// RepresentativeToPublicKey converts a uniform representative value for a
// curve25519 public key, as produced by ScalarBaseMult, to a curve25519 public
// key.
func RepresentativeToPublicKey(publicKey, representative *[32]byte) {
var rr2, v, e edwards25519.FieldElement
edwards25519.FeFromBytes(&rr2, representative)
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)
edwards25519.FeToBytes(publicKey, &v)
}
// extendedToBigAffine converts projective x, y, and z field elements into
// affine x and y coordinates, and returns whether or not the x value
// returned is negative.
func (curve *TwistedEdwardsCurve) extendedToBigAffine(xi, yi,
zi *edwards25519.FieldElement) (*big.Int, *big.Int, bool) {
var recip, x, y edwards25519.FieldElement
// Normalize to Z=1.
edwards25519.FeInvert(&recip, zi)
edwards25519.FeMul(&x, xi, &recip)
edwards25519.FeMul(&y, yi, &recip)
isNegative := edwards25519.FeIsNegative(&x) == 1
return FieldElementToBigInt(&x), FieldElementToBigInt(&y), isNegative
}
// Add adds two points represented by pairs of big integers on the elliptical
// curve.
func (curve *TwistedEdwardsCurve) Add(x1, y1, x2, y2 *big.Int) (x, y *big.Int) {
// Convert to extended from affine.
a := BigIntPointToEncodedBytes(x1, y1)
aEGE := new(edwards25519.ExtendedGroupElement)
aEGE.FromBytes(a)
b := BigIntPointToEncodedBytes(x2, y2)
bEGE := new(edwards25519.ExtendedGroupElement)
bEGE.FromBytes(b)
// Cache b for use in group element addition.
bCached := new(cachedGroupElement)
toCached(bCached, bEGE)
p := aEGE
q := bCached
// geAdd(r*CompletedGroupElement, p*ExtendedGroupElement,
// q*CachedGroupElement)
// r is the result.
r := new(edwards25519.CompletedGroupElement)
var t0 edwards25519.FieldElement
edwards25519.FeAdd(&r.X, &p.Y, &p.X)
edwards25519.FeSub(&r.Y, &p.Y, &p.X)
edwards25519.FeMul(&r.Z, &r.X, &q.yPlusX)
edwards25519.FeMul(&r.Y, &r.Y, &q.yMinusX)
edwards25519.FeMul(&r.T, &q.T2d, &p.T)
edwards25519.FeMul(&r.X, &p.Z, &q.Z)
edwards25519.FeAdd(&t0, &r.X, &r.X)
edwards25519.FeSub(&r.X, &r.Z, &r.Y)
edwards25519.FeAdd(&r.Y, &r.Z, &r.Y)
edwards25519.FeAdd(&r.Z, &t0, &r.T)
edwards25519.FeSub(&r.T, &t0, &r.T)
rEGE := new(edwards25519.ExtendedGroupElement)
r.ToExtended(rEGE)
s := new([32]byte)
rEGE.ToBytes(s)
x, y, _ = curve.EncodedBytesToBigIntPoint(s)
return
}
// IsOnCurve returns bool to say if the point (x,y) is on the curve by
// checking (y^2 - x^2 - 1 - dx^2y^2) % P == 0.
func (curve *TwistedEdwardsCurve) IsOnCurve(x *big.Int, y *big.Int) bool {
// Convert to field elements.
xB := BigIntToEncodedBytes(x)
yB := BigIntToEncodedBytes(y)
yfe := new(edwards25519.FieldElement)
xfe := new(edwards25519.FieldElement)
edwards25519.FeFromBytes(yfe, yB)
edwards25519.FeFromBytes(xfe, xB)
x2 := new(edwards25519.FieldElement)
edwards25519.FeSquare(x2, xfe)
y2 := new(edwards25519.FieldElement)
edwards25519.FeSquare(y2, yfe)
dx2y2 := new(edwards25519.FieldElement)
edwards25519.FeMul(dx2y2, &fed, x2)
edwards25519.FeMul(dx2y2, dx2y2, y2)
enum := new(edwards25519.FieldElement)
edwards25519.FeSub(enum, y2, x2)
edwards25519.FeSub(enum, enum, &feOne)
edwards25519.FeSub(enum, enum, dx2y2)
enumBig := FieldElementToBigInt(enum)
enumBig.Mod(enumBig, curve.P)
if enumBig.Cmp(zero) != 0 {
return false
}
// Check if we're in the cofactor of the curve (8).
modEight := new(big.Int)
modEight.Mod(enumBig, eight)
if modEight.Cmp(zero) != 0 {
return false
}
return true
}
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)
}
// RecoverXFieldElement recovers the X value for some Y value, for a coordinate
// on the Ed25519 curve given as a field element. Y value. Probably the fastest
// way to get your respective X from Y.
func (curve *TwistedEdwardsCurve) RecoverXFieldElement(xIsNeg bool,
y *edwards25519.FieldElement) *edwards25519.FieldElement {
// (y^2 - 1)
l := new(edwards25519.FieldElement)
edwards25519.FeSquare(l, y)
edwards25519.FeSub(l, l, &feOne)
// inv(d*y^2+1)
r := new(edwards25519.FieldElement)
edwards25519.FeSquare(r, y)
edwards25519.FeMul(r, r, &fed)
edwards25519.FeAdd(r, r, &feOne)
edwards25519.FeInvert(r, r)
x2 := new(edwards25519.FieldElement)
edwards25519.FeMul(x2, r, l)
// Get a big int so we can do the exponentiation.
x2Big := FieldElementToBigInt(x2)
// x = exp(x^2,(P+3)/8, P)
qp3 := new(big.Int).Add(curve.P, three)
qp3.Div(qp3, eight) // /= curve.H
xBig := new(big.Int).Exp(x2Big, qp3, curve.P)
// Convert back to a field element and do
// the rest.
x := BigIntToFieldElement(xBig)
// check (x^2 - x2) % q != 0
x22 := new(edwards25519.FieldElement)
edwards25519.FeSquare(x22, x)
xsub := new(edwards25519.FieldElement)
edwards25519.FeSub(xsub, x22, x2)
xsubBig := FieldElementToBigInt(xsub)
xsubBig.Mod(xsubBig, curve.P)
if xsubBig.Cmp(zero) != 0 {
xi := new(edwards25519.FieldElement)
edwards25519.FeMul(xi, x, &feI)
xiModBig := FieldElementToBigInt(xi)
xiModBig.Mod(xiModBig, curve.P)
xiMod := BigIntToFieldElement(xiModBig)
x = xiMod
}
xBig = FieldElementToBigInt(x)
xmod2 := new(big.Int).Mod(xBig, two)
if xmod2.Cmp(zero) != 0 {
// TODO replace this with FeSub
xBig.Sub(curve.P, xBig)
x = BigIntToFieldElement(xBig)
}
// We got the wrong x, negate it to get the right one.
isNegative := edwards25519.FeIsNegative(x) == 1
if xIsNeg != isNegative {
edwards25519.FeNeg(x, x)
}
return x
}
// toCached converts an extended group element to a useful intermediary
// containing precalculated values.
func toCached(r *cachedGroupElement, p *edwards25519.ExtendedGroupElement) {
edwards25519.FeAdd(&r.yPlusX, &p.Y, &p.X)
edwards25519.FeSub(&r.yMinusX, &p.Y, &p.X)
edwards25519.FeCopy(&r.Z, &p.Z)
edwards25519.FeMul(&r.T2d, &p.T, &fed2)
}
// 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
}