mirror of
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441 lines
15 KiB
Go
441 lines
15 KiB
Go
package internal
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type CurveOperations struct {
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Params *SidhParams
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}
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// Computes j-invariant for a curve y2=x3+A/Cx+x with A,C in F_(p^2). Result
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// is returned in jBytes buffer, encoded in little-endian format. Caller
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// provided jBytes buffer has to be big enough to j-invariant value. In case
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// of SIDH, buffer size must be at least size of shared secret.
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// Implementation corresponds to Algorithm 9 from SIKE.
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func (c *CurveOperations) Jinvariant(cparams *ProjectiveCurveParameters, jBytes []byte) {
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var j, t0, t1 Fp2Element
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op := c.Params.Op
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op.Square(&j, &cparams.A) // j = A^2
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op.Square(&t1, &cparams.C) // t1 = C^2
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op.Add(&t0, &t1, &t1) // t0 = t1 + t1
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op.Sub(&t0, &j, &t0) // t0 = j - t0
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op.Sub(&t0, &t0, &t1) // t0 = t0 - t1
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op.Sub(&j, &t0, &t1) // t0 = t0 - t1
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op.Square(&t1, &t1) // t1 = t1^2
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op.Mul(&j, &j, &t1) // j = j * t1
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op.Add(&t0, &t0, &t0) // t0 = t0 + t0
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op.Add(&t0, &t0, &t0) // t0 = t0 + t0
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op.Square(&t1, &t0) // t1 = t0^2
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op.Mul(&t0, &t0, &t1) // t0 = t0 * t1
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op.Add(&t0, &t0, &t0) // t0 = t0 + t0
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op.Add(&t0, &t0, &t0) // t0 = t0 + t0
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op.Inv(&j, &j) // j = 1/j
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op.Mul(&j, &t0, &j) // j = t0 * j
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c.Fp2ToBytes(jBytes, &j)
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}
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// Given affine points x(P), x(Q) and x(Q-P) in a extension field F_{p^2}, function
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// recorvers projective coordinate A of a curve. This is Algorithm 10 from SIKE.
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func (c *CurveOperations) RecoverCoordinateA(curve *ProjectiveCurveParameters, xp, xq, xr *Fp2Element) {
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var t0, t1 Fp2Element
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op := c.Params.Op
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op.Add(&t1, xp, xq) // t1 = Xp + Xq
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op.Mul(&t0, xp, xq) // t0 = Xp * Xq
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op.Mul(&curve.A, xr, &t1) // A = X(q-p) * t1
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op.Add(&curve.A, &curve.A, &t0) // A = A + t0
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op.Mul(&t0, &t0, xr) // t0 = t0 * X(q-p)
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op.Sub(&curve.A, &curve.A, &c.Params.OneFp2) // A = A - 1
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op.Add(&t0, &t0, &t0) // t0 = t0 + t0
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op.Add(&t1, &t1, xr) // t1 = t1 + X(q-p)
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op.Add(&t0, &t0, &t0) // t0 = t0 + t0
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op.Square(&curve.A, &curve.A) // A = A^2
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op.Inv(&t0, &t0) // t0 = 1/t0
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op.Mul(&curve.A, &curve.A, &t0) // A = A * t0
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op.Sub(&curve.A, &curve.A, &t1) // A = A - t1
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}
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// Computes equivalence (A:C) ~ (A+2C : A-2C)
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func (c *CurveOperations) CalcCurveParamsEquiv3(cparams *ProjectiveCurveParameters) CurveCoefficientsEquiv {
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var coef CurveCoefficientsEquiv
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var c2 Fp2Element
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var op = c.Params.Op
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op.Add(&c2, &cparams.C, &cparams.C)
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// A24p = A+2*C
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op.Add(&coef.A, &cparams.A, &c2)
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// A24m = A-2*C
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op.Sub(&coef.C, &cparams.A, &c2)
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return coef
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}
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// Computes equivalence (A:C) ~ (A+2C : 4C)
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func (c *CurveOperations) CalcCurveParamsEquiv4(cparams *ProjectiveCurveParameters) CurveCoefficientsEquiv {
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var coefEq CurveCoefficientsEquiv
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var op = c.Params.Op
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op.Add(&coefEq.C, &cparams.C, &cparams.C)
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// A24p = A+2C
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op.Add(&coefEq.A, &cparams.A, &coefEq.C)
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// C24 = 4*C
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op.Add(&coefEq.C, &coefEq.C, &coefEq.C)
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return coefEq
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}
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// Helper function for RightToLeftLadder(). Returns A+2C / 4.
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func (c *CurveOperations) CalcAplus2Over4(cparams *ProjectiveCurveParameters) (ret Fp2Element) {
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var tmp Fp2Element
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var op = c.Params.Op
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// 2C
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op.Add(&tmp, &cparams.C, &cparams.C)
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// A+2C
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op.Add(&ret, &cparams.A, &tmp)
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// 1/4C
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op.Add(&tmp, &tmp, &tmp)
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op.Inv(&tmp, &tmp)
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// A+2C/4C
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op.Mul(&ret, &ret, &tmp)
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return
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}
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// Recovers (A:C) curve parameters from projectively equivalent (A+2C:A-2C).
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func (c *CurveOperations) RecoverCurveCoefficients3(cparams *ProjectiveCurveParameters, coefEq *CurveCoefficientsEquiv) {
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var op = c.Params.Op
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op.Add(&cparams.A, &coefEq.A, &coefEq.C)
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// cparams.A = 2*(A+2C+A-2C) = 4A
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op.Add(&cparams.A, &cparams.A, &cparams.A)
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// cparams.C = (A+2C-A+2C) = 4C
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op.Sub(&cparams.C, &coefEq.A, &coefEq.C)
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return
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}
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// Recovers (A:C) curve parameters from projectively equivalent (A+2C:4C).
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func (c *CurveOperations) RecoverCurveCoefficients4(cparams *ProjectiveCurveParameters, coefEq *CurveCoefficientsEquiv) {
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var op = c.Params.Op
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// cparams.C = (4C)*1/2=2C
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op.Mul(&cparams.C, &coefEq.C, &c.Params.HalfFp2)
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// cparams.A = A+2C - 2C = A
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op.Sub(&cparams.A, &coefEq.A, &cparams.C)
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// cparams.C = 2C * 1/2 = C
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op.Mul(&cparams.C, &cparams.C, &c.Params.HalfFp2)
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return
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}
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// Combined coordinate doubling and differential addition. Takes projective points
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// P,Q,Q-P and (A+2C)/4C curve E coefficient. Returns 2*P and P+Q calculated on E.
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// Function is used only by RightToLeftLadder. Corresponds to Algorithm 5 of SIKE
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func (c *CurveOperations) xDblAdd(P, Q, QmP *ProjectivePoint, a24 *Fp2Element) (dblP, PaQ ProjectivePoint) {
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var t0, t1, t2 Fp2Element
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var op = c.Params.Op
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xQmP, zQmP := &QmP.X, &QmP.Z
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xPaQ, zPaQ := &PaQ.X, &PaQ.Z
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x2P, z2P := &dblP.X, &dblP.Z
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xP, zP := &P.X, &P.Z
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xQ, zQ := &Q.X, &Q.Z
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op.Add(&t0, xP, zP) // t0 = Xp+Zp
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op.Sub(&t1, xP, zP) // t1 = Xp-Zp
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op.Square(x2P, &t0) // 2P.X = t0^2
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op.Sub(&t2, xQ, zQ) // t2 = Xq-Zq
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op.Add(xPaQ, xQ, zQ) // Xp+q = Xq+Zq
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op.Mul(&t0, &t0, &t2) // t0 = t0 * t2
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op.Mul(z2P, &t1, &t1) // 2P.Z = t1 * t1
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op.Mul(&t1, &t1, xPaQ) // t1 = t1 * Xp+q
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op.Sub(&t2, x2P, z2P) // t2 = 2P.X - 2P.Z
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op.Mul(x2P, x2P, z2P) // 2P.X = 2P.X * 2P.Z
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op.Mul(xPaQ, a24, &t2) // Xp+q = A24 * t2
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op.Sub(zPaQ, &t0, &t1) // Zp+q = t0 - t1
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op.Add(z2P, xPaQ, z2P) // 2P.Z = Xp+q + 2P.Z
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op.Add(xPaQ, &t0, &t1) // Xp+q = t0 + t1
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op.Mul(z2P, z2P, &t2) // 2P.Z = 2P.Z * t2
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op.Square(zPaQ, zPaQ) // Zp+q = Zp+q ^ 2
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op.Square(xPaQ, xPaQ) // Xp+q = Xp+q ^ 2
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op.Mul(zPaQ, xQmP, zPaQ) // Zp+q = Xq-p * Zp+q
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op.Mul(xPaQ, zQmP, xPaQ) // Xp+q = Zq-p * Xp+q
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return
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}
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// Given the curve parameters, xP = x(P), computes xP = x([2^k]P)
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// Safe to overlap xP, x2P.
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func (c *CurveOperations) Pow2k(xP *ProjectivePoint, params *CurveCoefficientsEquiv, k uint32) {
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var t0, t1 Fp2Element
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var op = c.Params.Op
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x, z := &xP.X, &xP.Z
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for i := uint32(0); i < k; i++ {
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op.Sub(&t0, x, z) // t0 = Xp - Zp
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op.Add(&t1, x, z) // t1 = Xp + Zp
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op.Square(&t0, &t0) // t0 = t0 ^ 2
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op.Square(&t1, &t1) // t1 = t1 ^ 2
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op.Mul(z, ¶ms.C, &t0) // Z2p = C24 * t0
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op.Mul(x, z, &t1) // X2p = Z2p * t1
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op.Sub(&t1, &t1, &t0) // t1 = t1 - t0
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op.Mul(&t0, ¶ms.A, &t1) // t0 = A24+ * t1
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op.Add(z, z, &t0) // Z2p = Z2p + t0
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op.Mul(z, z, &t1) // Zp = Z2p * t1
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}
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}
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// Given the curve parameters, xP = x(P), and k >= 0, compute xP = x([3^k]P).
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//
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// Safe to overlap xP, xR.
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func (c *CurveOperations) Pow3k(xP *ProjectivePoint, params *CurveCoefficientsEquiv, k uint32) {
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var t0, t1, t2, t3, t4, t5, t6 Fp2Element
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var op = c.Params.Op
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x, z := &xP.X, &xP.Z
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for i := uint32(0); i < k; i++ {
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op.Sub(&t0, x, z) // t0 = Xp - Zp
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op.Square(&t2, &t0) // t2 = t0^2
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op.Add(&t1, x, z) // t1 = Xp + Zp
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op.Square(&t3, &t1) // t3 = t1^2
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op.Add(&t4, &t1, &t0) // t4 = t1 + t0
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op.Sub(&t0, &t1, &t0) // t0 = t1 - t0
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op.Square(&t1, &t4) // t1 = t4^2
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op.Sub(&t1, &t1, &t3) // t1 = t1 - t3
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op.Sub(&t1, &t1, &t2) // t1 = t1 - t2
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op.Mul(&t5, &t3, ¶ms.A) // t5 = t3 * A24+
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op.Mul(&t3, &t3, &t5) // t3 = t5 * t3
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op.Mul(&t6, &t2, ¶ms.C) // t6 = t2 * A24-
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op.Mul(&t2, &t2, &t6) // t2 = t2 * t6
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op.Sub(&t3, &t2, &t3) // t3 = t2 - t3
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op.Sub(&t2, &t5, &t6) // t2 = t5 - t6
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op.Mul(&t1, &t2, &t1) // t1 = t2 * t1
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op.Add(&t2, &t3, &t1) // t2 = t3 + t1
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op.Square(&t2, &t2) // t2 = t2^2
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op.Mul(x, &t2, &t4) // X3p = t2 * t4
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op.Sub(&t1, &t3, &t1) // t1 = t3 - t1
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op.Square(&t1, &t1) // t1 = t1^2
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op.Mul(z, &t1, &t0) // Z3p = t1 * t0
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}
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}
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// Set (y1, y2, y3) = (1/x1, 1/x2, 1/x3).
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//
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// All xi, yi must be distinct.
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func (c *CurveOperations) Fp2Batch3Inv(x1, x2, x3, y1, y2, y3 *Fp2Element) {
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var x1x2, t Fp2Element
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var op = c.Params.Op
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op.Mul(&x1x2, x1, x2) // x1*x2
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op.Mul(&t, &x1x2, x3) // 1/(x1*x2*x3)
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op.Inv(&t, &t)
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op.Mul(y1, &t, x2) // 1/x1
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op.Mul(y1, y1, x3)
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op.Mul(y2, &t, x1) // 1/x2
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op.Mul(y2, y2, x3)
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op.Mul(y3, &t, &x1x2) // 1/x3
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}
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// ScalarMul3Pt is a right-to-left point multiplication that given the
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// x-coordinate of P, Q and P-Q calculates the x-coordinate of R=Q+[scalar]P.
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// nbits must be smaller or equal to len(scalar).
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func (c *CurveOperations) ScalarMul3Pt(cparams *ProjectiveCurveParameters, P, Q, PmQ *ProjectivePoint, nbits uint, scalar []uint8) ProjectivePoint {
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var R0, R2, R1 ProjectivePoint
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var op = c.Params.Op
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aPlus2Over4 := c.CalcAplus2Over4(cparams)
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R1 = *P
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R2 = *PmQ
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R0 = *Q
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// Iterate over the bits of the scalar, bottom to top
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prevBit := uint8(0)
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for i := uint(0); i < nbits; i++ {
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bit := (scalar[i>>3] >> (i & 7) & 1)
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swap := prevBit ^ bit
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prevBit = bit
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op.CondSwap(&R1.X, &R1.Z, &R2.X, &R2.Z, swap)
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R0, R2 = c.xDblAdd(&R0, &R2, &R1, &aPlus2Over4)
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}
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op.CondSwap(&R1.X, &R1.Z, &R2.X, &R2.Z, prevBit)
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return R1
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}
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// Convert the input to wire format.
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//
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// The output byte slice must be at least 2*bytelen(p) bytes long.
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func (c *CurveOperations) Fp2ToBytes(output []byte, fp2 *Fp2Element) {
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if len(output) < 2*c.Params.Bytelen {
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panic("output byte slice too short")
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}
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var a Fp2Element
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c.Params.Op.FromMontgomery(fp2, &a)
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// convert to bytes in little endian form
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for i := 0; i < c.Params.Bytelen; i++ {
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// set i = j*8 + k
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fp2 := i / 8
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k := uint64(i % 8)
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output[i] = byte(a.A[fp2] >> (8 * k))
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output[i+c.Params.Bytelen] = byte(a.B[fp2] >> (8 * k))
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}
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}
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// Read 2*bytelen(p) bytes into the given ExtensionFieldElement.
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//
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// It is an error to call this function if the input byte slice is less than 2*bytelen(p) bytes long.
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func (c *CurveOperations) Fp2FromBytes(fp2 *Fp2Element, input []byte) {
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if len(input) < 2*c.Params.Bytelen {
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panic("input byte slice too short")
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}
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for i := 0; i < c.Params.Bytelen; i++ {
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j := i / 8
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k := uint64(i % 8)
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fp2.A[j] |= uint64(input[i]) << (8 * k)
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fp2.B[j] |= uint64(input[i+c.Params.Bytelen]) << (8 * k)
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}
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c.Params.Op.ToMontgomery(fp2)
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}
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/* -------------------------------------------------------------------------
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Mechnisms used for isogeny calculations
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-------------------------------------------------------------------------*/
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// Constructs isogeny3 objects
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func Newisogeny3(op FieldOps) Isogeny {
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return &isogeny3{Field: op}
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}
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// Constructs isogeny4 objects
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func Newisogeny4(op FieldOps) Isogeny {
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return &isogeny4{isogeny3: isogeny3{Field: op}}
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}
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// Given a three-torsion point p = x(PB) 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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// Input: (XP_3: ZP_3), where P_3 has exact order 3 on E_A/C
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// Output: * Curve coordinates (A' + 2C', A' - 2C') corresponding to E_A'/C' = A_E/C/<P3>
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// * Isogeny phi with constants in F_p^2
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func (phi *isogeny3) GenerateCurve(p *ProjectivePoint) CurveCoefficientsEquiv {
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var t0, t1, t2, t3, t4 Fp2Element
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var coefEq CurveCoefficientsEquiv
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var K1, K2 = &phi.K1, &phi.K2
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op := phi.Field
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op.Sub(K1, &p.X, &p.Z) // K1 = XP3 - ZP3
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op.Square(&t0, K1) // t0 = K1^2
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op.Add(K2, &p.X, &p.Z) // K2 = XP3 + ZP3
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op.Square(&t1, K2) // t1 = K2^2
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op.Add(&t2, &t0, &t1) // t2 = t0 + t1
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op.Add(&t3, K1, K2) // t3 = K1 + K2
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op.Square(&t3, &t3) // t3 = t3^2
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op.Sub(&t3, &t3, &t2) // t3 = t3 - t2
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op.Add(&t2, &t1, &t3) // t2 = t1 + t3
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op.Add(&t3, &t3, &t0) // t3 = t3 + t0
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op.Add(&t4, &t3, &t0) // t4 = t3 + t0
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op.Add(&t4, &t4, &t4) // t4 = t4 + t4
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op.Add(&t4, &t1, &t4) // t4 = t1 + t4
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op.Mul(&coefEq.C, &t2, &t4) // A24m = t2 * t4
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op.Add(&t4, &t1, &t2) // t4 = t1 + t2
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op.Add(&t4, &t4, &t4) // t4 = t4 + t4
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op.Add(&t4, &t0, &t4) // t4 = t0 + t4
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op.Mul(&t4, &t3, &t4) // t4 = t3 * t4
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op.Sub(&t0, &t4, &coefEq.C) // t0 = t4 - A24m
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op.Add(&coefEq.A, &coefEq.C, &t0) // A24p = A24m + t0
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return coefEq
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}
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// Given a 3-isogeny phi and a point pB = x(PB), compute x(QB), the x-coordinate
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// of the image QB = phi(PB) of PB 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 GenerateCurve function used to construct phi.
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func (phi *isogeny3) EvaluatePoint(p *ProjectivePoint) ProjectivePoint {
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var t0, t1, t2 Fp2Element
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var q ProjectivePoint
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var K1, K2 = &phi.K1, &phi.K2
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var px, pz = &p.X, &p.Z
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op := phi.Field
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op.Add(&t0, px, pz) // t0 = XQ + ZQ
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op.Sub(&t1, px, pz) // t1 = XQ - ZQ
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op.Mul(&t0, K1, &t0) // t2 = K1 * t0
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op.Mul(&t1, K2, &t1) // t1 = K2 * t1
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op.Add(&t2, &t0, &t1) // t2 = t0 + t1
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op.Sub(&t0, &t1, &t0) // t0 = t1 - t0
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op.Square(&t2, &t2) // t2 = t2 ^ 2
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op.Square(&t0, &t0) // t0 = t0 ^ 2
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op.Mul(&q.X, px, &t2) // XQ'= XQ * t2
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op.Mul(&q.Z, pz, &t0) // ZQ'= ZQ * t0
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return q
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}
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|
|
|
// Given a four-torsion point p = x(PB) on the curve E_(A:C), construct the
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// four-isogeny phi : E_(A:C) -> E_(A:C)/<P_4> = E_(A':C').
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//
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// Input: (XP_4: ZP_4), where P_4 has exact order 4 on E_A/C
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// Output: * Curve coordinates (A' + 2C', 4C') corresponding to E_A'/C' = A_E/C/<P4>
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// * Isogeny phi with constants in F_p^2
|
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func (phi *isogeny4) GenerateCurve(p *ProjectivePoint) CurveCoefficientsEquiv {
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|
var coefEq CurveCoefficientsEquiv
|
|
var xp4, zp4 = &p.X, &p.Z
|
|
var K1, K2, K3 = &phi.K1, &phi.K2, &phi.K3
|
|
|
|
op := phi.Field
|
|
op.Sub(K2, xp4, zp4)
|
|
op.Add(K3, xp4, zp4)
|
|
op.Square(K1, zp4)
|
|
op.Add(K1, K1, K1)
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|
op.Square(&coefEq.C, K1)
|
|
op.Add(K1, K1, K1)
|
|
op.Square(&coefEq.A, xp4)
|
|
op.Add(&coefEq.A, &coefEq.A, &coefEq.A)
|
|
op.Square(&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 Fp2Element
|
|
var q = *p
|
|
var xq, zq = &q.X, &q.Z
|
|
var K1, K2, K3 = &phi.K1, &phi.K2, &phi.K3
|
|
|
|
op := phi.Field
|
|
op.Add(&t0, xq, zq)
|
|
op.Sub(&t1, xq, zq)
|
|
op.Mul(xq, &t0, K2)
|
|
op.Mul(zq, &t1, K3)
|
|
op.Mul(&t0, &t0, &t1)
|
|
op.Mul(&t0, &t0, K1)
|
|
op.Add(&t1, xq, zq)
|
|
op.Sub(zq, xq, zq)
|
|
op.Square(&t1, &t1)
|
|
op.Square(zq, zq)
|
|
op.Add(xq, &t0, &t1)
|
|
op.Sub(&t0, zq, &t0)
|
|
op.Mul(xq, xq, &t1)
|
|
op.Mul(zq, zq, &t0)
|
|
return q
|
|
}
|
|
|
|
/* -------------------------------------------------------------------------
|
|
Utils
|
|
-------------------------------------------------------------------------*/
|
|
func (point *ProjectivePoint) ToAffine(c *CurveOperations) *Fp2Element {
|
|
var affine_x Fp2Element
|
|
c.Params.Op.Inv(&affine_x, &point.Z)
|
|
c.Params.Op.Mul(&affine_x, &affine_x, &point.X)
|
|
return &affine_x
|
|
}
|
|
|
|
// Cleans data in fp
|
|
func (fp *Fp2Element) Zeroize() {
|
|
// Zeroizing in 2 seperated loops tells compiler to
|
|
// use fast runtime.memclr()
|
|
for i := range fp.A {
|
|
fp.A[i] = 0
|
|
}
|
|
for i := range fp.B {
|
|
fp.B[i] = 0
|
|
}
|
|
}
|