// 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)
}
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)
}
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)
}
// 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
}
// chi calculates out = z^((p-1)/2). The result is either 1, 0, or -1 depending
// on whether z is a non-zero square, zero, or a non-square.
func chi(out, z *edwards25519.FieldElement) {
var t0, t1, t2, t3 edwards25519.FieldElement
var i int
edwards25519.FeSquare(&t0, z) // 2^1
edwards25519.FeMul(&t1, &t0, z) // 2^1 + 2^0
edwards25519.FeSquare(&t0, &t1) // 2^2 + 2^1
edwards25519.FeSquare(&t2, &t0) // 2^3 + 2^2
edwards25519.FeSquare(&t2, &t2) // 4,3
edwards25519.FeMul(&t2, &t2, &t0) // 4,3,2,1
edwards25519.FeMul(&t1, &t2, z) // 4..0
edwards25519.FeSquare(&t2, &t1) // 5..1
for i = 1; i < 5; i++ { // 9,8,7,6,5
edwards25519.FeSquare(&t2, &t2)
}
edwards25519.FeMul(&t1, &t2, &t1) // 9,8,7,6,5,4,3,2,1,0
edwards25519.FeSquare(&t2, &t1) // 10..1
for i = 1; i < 10; i++ { // 19..10
edwards25519.FeSquare(&t2, &t2)
}
edwards25519.FeMul(&t2, &t2, &t1) // 19..0
edwards25519.FeSquare(&t3, &t2) // 20..1
for i = 1; i < 20; i++ { // 39..20
edwards25519.FeSquare(&t3, &t3)
}
edwards25519.FeMul(&t2, &t3, &t2) // 39..0
edwards25519.FeSquare(&t2, &t2) // 40..1
for i = 1; i < 10; i++ { // 49..10
edwards25519.FeSquare(&t2, &t2)
}
edwards25519.FeMul(&t1, &t2, &t1) // 49..0
edwards25519.FeSquare(&t2, &t1) // 50..1
for i = 1; i < 50; i++ { // 99..50
edwards25519.FeSquare(&t2, &t2)
}
edwards25519.FeMul(&t2, &t2, &t1) // 99..0
edwards25519.FeSquare(&t3, &t2) // 100..1
for i = 1; i < 100; i++ { // 199..100
edwards25519.FeSquare(&t3, &t3)
}
edwards25519.FeMul(&t2, &t3, &t2) // 199..0
edwards25519.FeSquare(&t2, &t2) // 200..1
for i = 1; i < 50; i++ { // 249..50
edwards25519.FeSquare(&t2, &t2)
}
edwards25519.FeMul(&t1, &t2, &t1) // 249..0
edwards25519.FeSquare(&t1, &t1) // 250..1
for i = 1; i < 4; i++ { // 253..4
edwards25519.FeSquare(&t1, &t1)
}
edwards25519.FeMul(out, &t1, &t0) // 253..4,2,1
}