Source File
	p521.go

Belonging Package
	crypto/internal/nistec

// 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"
)

// 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),
	}
}

// SetGenerator sets p to the canonical generator and returns p.
func (p *P521Point) SetGenerator() *P521Point {
	p.x.SetBytes([]byte{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})
	p.y.SetBytes([]byte{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})
	p.z.One()
	return p
}

// 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")
	}
}

var _p521B *fiat.P521Element
var _p521BOnce sync.Once

func p521B() *fiat.P521Element {
	_p521BOnce.Do(func() {
		_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})
	})
	return _p521B
}

// 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
}

// 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 *P521Point) BytesX() ([]byte, error) {
	// This function is outlined to make the allocations inline in the caller
	// rather than happen on the heap.
	var out [p521ElementLength]byte
	return p.bytesX(&out)
}

func (p *P521Point) bytesX(out *[p521ElementLength]byte) ([]byte, error) {
	if p.z.IsZero() == 1 {
		return nil, errors.New("P521 point is the point at infinity")
	}

	zinv := new(fiat.P521Element).Invert(p.z)
	x := new(fiat.P521Element).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 *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 := NewP521Point().SetGenerator()
		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)
	}
}

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