Beispiel #1
0
// representativeToMontgomeryX consumes the rr2 input
func representativeToMontgomeryX(v, rr2 *edwards25519.FieldElement) {
	var e edwards25519.FieldElement
	edwards25519.FeSquare2(rr2, rr2)
	rr2[0]++
	edwards25519.FeInvert(rr2, rr2)
	edwards25519.FeMul(v, &edwards25519.A, rr2)
	edwards25519.FeNeg(v, v)

	var v2, v3 edwards25519.FieldElement
	edwards25519.FeSquare(&v2, v)
	edwards25519.FeMul(&v3, v, &v2)
	edwards25519.FeAdd(&e, &v3, v)
	edwards25519.FeMul(&v2, &v2, &edwards25519.A)
	edwards25519.FeAdd(&e, &v2, &e)
	chi(&e, &e)
	var eBytes [32]byte
	edwards25519.FeToBytes(&eBytes, &e)
	// eBytes[1] is either 0 (for e = 1) or 0xff (for e = -1)
	eIsMinus1 := int32(eBytes[1]) & 1
	var negV edwards25519.FieldElement
	edwards25519.FeNeg(&negV, v)
	edwards25519.FeCMove(v, &negV, eIsMinus1)

	edwards25519.FeZero(&v2)
	edwards25519.FeCMove(&v2, &edwards25519.A, eIsMinus1)
	edwards25519.FeSub(v, v, &v2)
}
Beispiel #2
0
func montgomeryXToEdwardsY(out, x *edwards25519.FieldElement) {
	var t, tt edwards25519.FieldElement
	edwards25519.FeOne(&t)
	edwards25519.FeAdd(&tt, x, &t)   // u+1
	edwards25519.FeInvert(&tt, &tt)  // 1/(u+1)
	edwards25519.FeSub(&t, x, &t)    // u-1
	edwards25519.FeMul(out, &tt, &t) // (u-1)/(u+1)
}
Beispiel #3
0
func edwardsToMontgomeryX(outX, y *edwards25519.FieldElement) {
	// We only need the x-coordinate of the curve25519 point, which I'll
	// call u. The isomorphism is u=(y+1)/(1-y), since y=Y/Z, this gives
	// u=(Y+Z)/(Z-Y). We know that Z=1, thus u=(Y+1)/(1-Y).
	var oneMinusY edwards25519.FieldElement
	edwards25519.FeOne(&oneMinusY)
	edwards25519.FeSub(&oneMinusY, &oneMinusY, y)
	edwards25519.FeInvert(&oneMinusY, &oneMinusY)

	edwards25519.FeOne(outX)
	edwards25519.FeAdd(outX, outX, y)

	edwards25519.FeMul(outX, outX, &oneMinusY)
}
Beispiel #4
0
// ScalarBaseMult computes a curve25519 public key from a private key and also
// a uniform representative for that public key. Note that this function will
// fail and return false for about half of private keys.
// See http://elligator.cr.yp.to/elligator-20130828.pdf.
func ScalarBaseMult(publicKey, representative, privateKey *[32]byte) bool {
	var maskedPrivateKey [32]byte
	copy(maskedPrivateKey[:], privateKey[:])

	maskedPrivateKey[0] &= 248
	maskedPrivateKey[31] &= 127
	maskedPrivateKey[31] |= 64

	var A edwards25519.ExtendedGroupElement
	edwards25519.GeScalarMultBase(&A, &maskedPrivateKey)

	var inv1 edwards25519.FieldElement
	edwards25519.FeSub(&inv1, &A.Z, &A.Y)
	edwards25519.FeMul(&inv1, &inv1, &A.X)
	edwards25519.FeInvert(&inv1, &inv1)

	var t0, u edwards25519.FieldElement
	edwards25519.FeMul(&u, &inv1, &A.X)
	edwards25519.FeAdd(&t0, &A.Y, &A.Z)
	edwards25519.FeMul(&u, &u, &t0)

	var v edwards25519.FieldElement
	edwards25519.FeMul(&v, &t0, &inv1)
	edwards25519.FeMul(&v, &v, &A.Z)
	edwards25519.FeMul(&v, &v, &sqrtMinusA)

	var b edwards25519.FieldElement
	edwards25519.FeAdd(&b, &u, &edwards25519.A)

	var c, b3, b8 edwards25519.FieldElement
	edwards25519.FeSquare(&b3, &b)   // 2
	edwards25519.FeMul(&b3, &b3, &b) // 3
	edwards25519.FeSquare(&c, &b3)   // 6
	edwards25519.FeMul(&c, &c, &b)   // 7
	edwards25519.FeMul(&b8, &c, &b)  // 8
	edwards25519.FeMul(&c, &c, &u)
	q58(&c, &c)

	var chi edwards25519.FieldElement
	edwards25519.FeSquare(&chi, &c)
	edwards25519.FeSquare(&chi, &chi)

	edwards25519.FeSquare(&t0, &u)
	edwards25519.FeMul(&chi, &chi, &t0)

	edwards25519.FeSquare(&t0, &b)   // 2
	edwards25519.FeMul(&t0, &t0, &b) // 3
	edwards25519.FeSquare(&t0, &t0)  // 6
	edwards25519.FeMul(&t0, &t0, &b) // 7
	edwards25519.FeSquare(&t0, &t0)  // 14
	edwards25519.FeMul(&chi, &chi, &t0)
	edwards25519.FeNeg(&chi, &chi)

	var chiBytes [32]byte
	edwards25519.FeToBytes(&chiBytes, &chi)
	// chi[1] is either 0 or 0xff
	if chiBytes[1] == 0xff {
		return false
	}

	// Calculate r1 = sqrt(-u/(2*(u+A)))
	var r1 edwards25519.FieldElement
	edwards25519.FeMul(&r1, &c, &u)
	edwards25519.FeMul(&r1, &r1, &b3)
	edwards25519.FeMul(&r1, &r1, &sqrtMinusHalf)

	var maybeSqrtM1 edwards25519.FieldElement
	edwards25519.FeSquare(&t0, &r1)
	edwards25519.FeMul(&t0, &t0, &b)
	edwards25519.FeAdd(&t0, &t0, &t0)
	edwards25519.FeAdd(&t0, &t0, &u)

	edwards25519.FeOne(&maybeSqrtM1)
	edwards25519.FeCMove(&maybeSqrtM1, &edwards25519.SqrtM1, edwards25519.FeIsNonZero(&t0))
	edwards25519.FeMul(&r1, &r1, &maybeSqrtM1)

	// Calculate r = sqrt(-(u+A)/(2u))
	var r edwards25519.FieldElement
	edwards25519.FeSquare(&t0, &c)   // 2
	edwards25519.FeMul(&t0, &t0, &c) // 3
	edwards25519.FeSquare(&t0, &t0)  // 6
	edwards25519.FeMul(&r, &t0, &c)  // 7

	edwards25519.FeSquare(&t0, &u)   // 2
	edwards25519.FeMul(&t0, &t0, &u) // 3
	edwards25519.FeMul(&r, &r, &t0)

	edwards25519.FeSquare(&t0, &b8)   // 16
	edwards25519.FeMul(&t0, &t0, &b8) // 24
	edwards25519.FeMul(&t0, &t0, &b)  // 25
	edwards25519.FeMul(&r, &r, &t0)
	edwards25519.FeMul(&r, &r, &sqrtMinusHalf)

	edwards25519.FeSquare(&t0, &r)
	edwards25519.FeMul(&t0, &t0, &u)
	edwards25519.FeAdd(&t0, &t0, &t0)
	edwards25519.FeAdd(&t0, &t0, &b)
	edwards25519.FeOne(&maybeSqrtM1)
	edwards25519.FeCMove(&maybeSqrtM1, &edwards25519.SqrtM1, edwards25519.FeIsNonZero(&t0))
	edwards25519.FeMul(&r, &r, &maybeSqrtM1)

	var vBytes [32]byte
	edwards25519.FeToBytes(&vBytes, &v)
	vInSquareRootImage := feBytesLE(&vBytes, &halfQMinus1Bytes)
	edwards25519.FeCMove(&r, &r1, vInSquareRootImage)

	edwards25519.FeToBytes(publicKey, &u)
	edwards25519.FeToBytes(representative, &r)
	return true
}