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- package internal
-
- type CurveOperations struct {
- Params *SidhParams
- }
-
- // Computes j-invariant for a curve y2=x3+A/Cx+x with A,C in F_(p^2). Result
- // is returned in jBytes buffer, encoded in little-endian format. Caller
- // provided jBytes buffer has to be big enough to j-invariant value. In case
- // of SIDH, buffer size must be at least size of shared secret.
- // Implementation corresponds to Algorithm 9 from SIKE.
- func (c *CurveOperations) Jinvariant(cparams *ProjectiveCurveParameters, jBytes []byte) {
- var j, t0, t1 Fp2Element
-
- op := c.Params.Op
- op.Square(&j, &cparams.A) // j = A^2
- op.Square(&t1, &cparams.C) // t1 = C^2
- op.Add(&t0, &t1, &t1) // t0 = t1 + t1
- op.Sub(&t0, &j, &t0) // t0 = j - t0
- op.Sub(&t0, &t0, &t1) // t0 = t0 - t1
- op.Sub(&j, &t0, &t1) // t0 = t0 - t1
- op.Square(&t1, &t1) // t1 = t1^2
- op.Mul(&j, &j, &t1) // j = j * t1
- op.Add(&t0, &t0, &t0) // t0 = t0 + t0
- op.Add(&t0, &t0, &t0) // t0 = t0 + t0
- op.Square(&t1, &t0) // t1 = t0^2
- op.Mul(&t0, &t0, &t1) // t0 = t0 * t1
- op.Add(&t0, &t0, &t0) // t0 = t0 + t0
- op.Add(&t0, &t0, &t0) // t0 = t0 + t0
- op.Inv(&j, &j) // j = 1/j
- op.Mul(&j, &t0, &j) // j = t0 * j
-
- c.Fp2ToBytes(jBytes, &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 (c *CurveOperations) RecoverCoordinateA(curve *ProjectiveCurveParameters, xp, xq, xr *Fp2Element) {
- var t0, t1 Fp2Element
-
- op := c.Params.Op
- op.Add(&t1, xp, xq) // t1 = Xp + Xq
- op.Mul(&t0, xp, xq) // t0 = Xp * Xq
- op.Mul(&curve.A, xr, &t1) // A = X(q-p) * t1
- op.Add(&curve.A, &curve.A, &t0) // A = A + t0
- op.Mul(&t0, &t0, xr) // t0 = t0 * X(q-p)
- op.Sub(&curve.A, &curve.A, &c.Params.OneFp2) // A = A - 1
- op.Add(&t0, &t0, &t0) // t0 = t0 + t0
- op.Add(&t1, &t1, xr) // t1 = t1 + X(q-p)
- op.Add(&t0, &t0, &t0) // t0 = t0 + t0
- op.Square(&curve.A, &curve.A) // A = A^2
- op.Inv(&t0, &t0) // t0 = 1/t0
- op.Mul(&curve.A, &curve.A, &t0) // A = A * t0
- op.Sub(&curve.A, &curve.A, &t1) // A = A - t1
- }
-
- // Computes equivalence (A:C) ~ (A+2C : A-2C)
- func (c *CurveOperations) CalcCurveParamsEquiv3(cparams *ProjectiveCurveParameters) CurveCoefficientsEquiv {
- var coef CurveCoefficientsEquiv
- var c2 Fp2Element
- var op = c.Params.Op
-
- op.Add(&c2, &cparams.C, &cparams.C)
- // A24p = A+2*C
- op.Add(&coef.A, &cparams.A, &c2)
- // A24m = A-2*C
- op.Sub(&coef.C, &cparams.A, &c2)
- return coef
- }
-
- // Computes equivalence (A:C) ~ (A+2C : 4C)
- func (c *CurveOperations) CalcCurveParamsEquiv4(cparams *ProjectiveCurveParameters) CurveCoefficientsEquiv {
- var coefEq CurveCoefficientsEquiv
- var op = c.Params.Op
-
- op.Add(&coefEq.C, &cparams.C, &cparams.C)
- // A24p = A+2C
- op.Add(&coefEq.A, &cparams.A, &coefEq.C)
- // C24 = 4*C
- op.Add(&coefEq.C, &coefEq.C, &coefEq.C)
- return coefEq
- }
-
- // Helper function for RightToLeftLadder(). Returns A+2C / 4.
- func (c *CurveOperations) CalcAplus2Over4(cparams *ProjectiveCurveParameters) (ret Fp2Element) {
- var tmp Fp2Element
- var op = c.Params.Op
-
- // 2C
- op.Add(&tmp, &cparams.C, &cparams.C)
- // A+2C
- op.Add(&ret, &cparams.A, &tmp)
- // 1/4C
- op.Add(&tmp, &tmp, &tmp)
- op.Inv(&tmp, &tmp)
- // A+2C/4C
- op.Mul(&ret, &ret, &tmp)
- return
- }
-
- // Recovers (A:C) curve parameters from projectively equivalent (A+2C:A-2C).
- func (c *CurveOperations) RecoverCurveCoefficients3(cparams *ProjectiveCurveParameters, coefEq *CurveCoefficientsEquiv) {
- var op = c.Params.Op
-
- op.Add(&cparams.A, &coefEq.A, &coefEq.C)
- // cparams.A = 2*(A+2C+A-2C) = 4A
- op.Add(&cparams.A, &cparams.A, &cparams.A)
- // cparams.C = (A+2C-A+2C) = 4C
- op.Sub(&cparams.C, &coefEq.A, &coefEq.C)
- return
- }
-
- // Recovers (A:C) curve parameters from projectively equivalent (A+2C:4C).
- func (c *CurveOperations) RecoverCurveCoefficients4(cparams *ProjectiveCurveParameters, coefEq *CurveCoefficientsEquiv) {
- var op = c.Params.Op
- // cparams.C = (4C)*1/2=2C
- op.Mul(&cparams.C, &coefEq.C, &c.Params.HalfFp2)
- // cparams.A = A+2C - 2C = A
- op.Sub(&cparams.A, &coefEq.A, &cparams.C)
- // cparams.C = 2C * 1/2 = C
- op.Mul(&cparams.C, &cparams.C, &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 (c *CurveOperations) xDblAdd(P, Q, QmP *ProjectivePoint, a24 *Fp2Element) (dblP, PaQ ProjectivePoint) {
- var t0, t1, t2 Fp2Element
- var op = c.Params.Op
-
- 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
-
- op.Add(&t0, xP, zP) // t0 = Xp+Zp
- op.Sub(&t1, xP, zP) // t1 = Xp-Zp
- op.Square(x2P, &t0) // 2P.X = t0^2
- op.Sub(&t2, xQ, zQ) // t2 = Xq-Zq
- op.Add(xPaQ, xQ, zQ) // Xp+q = Xq+Zq
- op.Mul(&t0, &t0, &t2) // t0 = t0 * t2
- op.Mul(z2P, &t1, &t1) // 2P.Z = t1 * t1
- op.Mul(&t1, &t1, xPaQ) // t1 = t1 * Xp+q
- op.Sub(&t2, x2P, z2P) // t2 = 2P.X - 2P.Z
- op.Mul(x2P, x2P, z2P) // 2P.X = 2P.X * 2P.Z
- op.Mul(xPaQ, a24, &t2) // Xp+q = A24 * t2
- op.Sub(zPaQ, &t0, &t1) // Zp+q = t0 - t1
- op.Add(z2P, xPaQ, z2P) // 2P.Z = Xp+q + 2P.Z
- op.Add(xPaQ, &t0, &t1) // Xp+q = t0 + t1
- op.Mul(z2P, z2P, &t2) // 2P.Z = 2P.Z * t2
- op.Square(zPaQ, zPaQ) // Zp+q = Zp+q ^ 2
- op.Square(xPaQ, xPaQ) // Xp+q = Xp+q ^ 2
- op.Mul(zPaQ, xQmP, zPaQ) // Zp+q = Xq-p * Zp+q
- op.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 (c *CurveOperations) Pow2k(xP *ProjectivePoint, params *CurveCoefficientsEquiv, k uint32) {
- var t0, t1 Fp2Element
- var op = c.Params.Op
-
- x, z := &xP.X, &xP.Z
- for i := uint32(0); i < k; i++ {
- op.Sub(&t0, x, z) // t0 = Xp - Zp
- op.Add(&t1, x, z) // t1 = Xp + Zp
- op.Square(&t0, &t0) // t0 = t0 ^ 2
- op.Square(&t1, &t1) // t1 = t1 ^ 2
- op.Mul(z, ¶ms.C, &t0) // Z2p = C24 * t0
- op.Mul(x, z, &t1) // X2p = Z2p * t1
- op.Sub(&t1, &t1, &t0) // t1 = t1 - t0
- op.Mul(&t0, ¶ms.A, &t1) // t0 = A24+ * t1
- op.Add(z, z, &t0) // Z2p = Z2p + t0
- op.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 (c *CurveOperations) Pow3k(xP *ProjectivePoint, params *CurveCoefficientsEquiv, k uint32) {
- var t0, t1, t2, t3, t4, t5, t6 Fp2Element
- var op = c.Params.Op
-
- x, z := &xP.X, &xP.Z
- for i := uint32(0); i < k; i++ {
- op.Sub(&t0, x, z) // t0 = Xp - Zp
- op.Square(&t2, &t0) // t2 = t0^2
- op.Add(&t1, x, z) // t1 = Xp + Zp
- op.Square(&t3, &t1) // t3 = t1^2
- op.Add(&t4, &t1, &t0) // t4 = t1 + t0
- op.Sub(&t0, &t1, &t0) // t0 = t1 - t0
- op.Square(&t1, &t4) // t1 = t4^2
- op.Sub(&t1, &t1, &t3) // t1 = t1 - t3
- op.Sub(&t1, &t1, &t2) // t1 = t1 - t2
- op.Mul(&t5, &t3, ¶ms.A) // t5 = t3 * A24+
- op.Mul(&t3, &t3, &t5) // t3 = t5 * t3
- op.Mul(&t6, &t2, ¶ms.C) // t6 = t2 * A24-
- op.Mul(&t2, &t2, &t6) // t2 = t2 * t6
- op.Sub(&t3, &t2, &t3) // t3 = t2 - t3
- op.Sub(&t2, &t5, &t6) // t2 = t5 - t6
- op.Mul(&t1, &t2, &t1) // t1 = t2 * t1
- op.Add(&t2, &t3, &t1) // t2 = t3 + t1
- op.Square(&t2, &t2) // t2 = t2^2
- op.Mul(x, &t2, &t4) // X3p = t2 * t4
- op.Sub(&t1, &t3, &t1) // t1 = t3 - t1
- op.Square(&t1, &t1) // t1 = t1^2
- op.Mul(z, &t1, &t0) // Z3p = t1 * t0
- }
- }
-
- // Set (y1, y2, y3) = (1/x1, 1/x2, 1/x3).
- //
- // All xi, yi must be distinct.
- func (c *CurveOperations) Fp2Batch3Inv(x1, x2, x3, y1, y2, y3 *Fp2Element) {
- var x1x2, t Fp2Element
- var op = c.Params.Op
-
- op.Mul(&x1x2, x1, x2) // x1*x2
- op.Mul(&t, &x1x2, x3) // 1/(x1*x2*x3)
- op.Inv(&t, &t)
- op.Mul(y1, &t, x2) // 1/x1
- op.Mul(y1, y1, x3)
- op.Mul(y2, &t, x1) // 1/x2
- op.Mul(y2, y2, x3)
- op.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 (c *CurveOperations) ScalarMul3Pt(cparams *ProjectiveCurveParameters, P, Q, PmQ *ProjectivePoint, nbits uint, scalar []uint8) ProjectivePoint {
- var R0, R2, R1 ProjectivePoint
- var op = c.Params.Op
- aPlus2Over4 := c.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
- op.CondSwap(&R1.X, &R1.Z, &R2.X, &R2.Z, swap)
- R0, R2 = c.xDblAdd(&R0, &R2, &R1, &aPlus2Over4)
- }
- op.CondSwap(&R1.X, &R1.Z, &R2.X, &R2.Z, prevBit)
- return R1
- }
-
- // Convert the input to wire format.
- //
- // The output byte slice must be at least 2*bytelen(p) bytes long.
- func (c *CurveOperations) Fp2ToBytes(output []byte, fp2 *Fp2Element) {
- if len(output) < 2*c.Params.Bytelen {
- panic("output byte slice too short")
- }
- var a Fp2Element
- c.Params.Op.FromMontgomery(fp2, &a)
-
- // convert to bytes in little endian form
- for i := 0; i < c.Params.Bytelen; i++ {
- // set i = j*8 + k
- fp2 := i / 8
- k := uint64(i % 8)
- output[i] = byte(a.A[fp2] >> (8 * k))
- output[i+c.Params.Bytelen] = byte(a.B[fp2] >> (8 * k))
- }
- }
-
- // Read 2*bytelen(p) bytes into the given ExtensionFieldElement.
- //
- // It is an error to call this function if the input byte slice is less than 2*bytelen(p) bytes long.
- func (c *CurveOperations) Fp2FromBytes(fp2 *Fp2Element, input []byte) {
- if len(input) < 2*c.Params.Bytelen {
- panic("input byte slice too short")
- }
-
- for i := 0; i < c.Params.Bytelen; i++ {
- j := i / 8
- k := uint64(i % 8)
- fp2.A[j] |= uint64(input[i]) << (8 * k)
- fp2.B[j] |= uint64(input[i+c.Params.Bytelen]) << (8 * k)
- }
- c.Params.Op.ToMontgomery(fp2)
- }
-
- /* -------------------------------------------------------------------------
- Mechnisms used for isogeny calculations
- -------------------------------------------------------------------------*/
-
- // Constructs isogeny3 objects
- func Newisogeny3(op FieldOps) Isogeny {
- return &isogeny3{Field: op}
- }
-
- // Constructs isogeny4 objects
- func Newisogeny4(op FieldOps) Isogeny {
- return &isogeny4{isogeny3: isogeny3{Field: op}}
- }
-
- // 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 Fp2Element
- var coefEq CurveCoefficientsEquiv
- var K1, K2 = &phi.K1, &phi.K2
-
- op := phi.Field
- op.Sub(K1, &p.X, &p.Z) // K1 = XP3 - ZP3
- op.Square(&t0, K1) // t0 = K1^2
- op.Add(K2, &p.X, &p.Z) // K2 = XP3 + ZP3
- op.Square(&t1, K2) // t1 = K2^2
- op.Add(&t2, &t0, &t1) // t2 = t0 + t1
- op.Add(&t3, K1, K2) // t3 = K1 + K2
- op.Square(&t3, &t3) // t3 = t3^2
- op.Sub(&t3, &t3, &t2) // t3 = t3 - t2
- op.Add(&t2, &t1, &t3) // t2 = t1 + t3
- op.Add(&t3, &t3, &t0) // t3 = t3 + t0
- op.Add(&t4, &t3, &t0) // t4 = t3 + t0
- op.Add(&t4, &t4, &t4) // t4 = t4 + t4
- op.Add(&t4, &t1, &t4) // t4 = t1 + t4
- op.Mul(&coefEq.C, &t2, &t4) // A24m = t2 * t4
- op.Add(&t4, &t1, &t2) // t4 = t1 + t2
- op.Add(&t4, &t4, &t4) // t4 = t4 + t4
- op.Add(&t4, &t0, &t4) // t4 = t0 + t4
- op.Mul(&t4, &t3, &t4) // t4 = t3 * t4
- op.Sub(&t0, &t4, &coefEq.C) // t0 = t4 - A24m
- op.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 Fp2Element
- var q ProjectivePoint
- var K1, K2 = &phi.K1, &phi.K2
- var px, pz = &p.X, &p.Z
-
- op := phi.Field
- op.Add(&t0, px, pz) // t0 = XQ + ZQ
- op.Sub(&t1, px, pz) // t1 = XQ - ZQ
- op.Mul(&t0, K1, &t0) // t2 = K1 * t0
- op.Mul(&t1, K2, &t1) // t1 = K2 * t1
- op.Add(&t2, &t0, &t1) // t2 = t0 + t1
- op.Sub(&t0, &t1, &t0) // t0 = t1 - t0
- op.Square(&t2, &t2) // t2 = t2 ^ 2
- op.Square(&t0, &t0) // t0 = t0 ^ 2
- op.Mul(&q.X, px, &t2) // XQ'= XQ * t2
- op.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
-
- op := phi.Field
- op.Sub(K2, xp4, zp4)
- op.Add(K3, xp4, zp4)
- op.Square(K1, zp4)
- op.Add(K1, K1, K1)
- 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
- }
- }
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