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Mini Shell
Mini Shell
// Copyright 2022 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.
// Code generated by generate.go. DO NOT EDIT.
package nistec
import (
"crypto/internal/nistec/fiat"
"crypto/subtle"
"errors"
"sync"
)
var p521B, _ = new(fiat.P521Element).SetBytes([]byte{0x0, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85, 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3, 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1, 0x9, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e, 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1, 0xbf, 0x7, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c, 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50, 0x3f, 0x0})
var p521G, _ = NewP521Point().SetBytes([]byte{0x4, 0x0, 0xc6, 0x85, 0x8e, 0x6, 0xb7, 0x4, 0x4, 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95, 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x5, 0x3f, 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d, 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7, 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff, 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a, 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5, 0xbd, 0x66, 0x1, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, 0xc0, 0x4, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d, 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b, 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e, 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4, 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x1, 0x3f, 0xad, 0x7, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72, 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1, 0x66, 0x50})
// p521ElementLength is the length of an element of the base or scalar field,
// which have the same bytes length for all NIST P curves.
const p521ElementLength = 66
// P521Point is a P521 point. The zero value is NOT valid.
type P521Point struct {
// The point is represented in projective coordinates (X:Y:Z),
// where x = X/Z and y = Y/Z.
x, y, z *fiat.P521Element
}
// NewP521Point returns a new P521Point representing the point at infinity point.
func NewP521Point() *P521Point {
return &P521Point{
x: new(fiat.P521Element),
y: new(fiat.P521Element).One(),
z: new(fiat.P521Element),
}
}
// NewP521Generator returns a new P521Point set to the canonical generator.
func NewP521Generator() *P521Point {
return (&P521Point{
x: new(fiat.P521Element),
y: new(fiat.P521Element),
z: new(fiat.P521Element),
}).Set(p521G)
}
// Set sets p = q and returns p.
func (p *P521Point) Set(q *P521Point) *P521Point {
p.x.Set(q.x)
p.y.Set(q.y)
p.z.Set(q.z)
return p
}
// SetBytes sets p to the compressed, uncompressed, or infinity value encoded in
// b, as specified in SEC 1, Version 2.0, Section 2.3.4. If the point is not on
// the curve, it returns nil and an error, and the receiver is unchanged.
// Otherwise, it returns p.
func (p *P521Point) SetBytes(b []byte) (*P521Point, error) {
switch {
// Point at infinity.
case len(b) == 1 && b[0] == 0:
return p.Set(NewP521Point()), nil
// Uncompressed form.
case len(b) == 1+2*p521ElementLength && b[0] == 4:
x, err := new(fiat.P521Element).SetBytes(b[1 : 1+p521ElementLength])
if err != nil {
return nil, err
}
y, err := new(fiat.P521Element).SetBytes(b[1+p521ElementLength:])
if err != nil {
return nil, err
}
if err := p521CheckOnCurve(x, y); err != nil {
return nil, err
}
p.x.Set(x)
p.y.Set(y)
p.z.One()
return p, nil
// Compressed form.
case len(b) == 1+p521ElementLength && (b[0] == 2 || b[0] == 3):
x, err := new(fiat.P521Element).SetBytes(b[1:])
if err != nil {
return nil, err
}
// y² = x³ - 3x + b
y := p521Polynomial(new(fiat.P521Element), x)
if !p521Sqrt(y, y) {
return nil, errors.New("invalid P521 compressed point encoding")
}
// Select the positive or negative root, as indicated by the least
// significant bit, based on the encoding type byte.
otherRoot := new(fiat.P521Element)
otherRoot.Sub(otherRoot, y)
cond := y.Bytes()[p521ElementLength-1]&1 ^ b[0]&1
y.Select(otherRoot, y, int(cond))
p.x.Set(x)
p.y.Set(y)
p.z.One()
return p, nil
default:
return nil, errors.New("invalid P521 point encoding")
}
}
// p521Polynomial sets y2 to x³ - 3x + b, and returns y2.
func p521Polynomial(y2, x *fiat.P521Element) *fiat.P521Element {
y2.Square(x)
y2.Mul(y2, x)
threeX := new(fiat.P521Element).Add(x, x)
threeX.Add(threeX, x)
y2.Sub(y2, threeX)
return y2.Add(y2, p521B)
}
func p521CheckOnCurve(x, y *fiat.P521Element) error {
// y² = x³ - 3x + b
rhs := p521Polynomial(new(fiat.P521Element), x)
lhs := new(fiat.P521Element).Square(y)
if rhs.Equal(lhs) != 1 {
return errors.New("P521 point not on curve")
}
return nil
}
// Bytes returns the uncompressed or infinity encoding of p, as specified in
// SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the point at
// infinity is shorter than all other encodings.
func (p *P521Point) Bytes() []byte {
// This function is outlined to make the allocations inline in the caller
// rather than happen on the heap.
var out [1 + 2*p521ElementLength]byte
return p.bytes(&out)
}
func (p *P521Point) bytes(out *[1 + 2*p521ElementLength]byte) []byte {
if p.z.IsZero() == 1 {
return append(out[:0], 0)
}
zinv := new(fiat.P521Element).Invert(p.z)
x := new(fiat.P521Element).Mul(p.x, zinv)
y := new(fiat.P521Element).Mul(p.y, zinv)
buf := append(out[:0], 4)
buf = append(buf, x.Bytes()...)
buf = append(buf, y.Bytes()...)
return buf
}
// BytesCompressed returns the compressed or infinity encoding of p, as
// specified in SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the
// point at infinity is shorter than all other encodings.
func (p *P521Point) BytesCompressed() []byte {
// This function is outlined to make the allocations inline in the caller
// rather than happen on the heap.
var out [1 + p521ElementLength]byte
return p.bytesCompressed(&out)
}
func (p *P521Point) bytesCompressed(out *[1 + p521ElementLength]byte) []byte {
if p.z.IsZero() == 1 {
return append(out[:0], 0)
}
zinv := new(fiat.P521Element).Invert(p.z)
x := new(fiat.P521Element).Mul(p.x, zinv)
y := new(fiat.P521Element).Mul(p.y, zinv)
// Encode the sign of the y coordinate (indicated by the least significant
// bit) as the encoding type (2 or 3).
buf := append(out[:0], 2)
buf[0] |= y.Bytes()[p521ElementLength-1] & 1
buf = append(buf, x.Bytes()...)
return buf
}
// Add sets q = p1 + p2, and returns q. The points may overlap.
func (q *P521Point) Add(p1, p2 *P521Point) *P521Point {
// Complete addition formula for a = -3 from "Complete addition formulas for
// prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2.
t0 := new(fiat.P521Element).Mul(p1.x, p2.x) // t0 := X1 * X2
t1 := new(fiat.P521Element).Mul(p1.y, p2.y) // t1 := Y1 * Y2
t2 := new(fiat.P521Element).Mul(p1.z, p2.z) // t2 := Z1 * Z2
t3 := new(fiat.P521Element).Add(p1.x, p1.y) // t3 := X1 + Y1
t4 := new(fiat.P521Element).Add(p2.x, p2.y) // t4 := X2 + Y2
t3.Mul(t3, t4) // t3 := t3 * t4
t4.Add(t0, t1) // t4 := t0 + t1
t3.Sub(t3, t4) // t3 := t3 - t4
t4.Add(p1.y, p1.z) // t4 := Y1 + Z1
x3 := new(fiat.P521Element).Add(p2.y, p2.z) // X3 := Y2 + Z2
t4.Mul(t4, x3) // t4 := t4 * X3
x3.Add(t1, t2) // X3 := t1 + t2
t4.Sub(t4, x3) // t4 := t4 - X3
x3.Add(p1.x, p1.z) // X3 := X1 + Z1
y3 := new(fiat.P521Element).Add(p2.x, p2.z) // Y3 := X2 + Z2
x3.Mul(x3, y3) // X3 := X3 * Y3
y3.Add(t0, t2) // Y3 := t0 + t2
y3.Sub(x3, y3) // Y3 := X3 - Y3
z3 := new(fiat.P521Element).Mul(p521B, t2) // Z3 := b * t2
x3.Sub(y3, z3) // X3 := Y3 - Z3
z3.Add(x3, x3) // Z3 := X3 + X3
x3.Add(x3, z3) // X3 := X3 + Z3
z3.Sub(t1, x3) // Z3 := t1 - X3
x3.Add(t1, x3) // X3 := t1 + X3
y3.Mul(p521B, y3) // Y3 := b * Y3
t1.Add(t2, t2) // t1 := t2 + t2
t2.Add(t1, t2) // t2 := t1 + t2
y3.Sub(y3, t2) // Y3 := Y3 - t2
y3.Sub(y3, t0) // Y3 := Y3 - t0
t1.Add(y3, y3) // t1 := Y3 + Y3
y3.Add(t1, y3) // Y3 := t1 + Y3
t1.Add(t0, t0) // t1 := t0 + t0
t0.Add(t1, t0) // t0 := t1 + t0
t0.Sub(t0, t2) // t0 := t0 - t2
t1.Mul(t4, y3) // t1 := t4 * Y3
t2.Mul(t0, y3) // t2 := t0 * Y3
y3.Mul(x3, z3) // Y3 := X3 * Z3
y3.Add(y3, t2) // Y3 := Y3 + t2
x3.Mul(t3, x3) // X3 := t3 * X3
x3.Sub(x3, t1) // X3 := X3 - t1
z3.Mul(t4, z3) // Z3 := t4 * Z3
t1.Mul(t3, t0) // t1 := t3 * t0
z3.Add(z3, t1) // Z3 := Z3 + t1
q.x.Set(x3)
q.y.Set(y3)
q.z.Set(z3)
return q
}
// Double sets q = p + p, and returns q. The points may overlap.
func (q *P521Point) Double(p *P521Point) *P521Point {
// Complete addition formula for a = -3 from "Complete addition formulas for
// prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2.
t0 := new(fiat.P521Element).Square(p.x) // t0 := X ^ 2
t1 := new(fiat.P521Element).Square(p.y) // t1 := Y ^ 2
t2 := new(fiat.P521Element).Square(p.z) // t2 := Z ^ 2
t3 := new(fiat.P521Element).Mul(p.x, p.y) // t3 := X * Y
t3.Add(t3, t3) // t3 := t3 + t3
z3 := new(fiat.P521Element).Mul(p.x, p.z) // Z3 := X * Z
z3.Add(z3, z3) // Z3 := Z3 + Z3
y3 := new(fiat.P521Element).Mul(p521B, t2) // Y3 := b * t2
y3.Sub(y3, z3) // Y3 := Y3 - Z3
x3 := new(fiat.P521Element).Add(y3, y3) // X3 := Y3 + Y3
y3.Add(x3, y3) // Y3 := X3 + Y3
x3.Sub(t1, y3) // X3 := t1 - Y3
y3.Add(t1, y3) // Y3 := t1 + Y3
y3.Mul(x3, y3) // Y3 := X3 * Y3
x3.Mul(x3, t3) // X3 := X3 * t3
t3.Add(t2, t2) // t3 := t2 + t2
t2.Add(t2, t3) // t2 := t2 + t3
z3.Mul(p521B, z3) // Z3 := b * Z3
z3.Sub(z3, t2) // Z3 := Z3 - t2
z3.Sub(z3, t0) // Z3 := Z3 - t0
t3.Add(z3, z3) // t3 := Z3 + Z3
z3.Add(z3, t3) // Z3 := Z3 + t3
t3.Add(t0, t0) // t3 := t0 + t0
t0.Add(t3, t0) // t0 := t3 + t0
t0.Sub(t0, t2) // t0 := t0 - t2
t0.Mul(t0, z3) // t0 := t0 * Z3
y3.Add(y3, t0) // Y3 := Y3 + t0
t0.Mul(p.y, p.z) // t0 := Y * Z
t0.Add(t0, t0) // t0 := t0 + t0
z3.Mul(t0, z3) // Z3 := t0 * Z3
x3.Sub(x3, z3) // X3 := X3 - Z3
z3.Mul(t0, t1) // Z3 := t0 * t1
z3.Add(z3, z3) // Z3 := Z3 + Z3
z3.Add(z3, z3) // Z3 := Z3 + Z3
q.x.Set(x3)
q.y.Set(y3)
q.z.Set(z3)
return q
}
// Select sets q to p1 if cond == 1, and to p2 if cond == 0.
func (q *P521Point) Select(p1, p2 *P521Point, cond int) *P521Point {
q.x.Select(p1.x, p2.x, cond)
q.y.Select(p1.y, p2.y, cond)
q.z.Select(p1.z, p2.z, cond)
return q
}
// A p521Table holds the first 15 multiples of a point at offset -1, so [1]P
// is at table[0], [15]P is at table[14], and [0]P is implicitly the identity
// point.
type p521Table [15]*P521Point
// Select selects the n-th multiple of the table base point into p. It works in
// constant time by iterating over every entry of the table. n must be in [0, 15].
func (table *p521Table) Select(p *P521Point, n uint8) {
if n >= 16 {
panic("nistec: internal error: p521Table called with out-of-bounds value")
}
p.Set(NewP521Point())
for i := uint8(1); i < 16; i++ {
cond := subtle.ConstantTimeByteEq(i, n)
p.Select(table[i-1], p, cond)
}
}
// ScalarMult sets p = scalar * q, and returns p.
func (p *P521Point) ScalarMult(q *P521Point, scalar []byte) (*P521Point, error) {
// Compute a p521Table for the base point q. The explicit NewP521Point
// calls get inlined, letting the allocations live on the stack.
var table = p521Table{NewP521Point(), NewP521Point(), NewP521Point(),
NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(),
NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(),
NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point()}
table[0].Set(q)
for i := 1; i < 15; i += 2 {
table[i].Double(table[i/2])
table[i+1].Add(table[i], q)
}
// Instead of doing the classic double-and-add chain, we do it with a
// four-bit window: we double four times, and then add [0-15]P.
t := NewP521Point()
p.Set(NewP521Point())
for i, byte := range scalar {
// No need to double on the first iteration, as p is the identity at
// this point, and [N]∞ = ∞.
if i != 0 {
p.Double(p)
p.Double(p)
p.Double(p)
p.Double(p)
}
windowValue := byte >> 4
table.Select(t, windowValue)
p.Add(p, t)
p.Double(p)
p.Double(p)
p.Double(p)
p.Double(p)
windowValue = byte & 0b1111
table.Select(t, windowValue)
p.Add(p, t)
}
return p, nil
}
var p521GeneratorTable *[p521ElementLength * 2]p521Table
var p521GeneratorTableOnce sync.Once
// generatorTable returns a sequence of p521Tables. The first table contains
// multiples of G. Each successive table is the previous table doubled four
// times.
func (p *P521Point) generatorTable() *[p521ElementLength * 2]p521Table {
p521GeneratorTableOnce.Do(func() {
p521GeneratorTable = new([p521ElementLength * 2]p521Table)
base := NewP521Generator()
for i := 0; i < p521ElementLength*2; i++ {
p521GeneratorTable[i][0] = NewP521Point().Set(base)
for j := 1; j < 15; j++ {
p521GeneratorTable[i][j] = NewP521Point().Add(p521GeneratorTable[i][j-1], base)
}
base.Double(base)
base.Double(base)
base.Double(base)
base.Double(base)
}
})
return p521GeneratorTable
}
// ScalarBaseMult sets p = scalar * B, where B is the canonical generator, and
// returns p.
func (p *P521Point) ScalarBaseMult(scalar []byte) (*P521Point, error) {
if len(scalar) != p521ElementLength {
return nil, errors.New("invalid scalar length")
}
tables := p.generatorTable()
// This is also a scalar multiplication with a four-bit window like in
// ScalarMult, but in this case the doublings are precomputed. The value
// [windowValue]G added at iteration k would normally get doubled
// (totIterations-k)×4 times, but with a larger precomputation we can
// instead add [2^((totIterations-k)×4)][windowValue]G and avoid the
// doublings between iterations.
t := NewP521Point()
p.Set(NewP521Point())
tableIndex := len(tables) - 1
for _, byte := range scalar {
windowValue := byte >> 4
tables[tableIndex].Select(t, windowValue)
p.Add(p, t)
tableIndex--
windowValue = byte & 0b1111
tables[tableIndex].Select(t, windowValue)
p.Add(p, t)
tableIndex--
}
return p, nil
}
// p521Sqrt sets e to a square root of x. If x is not a square, p521Sqrt returns
// false and e is unchanged. e and x can overlap.
func p521Sqrt(e, x *fiat.P521Element) (isSquare bool) {
candidate := new(fiat.P521Element)
p521SqrtCandidate(candidate, x)
square := new(fiat.P521Element).Square(candidate)
if square.Equal(x) != 1 {
return false
}
e.Set(candidate)
return true
}
// p521SqrtCandidate sets z to a square root candidate for x. z and x must not overlap.
func p521SqrtCandidate(z, x *fiat.P521Element) {
// Since p = 3 mod 4, exponentiation by (p + 1) / 4 yields a square root candidate.
//
// The sequence of 0 multiplications and 519 squarings is derived from the
// following addition chain generated with github.com/mmcloughlin/addchain v0.4.0.
//
// return 1 << 519
//
z.Square(x)
for s := 1; s < 519; s++ {
z.Square(z)
}
}
Zerion Mini Shell 1.0