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