p256.go

// 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.

//go:build purego || (!amd64 && !arm64 && !(ppc64le && go1.19) && !s390x)

package nistec

import (
	"crypto/subtle"
	"errors"
	"filippo.io/nistec/internal/fiat"
	"sync"
)

// p256ElementLength is the length of an element of the base or scalar field,
// which have the same bytes length for all NIST P curves.
const p256ElementLength = 32

// P256Point is a P256 point. The zero value is NOT valid.
type P256Point struct {
	// The point is represented in projective coordinates (X:Y:Z),
	// where x = X/Z and y = Y/Z.
	x, y, z *fiat.P256Element
}

// NewP256Point returns a new P256Point representing the point at infinity point.
func NewP256Point() *P256Point {
	return &P256Point{
		x: new(fiat.P256Element),
		y: new(fiat.P256Element).One(),
		z: new(fiat.P256Element),
	}
}

// SetGenerator sets p to the canonical generator and returns p.
func (p *P256Point) SetGenerator() *P256Point {
	p.x.SetBytes([]byte{0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2, 0x77, 0x3, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0, 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96})
	p.y.SetBytes([]byte{0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0xf, 0x9e, 0x16, 0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce, 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5})
	p.z.One()
	return p
}

// Set sets p = q and returns p.
func (p *P256Point) Set(q *P256Point) *P256Point {
	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 *P256Point) SetBytes(b []byte) (*P256Point, error) {
	switch {
	// Point at infinity.
	case len(b) == 1 && b[0] == 0:
		return p.Set(NewP256Point()), nil

	// Uncompressed form.
	case len(b) == 1+2*p256ElementLength && b[0] == 4:
		x, err := new(fiat.P256Element).SetBytes(b[1 : 1+p256ElementLength])
		if err != nil {
			return nil, err
		}
		y, err := new(fiat.P256Element).SetBytes(b[1+p256ElementLength:])
		if err != nil {
			return nil, err
		}
		if err := p256CheckOnCurve(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+p256ElementLength && (b[0] == 2 || b[0] == 3):
		x, err := new(fiat.P256Element).SetBytes(b[1:])
		if err != nil {
			return nil, err
		}

		// y² = x³ - 3x + b
		y := p256Polynomial(new(fiat.P256Element), x)
		if !p256Sqrt(y, y) {
			return nil, errors.New("invalid P256 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.P256Element)
		otherRoot.Sub(otherRoot, y)
		cond := y.Bytes()[p256ElementLength-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 P256 point encoding")
	}
}

var _p256B *fiat.P256Element
var _p256BOnce sync.Once

func p256B() *fiat.P256Element {
	_p256BOnce.Do(func() {
		_p256B, _ = new(fiat.P256Element).SetBytes([]byte{0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7, 0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc, 0x65, 0x1d, 0x6, 0xb0, 0xcc, 0x53, 0xb0, 0xf6, 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b})
	})
	return _p256B
}

// p256Polynomial sets y2 to x³ - 3x + b, and returns y2.
func p256Polynomial(y2, x *fiat.P256Element) *fiat.P256Element {
	y2.Square(x)
	y2.Mul(y2, x)

	threeX := new(fiat.P256Element).Add(x, x)
	threeX.Add(threeX, x)
	y2.Sub(y2, threeX)

	return y2.Add(y2, p256B())
}

func p256CheckOnCurve(x, y *fiat.P256Element) error {
	// y² = x³ - 3x + b
	rhs := p256Polynomial(new(fiat.P256Element), x)
	lhs := new(fiat.P256Element).Square(y)
	if rhs.Equal(lhs) != 1 {
		return errors.New("P256 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 *P256Point) Bytes() []byte {
	// This function is outlined to make the allocations inline in the caller
	// rather than happen on the heap.
	var out [1 + 2*p256ElementLength]byte
	return p.bytes(&out)
}

func (p *P256Point) bytes(out *[1 + 2*p256ElementLength]byte) []byte {
	if p.z.IsZero() == 1 {
		return append(out[:0], 0)
	}

	zinv := new(fiat.P256Element).Invert(p.z)
	x := new(fiat.P256Element).Mul(p.x, zinv)
	y := new(fiat.P256Element).Mul(p.y, zinv)

	buf := append(out[:0], 4)
	buf = append(buf, x.Bytes()...)
	buf = append(buf, y.Bytes()...)
	return buf
}

// BytesX returns the encoding of the x-coordinate of p, as specified in SEC 1,
// Version 2.0, Section 2.3.5, or an error if p is the point at infinity.
func (p *P256Point) BytesX() ([]byte, error) {
	// This function is outlined to make the allocations inline in the caller
	// rather than happen on the heap.
	var out [p256ElementLength]byte
	return p.bytesX(&out)
}

func (p *P256Point) bytesX(out *[p256ElementLength]byte) ([]byte, error) {
	if p.z.IsZero() == 1 {
		return nil, errors.New("P256 point is the point at infinity")
	}

	zinv := new(fiat.P256Element).Invert(p.z)
	x := new(fiat.P256Element).Mul(p.x, zinv)

	return append(out[:0], x.Bytes()...), nil
}

// 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 *P256Point) BytesCompressed() []byte {
	// This function is outlined to make the allocations inline in the caller
	// rather than happen on the heap.
	var out [1 + p256ElementLength]byte
	return p.bytesCompressed(&out)
}

func (p *P256Point) bytesCompressed(out *[1 + p256ElementLength]byte) []byte {
	if p.z.IsZero() == 1 {
		return append(out[:0], 0)
	}

	zinv := new(fiat.P256Element).Invert(p.z)
	x := new(fiat.P256Element).Mul(p.x, zinv)
	y := new(fiat.P256Element).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()[p256ElementLength-1] & 1
	buf = append(buf, x.Bytes()...)
	return buf
}

// Add sets q = p1 + p2, and returns q. The points may overlap.
func (q *P256Point) Add(p1, p2 *P256Point) *P256Point {
	// 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.P256Element).Mul(p1.x, p2.x)  // t0 := X1 * X2
	t1 := new(fiat.P256Element).Mul(p1.y, p2.y)  // t1 := Y1 * Y2
	t2 := new(fiat.P256Element).Mul(p1.z, p2.z)  // t2 := Z1 * Z2
	t3 := new(fiat.P256Element).Add(p1.x, p1.y)  // t3 := X1 + Y1
	t4 := new(fiat.P256Element).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.P256Element).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.P256Element).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.P256Element).Mul(p256B(), 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(p256B(), 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 *P256Point) Double(p *P256Point) *P256Point {
	// 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.P256Element).Square(p.x)      // t0 := X ^ 2
	t1 := new(fiat.P256Element).Square(p.y)      // t1 := Y ^ 2
	t2 := new(fiat.P256Element).Square(p.z)      // t2 := Z ^ 2
	t3 := new(fiat.P256Element).Mul(p.x, p.y)    // t3 := X * Y
	t3.Add(t3, t3)                               // t3 := t3 + t3
	z3 := new(fiat.P256Element).Mul(p.x, p.z)    // Z3 := X * Z
	z3.Add(z3, z3)                               // Z3 := Z3 + Z3
	y3 := new(fiat.P256Element).Mul(p256B(), t2) // Y3 := b * t2
	y3.Sub(y3, z3)                               // Y3 := Y3 - Z3
	x3 := new(fiat.P256Element).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(p256B(), 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 *P256Point) Select(p1, p2 *P256Point, cond int) *P256Point {
	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 p256Table 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 p256Table [15]*P256Point

// 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 *p256Table) Select(p *P256Point, n uint8) {
	if n >= 16 {
		panic("nistec: internal error: p256Table called with out-of-bounds value")
	}
	p.Set(NewP256Point())
	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 *P256Point) ScalarMult(q *P256Point, scalar []byte) (*P256Point, error) {
	// Compute a p256Table for the base point q. The explicit NewP256Point
	// calls get inlined, letting the allocations live on the stack.
	var table = p256Table{NewP256Point(), NewP256Point(), NewP256Point(),
		NewP256Point(), NewP256Point(), NewP256Point(), NewP256Point(),
		NewP256Point(), NewP256Point(), NewP256Point(), NewP256Point(),
		NewP256Point(), NewP256Point(), NewP256Point(), NewP256Point()}
	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 := NewP256Point()
	p.Set(NewP256Point())
	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 p256GeneratorTable *[p256ElementLength * 2]p256Table
var p256GeneratorTableOnce sync.Once

// generatorTable returns a sequence of p256Tables. The first table contains
// multiples of G. Each successive table is the previous table doubled four
// times.
func (p *P256Point) generatorTable() *[p256ElementLength * 2]p256Table {
	p256GeneratorTableOnce.Do(func() {
		p256GeneratorTable = new([p256ElementLength * 2]p256Table)
		base := NewP256Point().SetGenerator()
		for i := 0; i < p256ElementLength*2; i++ {
			p256GeneratorTable[i][0] = NewP256Point().Set(base)
			for j := 1; j < 15; j++ {
				p256GeneratorTable[i][j] = NewP256Point().Add(p256GeneratorTable[i][j-1], base)
			}
			base.Double(base)
			base.Double(base)
			base.Double(base)
			base.Double(base)
		}
	})
	return p256GeneratorTable
}

// ScalarBaseMult sets p = scalar * B, where B is the canonical generator, and
// returns p.
func (p *P256Point) ScalarBaseMult(scalar []byte) (*P256Point, error) {
	if len(scalar) != p256ElementLength {
		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 := NewP256Point()
	p.Set(NewP256Point())
	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
}

// p256Sqrt sets e to a square root of x. If x is not a square, p256Sqrt returns
// false and e is unchanged. e and x can overlap.
func p256Sqrt(e, x *fiat.P256Element) (isSquare bool) {
	candidate := new(fiat.P256Element)
	p256SqrtCandidate(candidate, x)
	square := new(fiat.P256Element).Square(candidate)
	if square.Equal(x) != 1 {
		return false
	}
	e.Set(candidate)
	return true
}

// p256SqrtCandidate sets z to a square root candidate for x. z and x must not overlap.
func p256SqrtCandidate(z, x *fiat.P256Element) {
	// Since p = 3 mod 4, exponentiation by (p + 1) / 4 yields a square root candidate.
	//
	// The sequence of 7 multiplications and 253 squarings is derived from the
	// following addition chain generated with github.com/mmcloughlin/addchain v0.4.0.
	//
	//	_10       = 2*1
	//	_11       = 1 + _10
	//	_1100     = _11 << 2
	//	_1111     = _11 + _1100
	//	_11110000 = _1111 << 4
	//	_11111111 = _1111 + _11110000
	//	x16       = _11111111 << 8 + _11111111
	//	x32       = x16 << 16 + x16
	//	return      ((x32 << 32 + 1) << 96 + 1) << 94
	//
	var t0 = new(fiat.P256Element)

	z.Square(x)
	z.Mul(x, z)
	t0.Square(z)
	for s := 1; s < 2; s++ {
		t0.Square(t0)
	}
	z.Mul(z, t0)
	t0.Square(z)
	for s := 1; s < 4; s++ {
		t0.Square(t0)
	}
	z.Mul(z, t0)
	t0.Square(z)
	for s := 1; s < 8; s++ {
		t0.Square(t0)
	}
	z.Mul(z, t0)
	t0.Square(z)
	for s := 1; s < 16; s++ {
		t0.Square(t0)
	}
	z.Mul(z, t0)
	for s := 0; s < 32; s++ {
		z.Square(z)
	}
	z.Mul(x, z)
	for s := 0; s < 96; s++ {
		z.Square(z)
	}
	z.Mul(x, z)
	for s := 0; s < 94; s++ {
		z.Square(z)
	}
}