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// Copyright (c) 2019, Cloudflare Inc.
//
// Permission to use, copy, modify, and/or distribute this software for any
// purpose with or without fee is hereby granted, provided that the above
// copyright notice and this permission notice appear in all copies.
//
// THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
// WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
// MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
// SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
// WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
// OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
// CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
package sike
// Interface for working with isogenies.
type isogeny interface {
// Given a torsion point on a curve computes isogenous curve.
// Returns curve coefficients (A:C), so that E_(A/C) = E_(A/C)/<P>,
// where P is a provided projective point. Sets also isogeny constants
// that are needed for isogeny evaluation.
GenerateCurve(*ProjectivePoint) CurveCoefficientsEquiv
// Evaluates isogeny at caller provided point. Requires isogeny curve constants
// to be earlier computed by GenerateCurve.
EvaluatePoint(*ProjectivePoint) ProjectivePoint
}
// Stores isogeny 3 curve constants
type isogeny3 struct {
K1 Fp2
K2 Fp2
}
// Stores isogeny 4 curve constants
type isogeny4 struct {
isogeny3
K3 Fp2
}
// Constructs isogeny3 objects
func NewIsogeny3() isogeny {
return &isogeny3{}
}
// Constructs isogeny4 objects
func NewIsogeny4() isogeny {
return &isogeny4{}
}
// Helper function for RightToLeftLadder(). Returns A+2C / 4.
func calcAplus2Over4(cparams *ProjectiveCurveParameters) (ret Fp2) {
var tmp Fp2
// 2C
add(&tmp, &cparams.C, &cparams.C)
// A+2C
add(&ret, &cparams.A, &tmp)
// 1/4C
add(&tmp, &tmp, &tmp)
inv(&tmp, &tmp)
// A+2C/4C
mul(&ret, &ret, &tmp)
return
}
// Converts values in x.A and x.B to Montgomery domain
// x.A = x.A * R mod p
// x.B = x.B * R mod p
// Performs v = v*R^2*R^(-1) mod p, for both x.A and x.B
func toMontDomain(x *Fp2) {
var aRR FpX2
// convert to montgomery domain
fpMul(&aRR, &x.A, &R2) // = a*R*R
fpMontRdc(&x.A, &aRR) // = a*R mod p
fpMul(&aRR, &x.B, &R2)
fpMontRdc(&x.B, &aRR)
}
// Converts values in x.A and x.B from Montgomery domain
// a = x.A mod p
// b = x.B mod p
//
// After returning from the call x is not modified.
func fromMontDomain(x *Fp2, out *Fp2) {
var aR FpX2
// convert from montgomery domain
copy(aR[:], x.A[:])
fpMontRdc(&out.A, &aR) // = a mod p in [0, 2p)
fpRdcP(&out.A) // = a mod p in [0, p)
for i := range aR {
aR[i] = 0
}
copy(aR[:], x.B[:])
fpMontRdc(&out.B, &aR)
fpRdcP(&out.B)
}
// Computes j-invariant for a curve y2=x3+A/Cx+x with A,C in F_(p^2). Result
// is returned in 'j'. Implementation corresponds to Algorithm 9 from SIKE.
func Jinvariant(cparams *ProjectiveCurveParameters, j *Fp2) {
var t0, t1 Fp2
sqr(j, &cparams.A) // j = A^2
sqr(&t1, &cparams.C) // t1 = C^2
add(&t0, &t1, &t1) // t0 = t1 + t1
sub(&t0, j, &t0) // t0 = j - t0
sub(&t0, &t0, &t1) // t0 = t0 - t1
sub(j, &t0, &t1) // t0 = t0 - t1
sqr(&t1, &t1) // t1 = t1^2
mul(j, j, &t1) // j = j * t1
add(&t0, &t0, &t0) // t0 = t0 + t0
add(&t0, &t0, &t0) // t0 = t0 + t0
sqr(&t1, &t0) // t1 = t0^2
mul(&t0, &t0, &t1) // t0 = t0 * t1
add(&t0, &t0, &t0) // t0 = t0 + t0
add(&t0, &t0, &t0) // t0 = t0 + t0
inv(j, j) // j = 1/j
mul(j, &t0, j) // j = t0 * j
}
// Given affine points x(P), x(Q) and x(Q-P) in a extension field F_{p^2}, function
// recorvers projective coordinate A of a curve. This is Algorithm 10 from SIKE.
func RecoverCoordinateA(curve *ProjectiveCurveParameters, xp, xq, xr *Fp2) {
var t0, t1 Fp2
add(&t1, xp, xq) // t1 = Xp + Xq
mul(&t0, xp, xq) // t0 = Xp * Xq
mul(&curve.A, xr, &t1) // A = X(q-p) * t1
add(&curve.A, &curve.A, &t0) // A = A + t0
mul(&t0, &t0, xr) // t0 = t0 * X(q-p)
sub(&curve.A, &curve.A, &Params.OneFp2) // A = A - 1
add(&t0, &t0, &t0) // t0 = t0 + t0
add(&t1, &t1, xr) // t1 = t1 + X(q-p)
add(&t0, &t0, &t0) // t0 = t0 + t0
sqr(&curve.A, &curve.A) // A = A^2
inv(&t0, &t0) // t0 = 1/t0
mul(&curve.A, &curve.A, &t0) // A = A * t0
sub(&curve.A, &curve.A, &t1) // A = A - t1
}
// Computes equivalence (A:C) ~ (A+2C : A-2C)
func CalcCurveParamsEquiv3(cparams *ProjectiveCurveParameters) CurveCoefficientsEquiv {
var coef CurveCoefficientsEquiv
var c2 Fp2
add(&c2, &cparams.C, &cparams.C)
// A24p = A+2*C
add(&coef.A, &cparams.A, &c2)
// A24m = A-2*C
sub(&coef.C, &cparams.A, &c2)
return coef
}
// Computes equivalence (A:C) ~ (A+2C : 4C)
func CalcCurveParamsEquiv4(cparams *ProjectiveCurveParameters) CurveCoefficientsEquiv {
var coefEq CurveCoefficientsEquiv
add(&coefEq.C, &cparams.C, &cparams.C)
// A24p = A+2C
add(&coefEq.A, &cparams.A, &coefEq.C)
// C24 = 4*C
add(&coefEq.C, &coefEq.C, &coefEq.C)
return coefEq
}
// Recovers (A:C) curve parameters from projectively equivalent (A+2C:A-2C).
func RecoverCurveCoefficients3(cparams *ProjectiveCurveParameters, coefEq *CurveCoefficientsEquiv) {
add(&cparams.A, &coefEq.A, &coefEq.C)
// cparams.A = 2*(A+2C+A-2C) = 4A
add(&cparams.A, &cparams.A, &cparams.A)
// cparams.C = (A+2C-A+2C) = 4C
sub(&cparams.C, &coefEq.A, &coefEq.C)
return
}
// Recovers (A:C) curve parameters from projectively equivalent (A+2C:4C).
func RecoverCurveCoefficients4(cparams *ProjectiveCurveParameters, coefEq *CurveCoefficientsEquiv) {
// cparams.C = (4C)*1/2=2C
mul(&cparams.C, &coefEq.C, &Params.HalfFp2)
// cparams.A = A+2C - 2C = A
sub(&cparams.A, &coefEq.A, &cparams.C)
// cparams.C = 2C * 1/2 = C
mul(&cparams.C, &cparams.C, &Params.HalfFp2)
return
}
// Combined coordinate doubling and differential addition. Takes projective points
// P,Q,Q-P and (A+2C)/4C curve E coefficient. Returns 2*P and P+Q calculated on E.
// Function is used only by RightToLeftLadder. Corresponds to Algorithm 5 of SIKE
func xDbladd(P, Q, QmP *ProjectivePoint, a24 *Fp2) (dblP, PaQ ProjectivePoint) {
var t0, t1, t2 Fp2
xQmP, zQmP := &QmP.X, &QmP.Z
xPaQ, zPaQ := &PaQ.X, &PaQ.Z
x2P, z2P := &dblP.X, &dblP.Z
xP, zP := &P.X, &P.Z
xQ, zQ := &Q.X, &Q.Z
add(&t0, xP, zP) // t0 = Xp+Zp
sub(&t1, xP, zP) // t1 = Xp-Zp
sqr(x2P, &t0) // 2P.X = t0^2
sub(&t2, xQ, zQ) // t2 = Xq-Zq
add(xPaQ, xQ, zQ) // Xp+q = Xq+Zq
mul(&t0, &t0, &t2) // t0 = t0 * t2
mul(z2P, &t1, &t1) // 2P.Z = t1 * t1
mul(&t1, &t1, xPaQ) // t1 = t1 * Xp+q
sub(&t2, x2P, z2P) // t2 = 2P.X - 2P.Z
mul(x2P, x2P, z2P) // 2P.X = 2P.X * 2P.Z
mul(xPaQ, a24, &t2) // Xp+q = A24 * t2
sub(zPaQ, &t0, &t1) // Zp+q = t0 - t1
add(z2P, xPaQ, z2P) // 2P.Z = Xp+q + 2P.Z
add(xPaQ, &t0, &t1) // Xp+q = t0 + t1
mul(z2P, z2P, &t2) // 2P.Z = 2P.Z * t2
sqr(zPaQ, zPaQ) // Zp+q = Zp+q ^ 2
sqr(xPaQ, xPaQ) // Xp+q = Xp+q ^ 2
mul(zPaQ, xQmP, zPaQ) // Zp+q = Xq-p * Zp+q
mul(xPaQ, zQmP, xPaQ) // Xp+q = Zq-p * Xp+q
return
}
// Given the curve parameters, xP = x(P), computes xP = x([2^k]P)
// Safe to overlap xP, x2P.
func Pow2k(xP *ProjectivePoint, params *CurveCoefficientsEquiv, k uint32) {
var t0, t1 Fp2
x, z := &xP.X, &xP.Z
for i := uint32(0); i < k; i++ {
sub(&t0, x, z) // t0 = Xp - Zp
add(&t1, x, z) // t1 = Xp + Zp
sqr(&t0, &t0) // t0 = t0 ^ 2
sqr(&t1, &t1) // t1 = t1 ^ 2
mul(z, &params.C, &t0) // Z2p = C24 * t0
mul(x, z, &t1) // X2p = Z2p * t1
sub(&t1, &t1, &t0) // t1 = t1 - t0
mul(&t0, &params.A, &t1) // t0 = A24+ * t1
add(z, z, &t0) // Z2p = Z2p + t0
mul(z, z, &t1) // Zp = Z2p * t1
}
}
// Given the curve parameters, xP = x(P), and k >= 0, compute xP = x([3^k]P).
//
// Safe to overlap xP, xR.
func Pow3k(xP *ProjectivePoint, params *CurveCoefficientsEquiv, k uint32) {
var t0, t1, t2, t3, t4, t5, t6 Fp2
x, z := &xP.X, &xP.Z
for i := uint32(0); i < k; i++ {
sub(&t0, x, z) // t0 = Xp - Zp
sqr(&t2, &t0) // t2 = t0^2
add(&t1, x, z) // t1 = Xp + Zp
sqr(&t3, &t1) // t3 = t1^2
add(&t4, &t1, &t0) // t4 = t1 + t0
sub(&t0, &t1, &t0) // t0 = t1 - t0
sqr(&t1, &t4) // t1 = t4^2
sub(&t1, &t1, &t3) // t1 = t1 - t3
sub(&t1, &t1, &t2) // t1 = t1 - t2
mul(&t5, &t3, &params.A) // t5 = t3 * A24+
mul(&t3, &t3, &t5) // t3 = t5 * t3
mul(&t6, &t2, &params.C) // t6 = t2 * A24-
mul(&t2, &t2, &t6) // t2 = t2 * t6
sub(&t3, &t2, &t3) // t3 = t2 - t3
sub(&t2, &t5, &t6) // t2 = t5 - t6
mul(&t1, &t2, &t1) // t1 = t2 * t1
add(&t2, &t3, &t1) // t2 = t3 + t1
sqr(&t2, &t2) // t2 = t2^2
mul(x, &t2, &t4) // X3p = t2 * t4
sub(&t1, &t3, &t1) // t1 = t3 - t1
sqr(&t1, &t1) // t1 = t1^2
mul(z, &t1, &t0) // Z3p = t1 * t0
}
}
// Set (y1, y2, y3) = (1/x1, 1/x2, 1/x3).
//
// All xi, yi must be distinct.
func Fp2Batch3Inv(x1, x2, x3, y1, y2, y3 *Fp2) {
var x1x2, t Fp2
mul(&x1x2, x1, x2) // x1*x2
mul(&t, &x1x2, x3) // 1/(x1*x2*x3)
inv(&t, &t)
mul(y1, &t, x2) // 1/x1
mul(y1, y1, x3)
mul(y2, &t, x1) // 1/x2
mul(y2, y2, x3)
mul(y3, &t, &x1x2) // 1/x3
}
// ScalarMul3Pt is a right-to-left point multiplication that given the
// x-coordinate of P, Q and P-Q calculates the x-coordinate of R=Q+[scalar]P.
// nbits must be smaller or equal to len(scalar).
func ScalarMul3Pt(cparams *ProjectiveCurveParameters, P, Q, PmQ *ProjectivePoint, nbits uint, scalar []uint8) ProjectivePoint {
var R0, R2, R1 ProjectivePoint
aPlus2Over4 := calcAplus2Over4(cparams)
R1 = *P
R2 = *PmQ
R0 = *Q
// Iterate over the bits of the scalar, bottom to top
prevBit := uint8(0)
for i := uint(0); i < nbits; i++ {
bit := (scalar[i>>3] >> (i & 7) & 1)
swap := prevBit ^ bit
prevBit = bit
condSwap(&R1.X, &R1.Z, &R2.X, &R2.Z, swap)
R0, R2 = xDbladd(&R0, &R2, &R1, &aPlus2Over4)
}
condSwap(&R1.X, &R1.Z, &R2.X, &R2.Z, prevBit)
return R1
}
// Given a three-torsion point p = x(PB) on the curve E_(A:C), construct the
// three-isogeny phi : E_(A:C) -> E_(A:C)/<P_3> = E_(A':C').
//
// Input: (XP_3: ZP_3), where P_3 has exact order 3 on E_A/C
// Output: * Curve coordinates (A' + 2C', A' - 2C') corresponding to E_A'/C' = A_E/C/<P3>
// * isogeny phi with constants in F_p^2
func (phi *isogeny3) GenerateCurve(p *ProjectivePoint) CurveCoefficientsEquiv {
var t0, t1, t2, t3, t4 Fp2
var coefEq CurveCoefficientsEquiv
var K1, K2 = &phi.K1, &phi.K2
sub(K1, &p.X, &p.Z) // K1 = XP3 - ZP3
sqr(&t0, K1) // t0 = K1^2
add(K2, &p.X, &p.Z) // K2 = XP3 + ZP3
sqr(&t1, K2) // t1 = K2^2
add(&t2, &t0, &t1) // t2 = t0 + t1
add(&t3, K1, K2) // t3 = K1 + K2
sqr(&t3, &t3) // t3 = t3^2
sub(&t3, &t3, &t2) // t3 = t3 - t2
add(&t2, &t1, &t3) // t2 = t1 + t3
add(&t3, &t3, &t0) // t3 = t3 + t0
add(&t4, &t3, &t0) // t4 = t3 + t0
add(&t4, &t4, &t4) // t4 = t4 + t4
add(&t4, &t1, &t4) // t4 = t1 + t4
mul(&coefEq.C, &t2, &t4) // A24m = t2 * t4
add(&t4, &t1, &t2) // t4 = t1 + t2
add(&t4, &t4, &t4) // t4 = t4 + t4
add(&t4, &t0, &t4) // t4 = t0 + t4
mul(&t4, &t3, &t4) // t4 = t3 * t4
sub(&t0, &t4, &coefEq.C) // t0 = t4 - A24m
add(&coefEq.A, &coefEq.C, &t0) // A24p = A24m + t0
return coefEq
}
// Given a 3-isogeny phi and a point pB = x(PB), compute x(QB), the x-coordinate
// of the image QB = phi(PB) of PB under phi : E_(A:C) -> E_(A':C').
//
// The output xQ = x(Q) is then a point on the curve E_(A':C'); the curve
// parameters are returned by the GenerateCurve function used to construct phi.
func (phi *isogeny3) EvaluatePoint(p *ProjectivePoint) ProjectivePoint {
var t0, t1, t2 Fp2
var q ProjectivePoint
var K1, K2 = &phi.K1, &phi.K2
var px, pz = &p.X, &p.Z
add(&t0, px, pz) // t0 = XQ + ZQ
sub(&t1, px, pz) // t1 = XQ - ZQ
mul(&t0, K1, &t0) // t2 = K1 * t0
mul(&t1, K2, &t1) // t1 = K2 * t1
add(&t2, &t0, &t1) // t2 = t0 + t1
sub(&t0, &t1, &t0) // t0 = t1 - t0
sqr(&t2, &t2) // t2 = t2 ^ 2
sqr(&t0, &t0) // t0 = t0 ^ 2
mul(&q.X, px, &t2) // XQ'= XQ * t2
mul(&q.Z, pz, &t0) // ZQ'= ZQ * t0
return q
}
// Given a four-torsion point p = x(PB) on the curve E_(A:C), construct the
// four-isogeny phi : E_(A:C) -> E_(A:C)/<P_4> = E_(A':C').
//
// Input: (XP_4: ZP_4), where P_4 has exact order 4 on E_A/C
// Output: * Curve coordinates (A' + 2C', 4C') corresponding to E_A'/C' = A_E/C/<P4>
// * isogeny phi with constants in F_p^2
func (phi *isogeny4) GenerateCurve(p *ProjectivePoint) CurveCoefficientsEquiv {
var coefEq CurveCoefficientsEquiv
var xp4, zp4 = &p.X, &p.Z
var K1, K2, K3 = &phi.K1, &phi.K2, &phi.K3
sub(K2, xp4, zp4)
add(K3, xp4, zp4)
sqr(K1, zp4)
add(K1, K1, K1)
sqr(&coefEq.C, K1)
add(K1, K1, K1)
sqr(&coefEq.A, xp4)
add(&coefEq.A, &coefEq.A, &coefEq.A)
sqr(&coefEq.A, &coefEq.A)
return coefEq
}
// Given a 4-isogeny phi and a point xP = x(P), compute x(Q), the x-coordinate
// of the image Q = phi(P) of P under phi : E_(A:C) -> E_(A':C').
//
// Input: isogeny returned by GenerateCurve and point q=(Qx,Qz) from E0_A/C
// Output: Corresponding point q from E1_A'/C', where E1 is 4-isogenous to E0
func (phi *isogeny4) EvaluatePoint(p *ProjectivePoint) ProjectivePoint {
var t0, t1 Fp2
var q = *p
var xq, zq = &q.X, &q.Z
var K1, K2, K3 = &phi.K1, &phi.K2, &phi.K3
add(&t0, xq, zq)
sub(&t1, xq, zq)
mul(xq, &t0, K2)
mul(zq, &t1, K3)
mul(&t0, &t0, &t1)
mul(&t0, &t0, K1)
add(&t1, xq, zq)
sub(zq, xq, zq)
sqr(&t1, &t1)
sqr(zq, zq)
add(xq, &t0, &t1)
sub(&t0, zq, &t0)
mul(xq, xq, &t1)
mul(zq, zq, &t0)
return q
}