x(P_4) on the curve E_(A:C), compute the
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// coefficients of the codomain E_(A':C') of the four-isogeny phi : E_(A:C) ->
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// E_(A:C)/<P_4>.
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//
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// Returns a tuple (codomain, isogeny) = (E_(A':C') : phi).
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//
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// There are two sets of formulas in Costello-Longa-Naehrig for computing
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// four-isogenies. One set is for the case where (1,...) lies in the kernel of
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// the isogeny (this is the FirstFourIsogeny), and the other (this set) is for
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// the case that (1,...) is *not* in the kernel.
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func ComputeFourIsogeny(x4 *ProjectivePoint) (ProjectiveCurveParameters, FourIsogeny) {
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var codomain ProjectiveCurveParameters
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var isogeny FourIsogeny
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var v0, v1 ExtensionFieldElement
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v0.Square(&x4.X) // = X4^2
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v1.Square(&x4.Z) // = Z4^2
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isogeny.Xsq_plus_Zsq.Add(&v0, &v1) // = X4^2 + Z4^2
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isogeny.Xsq_minus_Zsq.Sub(&v0, &v1) // = X4^2 - Z4^2
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isogeny.XZ2.Add(&x4.X, &x4.Z) // = X4 + Z4
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isogeny.XZ2.Square(&isogeny.XZ2) // = X4^2 + Z4^2 + 2X4Z4
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isogeny.XZ2.Sub(&isogeny.XZ2, &isogeny.Xsq_plus_Zsq) // = 2X4Z4
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isogeny.Xpow4.Square(&v0) // = X4^4
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isogeny.Zpow4.Square(&v1) // = Z4^4
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v0.Add(&isogeny.Xpow4, &isogeny.Xpow4) // = 2X4^4
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v0.Sub(&v0, &isogeny.Zpow4) // = 2X4^4 - Z4^4
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codomain.A.Add(&v0, &v0) // = 2(2X4^4 - Z4^4)
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codomain.C = isogeny.Zpow4 // = Z4^4
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return codomain, isogeny
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}
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// Given a 4-isogeny phi and a point xP = x(P), compute x(Q), the x-coordinate
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// of the image Q = phi(P) of P under phi : E_(A:C) -> E_(A':C').
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//
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// The output xQ = x(Q) is then a point on the curve E_(A':C'); the curve
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// parameters are returned by the ComputeFourIsogeny function used to construct
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// phi.
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func (phi *FourIsogeny) Eval(xP *ProjectivePoint) ProjectivePoint {
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var xQ ProjectivePoint
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var t0, t1, t2 ExtensionFieldElement
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// We want to compute formula (7) of Costello-Longa-Naehrig, namely
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//
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// Xprime = (2*X_4*Z*Z_4 - (X_4^2 + Z_4^2)*X)*(X*X_4 - Z*Z_4)^2*X
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// Zprime = (2*X*X_4*Z_4 - (X_4^2 + Z_4^2)*Z)*(X_4*Z - X*Z_4)^2*Z
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//
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// To do this we adapt the method in the MSR implementation, which computes
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//
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// X_Q = Xprime*( 16*(X_4 + Z_4)*(X_4 - Z_4)*X_4^2*Z_4^4 )
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// Z_Q = Zprime*( 16*(X_4 + Z_4)*(X_4 - Z_4)*X_4^2*Z_4^4 )
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//
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t0.Mul(&xP.X, &phi.XZ2) // = 2*X*X_4*Z_4
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t1.Mul(&xP.Z, &phi.Xsq_plus_Zsq) // = (X_4^2 + Z_4^2)*Z
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t0.Sub(&t0, &t1) // = -X_4^2*Z + 2*X*X_4*Z_4 - Z*Z_4^2
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t1.Mul(&xP.Z, &phi.Xsq_minus_Zsq) // = (X_4^2 - Z_4^2)*Z
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t2.Sub(&t0, &t1).Square(&t2) // = 4*(X_4*Z - X*Z_4)^2*X_4^2
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t0.Mul(&t0, &t1).Add(&t0, &t0).Add(&t0, &t0) // = 4*(2*X*X_4*Z_4 - (X_4^2 + Z_4^2)*Z)*(X_4^2 - Z_4^2)*Z
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t1.Add(&t0, &t2) // = 4*(X*X_4 - Z*Z_4)^2*Z_4^2
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t0.Mul(&t0, &t2) // = Zprime * 16*(X_4 + Z_4)*(X_4 - Z_4)*X_4^2
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xQ.Z.Mul(&t0, &phi.Zpow4) // = Zprime * 16*(X_4 + Z_4)*(X_4 - Z_4)*X_4^2*Z_4^4
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t2.Mul(&t2, &phi.Zpow4) // = 4*(X_4*Z - X*Z_4)^2*X_4^2*Z_4^4
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t0.Mul(&t1, &phi.Xpow4) // = 4*(X*X_4 - Z*Z_4)^2*X_4^4*Z_4^2
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t0.Sub(&t2, &t0) // = -4*(X*X_4^2 - 2*X_4*Z*Z_4 + X*Z_4^2)*X*(X_4^2 - Z_4^2)*X_4^2*Z_4^2
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xQ.X.Mul(&t1, &t0) // = Xprime * 16*(X_4 + Z_4)*(X_4 - Z_4)*X_4^2*Z_4^4
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return xQ
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}
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// Represents a 4-isogeny phi. See ComputeFourIsogeny for details.
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type FirstFourIsogeny struct {
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A ExtensionFieldElement
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C ExtensionFieldElement
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}
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// Compute the "first" four-isogeny from the given curve. See also
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// ComputeFourIsogeny and Costello-Longa-Naehrig for more details.
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func ComputeFirstFourIsogeny(domain *ProjectiveCurveParameters) (ProjectiveCurveParameters, FirstFourIsogeny) {
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var codomain ProjectiveCurveParameters
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var isogeny FirstFourIsogeny
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var t0, t1 ExtensionFieldElement
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t0.Add(&domain.C, &domain.C) // = 2*C
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codomain.C.Sub(&domain.A, &t0) // = A - 2*C
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t1.Add(&t0, &t0) // = 4*C
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t1.Add(&t1, &t0) // = 6*C
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t0.Add(&t1, &domain.A) // = A + 6*C
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codomain.A.Add(&t0, &t0) // = 2*(A + 6*C)
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isogeny.A = domain.A
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isogeny.C = domain.C
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return codomain, isogeny
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}
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// Given a 4-isogeny phi and a point xP = x(P), compute x(Q), the x-coordinate
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// of the image Q = phi(P) of P under phi : E_(A:C) -> E_(A':C').
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//
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// The output xQ = x(Q) is then a point on the curve E_(A':C'); the curve
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// parameters are returned by the ComputeFirstFourIsogeny function used to construct
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// phi.
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func (phi *FirstFourIsogeny) Eval(xP *ProjectivePoint) ProjectivePoint {
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var xQ ProjectivePoint
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var t0, t1, t2, t3 ExtensionFieldElement
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t0.Add(&xP.X, &xP.Z).Square(&t0) // = (X+Z)^2
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t2.Mul(&xP.X, &xP.Z) // = X*Z
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t1.Add(&t2, &t2) // = 2*X*Z
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t1.Sub(&t0, &t1) // = X^2 + Z^2
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xQ.X.Mul(&phi.A, &t2) // = A*X*Z
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t3.Mul(&phi.C, &t1) // = C*(X^2 + Z^2)
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xQ.X.Add(&xQ.X, &t3) // = A*X*Z + C*(X^2 + Z^2)
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xQ.X.Mul(&xQ.X, &t0) // = (X+Z)^2 * (A*X*Z + C*(X^2 + Z^2))
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t0.Sub(&xP.X, &xP.Z).Square(&t0) // = (X-Z)^2
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t0.Mul(&t0, &t2) // = X*Z*(X-Z)^2
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t1.Add(&phi.C, &phi.C) // = 2*C
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t1.Sub(&t1, &phi.A) // = 2*C - A
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xQ.Z.Mul(&t1, &t0) // = (2*C - A)*X*Z*(X-Z)^2
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return xQ
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}
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