428 lines
10 KiB
Go
428 lines
10 KiB
Go
// Copyright (c) 2017 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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import (
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"errors"
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"filippo.io/edwards25519/field"
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)
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// Point types.
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type projP1xP1 struct {
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X, Y, Z, T field.Element
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}
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type projP2 struct {
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X, Y, Z field.Element
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}
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// Point represents a point on the edwards25519 curve.
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//
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// This type works similarly to math/big.Int, and all arguments and receivers
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// are allowed to alias.
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//
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// The zero value is NOT valid, and it may be used only as a receiver.
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type Point struct {
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// Make the type not comparable (i.e. used with == or as a map key), as
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// equivalent points can be represented by different Go values.
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_ incomparable
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// The point is internally represented in extended coordinates (X, Y, Z, T)
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// where x = X/Z, y = Y/Z, and xy = T/Z per https://eprint.iacr.org/2008/522.
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x, y, z, t field.Element
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}
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type incomparable [0]func()
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func checkInitialized(points ...*Point) {
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for _, p := range points {
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if p.x == (field.Element{}) && p.y == (field.Element{}) {
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panic("edwards25519: use of uninitialized Point")
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}
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}
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}
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type projCached struct {
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YplusX, YminusX, Z, T2d field.Element
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}
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type affineCached struct {
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YplusX, YminusX, T2d field.Element
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}
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// Constructors.
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func (v *projP2) Zero() *projP2 {
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v.X.Zero()
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v.Y.One()
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v.Z.One()
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return v
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}
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// identity is the point at infinity.
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var identity, _ = new(Point).SetBytes([]byte{
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1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0})
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// NewIdentityPoint returns a new Point set to the identity.
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func NewIdentityPoint() *Point {
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return new(Point).Set(identity)
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}
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// generator is the canonical curve basepoint. See TestGenerator for the
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// correspondence of this encoding with the values in RFC 8032.
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var generator, _ = new(Point).SetBytes([]byte{
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0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
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0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
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0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66,
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0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66})
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// NewGeneratorPoint returns a new Point set to the canonical generator.
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func NewGeneratorPoint() *Point {
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return new(Point).Set(generator)
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}
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func (v *projCached) Zero() *projCached {
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v.YplusX.One()
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v.YminusX.One()
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v.Z.One()
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v.T2d.Zero()
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return v
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}
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func (v *affineCached) Zero() *affineCached {
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v.YplusX.One()
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v.YminusX.One()
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v.T2d.Zero()
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return v
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}
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// Assignments.
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// Set sets v = u, and returns v.
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func (v *Point) Set(u *Point) *Point {
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*v = *u
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return v
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}
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// Encoding.
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// Bytes returns the canonical 32-byte encoding of v, according to RFC 8032,
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// Section 5.1.2.
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func (v *Point) Bytes() []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.bytes(&buf)
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}
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func (v *Point) bytes(buf *[32]byte) []byte {
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checkInitialized(v)
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var zInv, x, y field.Element
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zInv.Invert(&v.z) // zInv = 1 / Z
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x.Multiply(&v.x, &zInv) // x = X / Z
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y.Multiply(&v.y, &zInv) // y = Y / Z
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out := copyFieldElement(buf, &y)
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out[31] |= byte(x.IsNegative() << 7)
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return out
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}
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var feOne = new(field.Element).One()
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// SetBytes sets v = x, where x is a 32-byte encoding of v. If x does not
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// represent a valid point on the curve, SetBytes returns nil and an error and
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// the receiver is unchanged. Otherwise, SetBytes returns v.
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//
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// Note that SetBytes accepts all non-canonical encodings of valid points.
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// That is, it follows decoding rules that match most implementations in
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// the ecosystem rather than RFC 8032.
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func (v *Point) SetBytes(x []byte) (*Point, error) {
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// Specifically, the non-canonical encodings that are accepted are
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// 1) the ones where the field element is not reduced (see the
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// (*field.Element).SetBytes docs) and
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// 2) the ones where the x-coordinate is zero and the sign bit is set.
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//
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// Read more at https://hdevalence.ca/blog/2020-10-04-its-25519am,
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// specifically the "Canonical A, R" section.
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y, err := new(field.Element).SetBytes(x)
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if err != nil {
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return nil, errors.New("edwards25519: invalid point encoding length")
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}
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// -x² + y² = 1 + dx²y²
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// x² + dx²y² = x²(dy² + 1) = y² - 1
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// x² = (y² - 1) / (dy² + 1)
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// u = y² - 1
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y2 := new(field.Element).Square(y)
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u := new(field.Element).Subtract(y2, feOne)
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// v = dy² + 1
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vv := new(field.Element).Multiply(y2, d)
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vv = vv.Add(vv, feOne)
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// x = +√(u/v)
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xx, wasSquare := new(field.Element).SqrtRatio(u, vv)
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if wasSquare == 0 {
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return nil, errors.New("edwards25519: invalid point encoding")
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}
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// Select the negative square root if the sign bit is set.
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xxNeg := new(field.Element).Negate(xx)
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xx = xx.Select(xxNeg, xx, int(x[31]>>7))
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v.x.Set(xx)
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v.y.Set(y)
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v.z.One()
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v.t.Multiply(xx, y) // xy = T / Z
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return v, nil
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}
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func copyFieldElement(buf *[32]byte, v *field.Element) []byte {
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copy(buf[:], v.Bytes())
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return buf[:]
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}
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// Conversions.
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func (v *projP2) FromP1xP1(p *projP1xP1) *projP2 {
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v.X.Multiply(&p.X, &p.T)
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v.Y.Multiply(&p.Y, &p.Z)
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v.Z.Multiply(&p.Z, &p.T)
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return v
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}
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func (v *projP2) FromP3(p *Point) *projP2 {
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v.X.Set(&p.x)
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v.Y.Set(&p.y)
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v.Z.Set(&p.z)
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return v
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}
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func (v *Point) fromP1xP1(p *projP1xP1) *Point {
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v.x.Multiply(&p.X, &p.T)
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v.y.Multiply(&p.Y, &p.Z)
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v.z.Multiply(&p.Z, &p.T)
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v.t.Multiply(&p.X, &p.Y)
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return v
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}
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func (v *Point) fromP2(p *projP2) *Point {
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v.x.Multiply(&p.X, &p.Z)
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v.y.Multiply(&p.Y, &p.Z)
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v.z.Square(&p.Z)
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v.t.Multiply(&p.X, &p.Y)
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return v
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}
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// d is a constant in the curve equation.
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var d, _ = new(field.Element).SetBytes([]byte{
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0xa3, 0x78, 0x59, 0x13, 0xca, 0x4d, 0xeb, 0x75,
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0xab, 0xd8, 0x41, 0x41, 0x4d, 0x0a, 0x70, 0x00,
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0x98, 0xe8, 0x79, 0x77, 0x79, 0x40, 0xc7, 0x8c,
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0x73, 0xfe, 0x6f, 0x2b, 0xee, 0x6c, 0x03, 0x52})
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var d2 = new(field.Element).Add(d, d)
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func (v *projCached) FromP3(p *Point) *projCached {
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v.YplusX.Add(&p.y, &p.x)
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v.YminusX.Subtract(&p.y, &p.x)
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v.Z.Set(&p.z)
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v.T2d.Multiply(&p.t, d2)
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return v
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}
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func (v *affineCached) FromP3(p *Point) *affineCached {
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v.YplusX.Add(&p.y, &p.x)
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v.YminusX.Subtract(&p.y, &p.x)
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v.T2d.Multiply(&p.t, d2)
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var invZ field.Element
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invZ.Invert(&p.z)
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v.YplusX.Multiply(&v.YplusX, &invZ)
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v.YminusX.Multiply(&v.YminusX, &invZ)
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v.T2d.Multiply(&v.T2d, &invZ)
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return v
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}
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// (Re)addition and subtraction.
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// Add sets v = p + q, and returns v.
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func (v *Point) Add(p, q *Point) *Point {
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checkInitialized(p, q)
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qCached := new(projCached).FromP3(q)
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result := new(projP1xP1).Add(p, qCached)
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return v.fromP1xP1(result)
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}
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// Subtract sets v = p - q, and returns v.
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func (v *Point) Subtract(p, q *Point) *Point {
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checkInitialized(p, q)
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qCached := new(projCached).FromP3(q)
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result := new(projP1xP1).Sub(p, qCached)
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return v.fromP1xP1(result)
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}
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func (v *projP1xP1) Add(p *Point, q *projCached) *projP1xP1 {
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var YplusX, YminusX, PP, MM, TT2d, ZZ2 field.Element
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YplusX.Add(&p.y, &p.x)
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YminusX.Subtract(&p.y, &p.x)
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PP.Multiply(&YplusX, &q.YplusX)
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MM.Multiply(&YminusX, &q.YminusX)
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TT2d.Multiply(&p.t, &q.T2d)
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ZZ2.Multiply(&p.z, &q.Z)
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ZZ2.Add(&ZZ2, &ZZ2)
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v.X.Subtract(&PP, &MM)
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v.Y.Add(&PP, &MM)
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v.Z.Add(&ZZ2, &TT2d)
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v.T.Subtract(&ZZ2, &TT2d)
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return v
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}
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func (v *projP1xP1) Sub(p *Point, q *projCached) *projP1xP1 {
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var YplusX, YminusX, PP, MM, TT2d, ZZ2 field.Element
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YplusX.Add(&p.y, &p.x)
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YminusX.Subtract(&p.y, &p.x)
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PP.Multiply(&YplusX, &q.YminusX) // flipped sign
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MM.Multiply(&YminusX, &q.YplusX) // flipped sign
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TT2d.Multiply(&p.t, &q.T2d)
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ZZ2.Multiply(&p.z, &q.Z)
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ZZ2.Add(&ZZ2, &ZZ2)
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v.X.Subtract(&PP, &MM)
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v.Y.Add(&PP, &MM)
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v.Z.Subtract(&ZZ2, &TT2d) // flipped sign
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v.T.Add(&ZZ2, &TT2d) // flipped sign
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return v
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}
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func (v *projP1xP1) AddAffine(p *Point, q *affineCached) *projP1xP1 {
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var YplusX, YminusX, PP, MM, TT2d, Z2 field.Element
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YplusX.Add(&p.y, &p.x)
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YminusX.Subtract(&p.y, &p.x)
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PP.Multiply(&YplusX, &q.YplusX)
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MM.Multiply(&YminusX, &q.YminusX)
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TT2d.Multiply(&p.t, &q.T2d)
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Z2.Add(&p.z, &p.z)
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v.X.Subtract(&PP, &MM)
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v.Y.Add(&PP, &MM)
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v.Z.Add(&Z2, &TT2d)
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v.T.Subtract(&Z2, &TT2d)
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return v
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}
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func (v *projP1xP1) SubAffine(p *Point, q *affineCached) *projP1xP1 {
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var YplusX, YminusX, PP, MM, TT2d, Z2 field.Element
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YplusX.Add(&p.y, &p.x)
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YminusX.Subtract(&p.y, &p.x)
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PP.Multiply(&YplusX, &q.YminusX) // flipped sign
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MM.Multiply(&YminusX, &q.YplusX) // flipped sign
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TT2d.Multiply(&p.t, &q.T2d)
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Z2.Add(&p.z, &p.z)
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v.X.Subtract(&PP, &MM)
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v.Y.Add(&PP, &MM)
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v.Z.Subtract(&Z2, &TT2d) // flipped sign
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v.T.Add(&Z2, &TT2d) // flipped sign
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return v
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}
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// Doubling.
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func (v *projP1xP1) Double(p *projP2) *projP1xP1 {
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var XX, YY, ZZ2, XplusYsq field.Element
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XX.Square(&p.X)
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YY.Square(&p.Y)
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ZZ2.Square(&p.Z)
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ZZ2.Add(&ZZ2, &ZZ2)
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XplusYsq.Add(&p.X, &p.Y)
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XplusYsq.Square(&XplusYsq)
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v.Y.Add(&YY, &XX)
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v.Z.Subtract(&YY, &XX)
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v.X.Subtract(&XplusYsq, &v.Y)
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v.T.Subtract(&ZZ2, &v.Z)
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return v
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}
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// Negation.
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// Negate sets v = -p, and returns v.
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func (v *Point) Negate(p *Point) *Point {
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checkInitialized(p)
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v.x.Negate(&p.x)
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v.y.Set(&p.y)
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v.z.Set(&p.z)
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v.t.Negate(&p.t)
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return v
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}
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// Equal returns 1 if v is equivalent to u, and 0 otherwise.
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func (v *Point) Equal(u *Point) int {
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checkInitialized(v, u)
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var t1, t2, t3, t4 field.Element
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t1.Multiply(&v.x, &u.z)
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t2.Multiply(&u.x, &v.z)
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t3.Multiply(&v.y, &u.z)
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t4.Multiply(&u.y, &v.z)
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return t1.Equal(&t2) & t3.Equal(&t4)
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}
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// Constant-time operations
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// Select sets v to a if cond == 1 and to b if cond == 0.
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func (v *projCached) Select(a, b *projCached, cond int) *projCached {
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v.YplusX.Select(&a.YplusX, &b.YplusX, cond)
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v.YminusX.Select(&a.YminusX, &b.YminusX, cond)
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v.Z.Select(&a.Z, &b.Z, cond)
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v.T2d.Select(&a.T2d, &b.T2d, cond)
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return v
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}
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// Select sets v to a if cond == 1 and to b if cond == 0.
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func (v *affineCached) Select(a, b *affineCached, cond int) *affineCached {
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v.YplusX.Select(&a.YplusX, &b.YplusX, cond)
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v.YminusX.Select(&a.YminusX, &b.YminusX, cond)
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v.T2d.Select(&a.T2d, &b.T2d, cond)
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return v
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}
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// CondNeg negates v if cond == 1 and leaves it unchanged if cond == 0.
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func (v *projCached) CondNeg(cond int) *projCached {
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v.YplusX.Swap(&v.YminusX, cond)
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v.T2d.Select(new(field.Element).Negate(&v.T2d), &v.T2d, cond)
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return v
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}
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// CondNeg negates v if cond == 1 and leaves it unchanged if cond == 0.
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func (v *affineCached) CondNeg(cond int) *affineCached {
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v.YplusX.Swap(&v.YminusX, cond)
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v.T2d.Select(new(field.Element).Negate(&v.T2d), &v.T2d, cond)
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return v
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}
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