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// Copyright (c) 2021 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package edwards25519
// This file contains additional functionality that is not included in the
// upstream crypto/internal/edwards25519 package.
import (
"errors"
"filippo.io/edwards25519/field"
)
// ExtendedCoordinates returns v in extended coordinates (X:Y:Z:T) where
// x = X/Z, y = Y/Z, and xy = T/Z as in https://eprint.iacr.org/2008/522.
func (v *Point) ExtendedCoordinates() (X, Y, Z, T *field.Element) {
// This function is outlined to make the allocations inline in the caller
// rather than happen on the heap. Don't change the style without making
// sure it doesn't increase the inliner cost.
var e [4]field.Element
X, Y, Z, T = v.extendedCoordinates(&e)
return
}
func (v *Point) extendedCoordinates(e *[4]field.Element) (X, Y, Z, T *field.Element) {
checkInitialized(v)
X = e[0].Set(&v.x)
Y = e[1].Set(&v.y)
Z = e[2].Set(&v.z)
T = e[3].Set(&v.t)
return
}
// SetExtendedCoordinates sets v = (X:Y:Z:T) in extended coordinates where
// x = X/Z, y = Y/Z, and xy = T/Z as in https://eprint.iacr.org/2008/522.
//
// If the coordinates are invalid or don't represent a valid point on the curve,
// SetExtendedCoordinates returns nil and an error and the receiver is
// unchanged. Otherwise, SetExtendedCoordinates returns v.
func (v *Point) SetExtendedCoordinates(X, Y, Z, T *field.Element) (*Point, error) {
if !isOnCurve(X, Y, Z, T) {
return nil, errors.New("edwards25519: invalid point coordinates")
}
v.x.Set(X)
v.y.Set(Y)
v.z.Set(Z)
v.t.Set(T)
return v, nil
}
func isOnCurve(X, Y, Z, T *field.Element) bool {
var lhs, rhs field.Element
XX := new(field.Element).Square(X)
YY := new(field.Element).Square(Y)
ZZ := new(field.Element).Square(Z)
TT := new(field.Element).Square(T)
// -x² + y² = 1 + dx²y²
// -(X/Z)² + (Y/Z)² = 1 + d(T/Z)²
// -X² + Y² = Z² + dT²
lhs.Subtract(YY, XX)
rhs.Multiply(d, TT).Add(&rhs, ZZ)
if lhs.Equal(&rhs) != 1 {
return false
}
// xy = T/Z
// XY/Z² = T/Z
// XY = TZ
lhs.Multiply(X, Y)
rhs.Multiply(T, Z)
return lhs.Equal(&rhs) == 1
}
// BytesMontgomery converts v to a point on the birationally-equivalent
// Curve25519 Montgomery curve, and returns its canonical 32 bytes encoding
// according to RFC 7748.
//
// Note that BytesMontgomery only encodes the u-coordinate, so v and -v encode
// to the same value. If v is the identity point, BytesMontgomery returns 32
// zero bytes, analogously to the X25519 function.
//
// The lack of an inverse operation (such as SetMontgomeryBytes) is deliberate:
// while every valid edwards25519 point has a unique u-coordinate Montgomery
// encoding, X25519 accepts inputs on the quadratic twist, which don't correspond
// to any edwards25519 point, and every other X25519 input corresponds to two
// edwards25519 points.
func (v *Point) BytesMontgomery() []byte {
// This function is outlined to make the allocations inline in the caller
// rather than happen on the heap.
var buf [32]byte
return v.bytesMontgomery(&buf)
}
func (v *Point) bytesMontgomery(buf *[32]byte) []byte {
checkInitialized(v)
// RFC 7748, Section 4.1 provides the bilinear map to calculate the
// Montgomery u-coordinate
//
// u = (1 + y) / (1 - y)
//
// where y = Y / Z.
var y, recip, u field.Element
y.Multiply(&v.y, y.Invert(&v.z)) // y = Y / Z
recip.Invert(recip.Subtract(feOne, &y)) // r = 1/(1 - y)
u.Multiply(u.Add(feOne, &y), &recip) // u = (1 + y)*r
return copyFieldElement(buf, &u)
}
// MultByCofactor sets v = 8 * p, and returns v.
func (v *Point) MultByCofactor(p *Point) *Point {
checkInitialized(p)
result := projP1xP1{}
pp := (&projP2{}).FromP3(p)
result.Double(pp)
pp.FromP1xP1(&result)
result.Double(pp)
pp.FromP1xP1(&result)
result.Double(pp)
return v.fromP1xP1(&result)
}
// Given k > 0, set s = s**(2*i).
func (s *Scalar) pow2k(k int) {
for i := 0; i < k; i++ {
s.Multiply(s, s)
}
}
// Invert sets s to the inverse of a nonzero scalar v, and returns s.
//
// If t is zero, Invert returns zero.
func (s *Scalar) Invert(t *Scalar) *Scalar {
// Uses a hardcoded sliding window of width 4.
var table [8]Scalar
var tt Scalar
tt.Multiply(t, t)
table[0] = *t
for i := 0; i < 7; i++ {
table[i+1].Multiply(&table[i], &tt)
}
// Now table = [t**1, t**3, t**5, t**7, t**9, t**11, t**13, t**15]
// so t**k = t[k/2] for odd k
// To compute the sliding window digits, use the following Sage script:
// sage: import itertools
// sage: def sliding_window(w,k):
// ....: digits = []
// ....: while k > 0:
// ....: if k % 2 == 1:
// ....: kmod = k % (2**w)
// ....: digits.append(kmod)
// ....: k = k - kmod
// ....: else:
// ....: digits.append(0)
// ....: k = k // 2
// ....: return digits
// Now we can compute s roughly as follows:
// sage: s = 1
// sage: for coeff in reversed(sliding_window(4,l-2)):
// ....: s = s*s
// ....: if coeff > 0 :
// ....: s = s*t**coeff
// This works on one bit at a time, with many runs of zeros.
// The digits can be collapsed into [(count, coeff)] as follows:
// sage: [(len(list(group)),d) for d,group in itertools.groupby(sliding_window(4,l-2))]
// Entries of the form (k, 0) turn into pow2k(k)
// Entries of the form (1, coeff) turn into a squaring and then a table lookup.
// We can fold the squaring into the previous pow2k(k) as pow2k(k+1).
*s = table[1/2]
s.pow2k(127 + 1)
s.Multiply(s, &table[1/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[9/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[11/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[13/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[15/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[7/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[15/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[5/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[1/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[15/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[15/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[7/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[3/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[11/2])
s.pow2k(5 + 1)
s.Multiply(s, &table[11/2])
s.pow2k(9 + 1)
s.Multiply(s, &table[9/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[3/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[3/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[3/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[9/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[7/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[3/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[13/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[7/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[9/2])
s.pow2k(3 + 1)
s.Multiply(s, &table[15/2])
s.pow2k(4 + 1)
s.Multiply(s, &table[11/2])
return s
}
// MultiScalarMult sets v = sum(scalars[i] * points[i]), and returns v.
//
// Execution time depends only on the lengths of the two slices, which must match.
func (v *Point) MultiScalarMult(scalars []*Scalar, points []*Point) *Point {
if len(scalars) != len(points) {
panic("edwards25519: called MultiScalarMult with different size inputs")
}
checkInitialized(points...)
// Proceed as in the single-base case, but share doublings
// between each point in the multiscalar equation.
// Build lookup tables for each point
tables := make([]projLookupTable, len(points))
for i := range tables {
tables[i].FromP3(points[i])
}
// Compute signed radix-16 digits for each scalar
digits := make([][64]int8, len(scalars))
for i := range digits {
digits[i] = scalars[i].signedRadix16()
}
// Unwrap first loop iteration to save computing 16*identity
multiple := &projCached{}
tmp1 := &projP1xP1{}
tmp2 := &projP2{}
// Lookup-and-add the appropriate multiple of each input point
for j := range tables {
tables[j].SelectInto(multiple, digits[j][63])
tmp1.Add(v, multiple) // tmp1 = v + x_(j,63)*Q in P1xP1 coords
v.fromP1xP1(tmp1) // update v
}
tmp2.FromP3(v) // set up tmp2 = v in P2 coords for next iteration
for i := 62; i >= 0; i-- {
tmp1.Double(tmp2) // tmp1 = 2*(prev) in P1xP1 coords
tmp2.FromP1xP1(tmp1) // tmp2 = 2*(prev) in P2 coords
tmp1.Double(tmp2) // tmp1 = 4*(prev) in P1xP1 coords
tmp2.FromP1xP1(tmp1) // tmp2 = 4*(prev) in P2 coords
tmp1.Double(tmp2) // tmp1 = 8*(prev) in P1xP1 coords
tmp2.FromP1xP1(tmp1) // tmp2 = 8*(prev) in P2 coords
tmp1.Double(tmp2) // tmp1 = 16*(prev) in P1xP1 coords
v.fromP1xP1(tmp1) // v = 16*(prev) in P3 coords
// Lookup-and-add the appropriate multiple of each input point
for j := range tables {
tables[j].SelectInto(multiple, digits[j][i])
tmp1.Add(v, multiple) // tmp1 = v + x_(j,i)*Q in P1xP1 coords
v.fromP1xP1(tmp1) // update v
}
tmp2.FromP3(v) // set up tmp2 = v in P2 coords for next iteration
}
return v
}
// VarTimeMultiScalarMult sets v = sum(scalars[i] * points[i]), and returns v.
//
// Execution time depends on the inputs.
func (v *Point) VarTimeMultiScalarMult(scalars []*Scalar, points []*Point) *Point {
if len(scalars) != len(points) {
panic("edwards25519: called VarTimeMultiScalarMult with different size inputs")
}
checkInitialized(points...)
// Generalize double-base NAF computation to arbitrary sizes.
// Here all the points are dynamic, so we only use the smaller
// tables.
// Build lookup tables for each point
tables := make([]nafLookupTable5, len(points))
for i := range tables {
tables[i].FromP3(points[i])
}
// Compute a NAF for each scalar
nafs := make([][256]int8, len(scalars))
for i := range nafs {
nafs[i] = scalars[i].nonAdjacentForm(5)
}
multiple := &projCached{}
tmp1 := &projP1xP1{}
tmp2 := &projP2{}
tmp2.Zero()
// Move from high to low bits, doubling the accumulator
// at each iteration and checking whether there is a nonzero
// coefficient to look up a multiple of.
//
// Skip trying to find the first nonzero coefficent, because
// searching might be more work than a few extra doublings.
for i := 255; i >= 0; i-- {
tmp1.Double(tmp2)
for j := range nafs {
if nafs[j][i] > 0 {
v.fromP1xP1(tmp1)
tables[j].SelectInto(multiple, nafs[j][i])
tmp1.Add(v, multiple)
} else if nafs[j][i] < 0 {
v.fromP1xP1(tmp1)
tables[j].SelectInto(multiple, -nafs[j][i])
tmp1.Sub(v, multiple)
}
}
tmp2.FromP1xP1(tmp1)
}
v.fromP2(tmp2)
return v
}
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