// Equality test for two Points on the same curve.
// We can avoid inversions here because:
//
// (X1/Z1,Y1/Z1) == (X2/Z2,Y2/Z2)
// iff
// (X1*Z2,Y1*Z2) == (X2*Z1,Y2*Z1)
//
func (P1 *projPoint) Equal(CP2 abstract.Point) bool {
P2 := CP2.(*projPoint)
var t1, t2 nist.Int
xeq := t1.Mul(&P1.X, &P2.Z).Equal(t2.Mul(&P2.X, &P1.Z))
yeq := t1.Mul(&P1.Y, &P2.Z).Equal(t2.Mul(&P2.Y, &P1.Z))
return xeq && yeq
}
// Given a y-coordinate, solve for the x-coordinate on the curve,
// using the characteristic equation rewritten as:
//
// x^2 = (1 - y^2)/(a - d*y^2)
//
// Returns true on success,
// false if there is no x-coordinate corresponding to the chosen y-coordinate.
//
func (c *curve) solveForX(x, y *nist.Int) bool {
var yy, t1, t2 nist.Int
yy.Mul(y, y) // yy = y^2
t1.Sub(&c.one, &yy) // t1 = 1 - y^-2
t2.Mul(&c.d, &yy).Sub(&c.a, &t2) // t2 = a - d*y^2
t2.Div(&t1, &t2) // t2 = x^2
return x.Sqrt(&t2) // may fail if not a square
}
// Elligator 1 reverse-map from point to uniform representative.
// Returns nil if point has no uniform representative.
// See section 3.3 of the Elligator paper.
func (el *el1param) HideEncode(P point, rand cipher.Stream) []byte {
ec := el.ec
x, y := P.getXY()
var a, b, etar, etarp1, X, z, u, t, t1 nist.Int
// condition 1: a = y+1 is nonzero
a.Add(y, &ec.one)
if a.V.Sign() == 0 {
return nil // y+1 = 0, no representative
}
// etar = r(y-1)/2(y+1)
t1.Add(y, &ec.one).Add(&t1, &t1) // 2(y+1)
etar.Sub(y, &ec.one).Mul(&etar, &el.r).Div(&etar, &t1)
// condition 2: b = (1 + eta r)^2 - 1 is a square
etarp1.Add(&ec.one, &etar) // etarp1 = (1 + eta r)
b.Mul(&etarp1, &etarp1).Sub(&b, &ec.one)
if math.Jacobi(&b.V, b.M) < 0 {
return nil // b not a square, no representative
}
// condition 3: if etar = -2 then x=2s(c-1)Chi(c)/r
if etar.Equal(&el.m2) && !x.Equal(&el.c3x) {
return nil
}
// X = -(1+eta r)+((1+eta r)^2-1)^((q+1)/4)
X.Exp(&b, &el.pp1d4).Sub(&X, &etarp1)
// z = Chi((c-1)sX(1+X)x(X^2+1/c^2))
z.Mul(&el.cm1s, &X).Mul(&z, t.Add(&ec.one, &X)).Mul(&z, x)
z.Mul(&z, t.Mul(&X, &X).Add(&t, &el.invc2))
chi(&z, &z)
// u = zX
u.Mul(&z, &X)
// t = (1-u)/(1+u)
t.Div(a.Sub(&ec.one, &u), b.Add(&ec.one, &u))
// Map representative to a byte-string by padding the upper byte.
// This assumes that the prime c.P is close enough to a power of 2
// that the adversary will never notice the "missing" values;
// this is true for the class of curves Elligator1 was designed for.
rep, _ := t.MarshalBinary()
padmask := el.padmask()
if padmask != 0 {
var pad [1]byte
rand.XORKeyStream(pad[:], pad[:])
rep[0] |= pad[0] & padmask
}
return rep
}
// Compute the square root function,
// specified in section 5.5 of the Elligator paper.
func (el *el2param) sqrt(r, a *nist.Int) {
var b, b2 nist.Int
b.Exp(a, &el.pp3d8) // b = a^((p+3)/8); b in {a,-a}
b2.Mul(&b, &b) // b^2 = a?
if !b2.Equal(a) {
b.Mul(&b, &el.sqrtm1) // b*sqrt(-1)
}
if b.V.Cmp(&el.pm1d2) > 0 { // |b|
b.Neg(&b)
}
r.Set(&b)
}
// Elligator 2 forward-map from representative to Edwards curve point.
// Currently a straightforward, unoptimized implementation.
// See section 5.2 of the Elligator paper.
func (el *el2param) HideDecode(P point, rep []byte) {
ec := el.ec
var r, v, x, y, t1, edx, edy nist.Int
l := ec.PointLen()
if len(rep) != l {
panic("el2Map: wrong representative length")
}
// Take the appropriate number of bits from the representative.
buf := make([]byte, l)
copy(buf, rep)
buf[0] &^= el.padmask() // mask off the padding bits
r.InitBytes(buf, &ec.P)
// v = -A/(1+ur^2)
v.Mul(&r, &r).Mul(&el.u, &v).Add(&ec.one, &v).Div(&el.negA, &v)
// e = Chi(v^3+Av^2+Bv), where B=1 because of ed2mont equivalence
t1.Add(&v, &el.A).Mul(&t1, &v).Add(&t1, &ec.one).Mul(&t1, &v)
e := math.Jacobi(&t1.V, t1.M)
// x = ev-(1-e)A/2
if e == 1 {
x.Set(&v)
} else {
x.Add(&v, &el.A).Neg(&x)
}
// y = -e sqrt(x^3+Ax^2+Bx), where B=1
y.Add(&x, &el.A).Mul(&y, &x).Add(&y, &ec.one).Mul(&y, &x)
el.sqrt(&y, &y)
if e == 1 {
y.Neg(&y) // -e factor
}
// Convert Montgomery to Edwards coordinates
el.mont2ed(&edx, &edy, &x, &y)
// Sanity-check
if !ec.onCurve(&edx, &edy) {
panic("elligator2 produced invalid point")
}
P.initXY(&edx.V, &edy.V, ec.self)
}
// Test if a supposed point is on the curve,
// by checking the characteristic equation for Edwards curves:
//
// a*x^2 + y^2 = 1 + d*x^2*y^2
//
func (c *curve) onCurve(x, y *nist.Int) bool {
var xx, yy, l, r nist.Int
xx.Mul(x, x) // xx = x^2
yy.Mul(y, y) // yy = y^2
l.Mul(&c.a, &xx).Add(&l, &yy) // l = a*x^2 + y^2
r.Mul(&c.d, &xx).Mul(&r, &yy).Add(&c.one, &r)
// r = 1 + d*x^2*y^2
return l.Equal(&r)
}
// Subtract points so that their scalars subtract homomorphically
func (P *projPoint) Sub(CP1, CP2 abstract.Point) abstract.Point {
P1 := CP1.(*projPoint)
P2 := CP2.(*projPoint)
X1, Y1, Z1 := &P1.X, &P1.Y, &P1.Z
X2, Y2, Z2 := &P2.X, &P2.Y, &P2.Z
X3, Y3, Z3 := &P.X, &P.Y, &P.Z
var A, B, C, D, E, F, G nist.Int
A.Mul(Z1, Z2)
B.Mul(&A, &A)
C.Mul(X1, X2)
D.Mul(Y1, Y2)
E.Mul(&C, &D).Mul(&P.c.d, &E)
F.Add(&B, &E)
G.Sub(&B, &E)
X3.Add(X1, Y1).Mul(X3, Z3.Sub(Y2, X2)).Add(X3, &C).Sub(X3, &D).
Mul(&F, X3).Mul(&A, X3)
Y3.Mul(&P.c.a, &C).Add(&D, Y3).Mul(&G, Y3).Mul(&A, Y3)
Z3.Mul(&F, &G)
return P
}
// Optimized point doubling for use in scalar multiplication.
func (P *projPoint) double() {
var B, C, D, E, F, H, J nist.Int
B.Add(&P.X, &P.Y).Mul(&B, &B)
C.Mul(&P.X, &P.X)
D.Mul(&P.Y, &P.Y)
E.Mul(&P.c.a, &C)
F.Add(&E, &D)
H.Mul(&P.Z, &P.Z)
J.Add(&H, &H).Sub(&F, &J)
P.X.Sub(&B, &C).Sub(&P.X, &D).Mul(&P.X, &J)
P.Y.Sub(&E, &D).Mul(&F, &P.Y)
P.Z.Mul(&F, &J)
}
// Subtract points.
func (P *extPoint) Sub(CP1, CP2 abstract.Point) abstract.Point {
P1 := CP1.(*extPoint)
P2 := CP2.(*extPoint)
X1, Y1, Z1, T1 := &P1.X, &P1.Y, &P1.Z, &P1.T
X2, Y2, Z2, T2 := &P2.X, &P2.Y, &P2.Z, &P2.T
X3, Y3, Z3, T3 := &P.X, &P.Y, &P.Z, &P.T
var A, B, C, D, E, F, G, H nist.Int
A.Mul(X1, X2)
B.Mul(Y1, Y2)
C.Mul(T1, T2).Mul(&C, &P.c.d)
D.Mul(Z1, Z2)
E.Add(X1, Y1).Mul(&E, F.Sub(Y2, X2)).Add(&E, &A).Sub(&E, &B)
F.Add(&D, &C)
G.Sub(&D, &C)
H.Mul(&P.c.a, &A).Add(&B, &H)
X3.Mul(&E, &F)
Y3.Mul(&G, &H)
T3.Mul(&E, &H)
Z3.Mul(&F, &G)
return P
}
// Add two points using the basic unified addition laws for Edwards curves:
//
// x' = ((x1*y2 + x2*y1) / (1 + d*x1*x2*y1*y2))
// y' = ((y1*y2 - a*x1*x2) / (1 - d*x1*x2*y1*y2))
//
func (P *basicPoint) Add(P1, P2 abstract.Point) abstract.Point {
E1 := P1.(*basicPoint)
E2 := P2.(*basicPoint)
x1, y1 := E1.x, E1.y
x2, y2 := E2.x, E2.y
var t1, t2, dm, nx, dx, ny, dy nist.Int
// Reused part of denominator: dm = d*x1*x2*y1*y2
dm.Mul(&P.c.d, &x1).Mul(&dm, &x2).Mul(&dm, &y1).Mul(&dm, &y2)
// x' numerator/denominator
nx.Add(t1.Mul(&x1, &y2), t2.Mul(&x2, &y1))
dx.Add(&P.c.one, &dm)
// y' numerator/denominator
ny.Sub(t1.Mul(&y1, &y2), t2.Mul(&x1, &x2).Mul(&P.c.a, &t2))
dy.Sub(&P.c.one, &dm)
// result point
P.x.Div(&nx, &dx)
P.y.Div(&ny, &dy)
return P
}
// Optimized point doubling for use in scalar multiplication.
// Uses the formulae in section 3.3 of:
// https://www.iacr.org/archive/asiacrypt2008/53500329/53500329.pdf
func (P *extPoint) double() {
X1, Y1, Z1, T1 := &P.X, &P.Y, &P.Z, &P.T
var A, B, C, D, E, F, G, H nist.Int
A.Mul(X1, X1)
B.Mul(Y1, Y1)
C.Mul(Z1, Z1).Add(&C, &C)
D.Mul(&P.c.a, &A)
E.Add(X1, Y1).Mul(&E, &E).Sub(&E, &A).Sub(&E, &B)
G.Add(&D, &B)
F.Sub(&G, &C)
H.Sub(&D, &B)
X1.Mul(&E, &F)
Y1.Mul(&G, &H)
T1.Mul(&E, &H)
Z1.Mul(&F, &G)
}
// Elligator 2 reverse-map from point to uniform representative.
// Returns nil if point has no uniform representative.
// See section 5.3 of the Elligator paper.
func (el *el2param) HideEncode(P point, rand cipher.Stream) []byte {
edx, edy := P.getXY()
var x, y, r, xpA, t1 nist.Int
// convert Edwards to Montgomery coordinates
el.ed2mont(&x, &y, edx, edy)
// condition 1: x != -A
if x.Equal(&el.negA) {
return nil // x = -A, no representative
}
// condition 2: if y=0, then x=0
if y.V.Sign() == 0 && x.V.Sign() != 0 {
return nil // y=0 but x!=0, no representative
}
// condition 3: -ux(x+A) is a square
xpA.Add(&x, &el.A)
t1.Mul(&el.u, &x).Mul(&t1, &xpA).Neg(&t1)
if math.Jacobi(&t1.V, t1.M) < 0 {
return nil // not a square, no representative
}
if y.V.Cmp(&el.pm1d2) <= 0 { // y in image of sqrt function
r.Mul(&xpA, &el.u).Div(&x, &r)
} else { // y not in image of sqrt function
r.Mul(&el.u, &x).Div(&xpA, &r)
}
r.Neg(&r)
el.sqrt(&r, &r)
// Sanity check on result
if r.V.Cmp(&el.pm1d2) > 0 {
panic("el2: r too big")
}
// Map representative to a byte-string by padding the upper byte.
// This assumes that the prime c.P is close enough to a power of 2
// that the adversary will never notice the "missing" values;
// this is true for the class of curves Elligator1 was designed for.
rep, _ := r.MarshalBinary()
padmask := el.padmask()
if padmask != 0 {
var pad [1]byte
rand.XORKeyStream(pad[:], pad[:])
rep[0] |= pad[0] & padmask
}
return rep
}
// Elligator 1 forward-map from representative to Edwards curve point.
// Currently a straightforward, unoptimized implementation.
// See section 3.2 of the Elligator paper.
func (el *el1param) HideDecode(P point, rep []byte) {
ec := el.ec
var t, u, u2, v, Chiv, X, Y, x, y, t1, t2 nist.Int
l := ec.PointLen()
if len(rep) != l {
panic("el1Map: wrong representative length")
}
// Take the appropriate number of bits from the representative.
b := make([]byte, l)
copy(b, rep)
b[0] &^= el.padmask() // mask off the padding bits
t.InitBytes(b, &ec.P)
// u = (1-t)/(1+t)
u.Div(t1.Sub(&ec.one, &t), t2.Add(&ec.one, &t))
// v = u^5 + (r^2-2)u^3 + u
u2.Mul(&u, &u) // u2 = u^2
v.Mul(&u2, &u2) // v = u^4
v.Add(&v, t1.Mul(&el.r2m2, &u2)) // v = u^4 + (r^2-2)u^2
v.Add(&v, &ec.one).Mul(&v, &u) // v = u^5 + (r^2-2)u^3 + u
// X = Chi(v)u
chi(&Chiv, &v)
X.Mul(&Chiv, &u)
// Y = (Chi(v)v)^((q+1)/4) Chi(v) Chi(u^2+1/c^2)
t1.Add(&u2, &el.invc2)
chi(&t1, &t1) // t1 = Chi(u^2+1/c^2)
Y.Mul(&Chiv, &v)
Y.Exp(&Y, &el.pp1d4).Mul(&Y, &Chiv).Mul(&Y, &t1)
// x = (c-1)sX(1+X)/Y
x.Add(&ec.one, &X).Mul(&X, &x).Mul(&el.cm1s, &x).Div(&x, &Y)
// y = (rX-(1+X)^2)/(rX+(1+X)^2)
t1.Mul(&el.r, &X) // t1 = rX
t2.Add(&ec.one, &X).Mul(&t2, &t2) // t2 = (1+X)^2
y.Div(u.Sub(&t1, &t2), v.Add(&t1, &t2))
// Sanity-check
if !ec.onCurve(&x, &y) {
panic("elligator1 produced invalid point")
}
P.initXY(&x.V, &y.V, ec.self)
}
// Check the validity of the T coordinate
func (P *extPoint) checkT() {
var t1, t2 nist.Int
if !t1.Mul(&P.X, &P.Y).Equal(t2.Mul(&P.Z, &P.T)) {
panic("oops")
}
}
// Convert from Montgomery form (u,v) to Edwards (x,y) via:
//
// x = sqrt(B)u/v
// y = (u-1)/(u+1)
//
func (el *el2param) mont2ed(x, y, u, v *nist.Int) {
ec := el.ec
var t1, t2 nist.Int
x.Mul(u, &el.sqrtB).Div(x, v)
y.Div(t1.Sub(u, &ec.one), t2.Add(u, &ec.one))
}
// Convert point from Twisted Edwards form: ax^2+y^2 = 1+dx^2y^2
// to Montgomery form: v^2 = u^3+Au^2+u
// via the equivalence:
//
// u = (1+y)/(1-y)
// v = sqrt(B)u/x
//
// where A=2(a+d)/(a-d) and B=4(a-d)
//
// Beware: the Twisted Edwards Curves paper uses B as a factor for v^2,
// whereas the Elligator 2 paper uses B as a factor for the last u term.
//
func (el *el2param) ed2mont(u, v, x, y *nist.Int) {
ec := el.ec
var t1, t2 nist.Int
u.Div(t1.Add(&ec.one, y), t2.Sub(&ec.one, y))
v.Mul(u, &el.sqrtB).Div(v, x)
}
