Source File
	p384.go

Belonging Package
	crypto/internal/fips140/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/fips140/nistec/fiat"
	"crypto/internal/fips140/subtle"
	"errors"
	"sync"
)

// p384ElementLength is the length of an element of the base or scalar field,
// which have the same bytes length for all NIST P curves.
const p384ElementLength = 48

// P384Point is a P384 point. The zero value is NOT valid.
type P384Point struct {
	// The point is represented in projective coordinates (X:Y:Z),
	// where x = X/Z and y = Y/Z.
	x, y, z *fiat.P384Element
}

// NewP384Point returns a new P384Point representing the point at infinity point.
func NewP384Point() *P384Point {
	return &P384Point{
		x: new(fiat.P384Element),
		y: new(fiat.P384Element).One(),
		z: new(fiat.P384Element),
	}
}

// SetGenerator sets p to the canonical generator and returns p.
func (p *P384Point) SetGenerator() *P384Point {
	p.x.SetBytes([]byte{0xaa, 0x87, 0xca, 0x22, 0xbe, 0x8b, 0x5, 0x37, 0x8e, 0xb1, 0xc7, 0x1e, 0xf3, 0x20, 0xad, 0x74, 0x6e, 0x1d, 0x3b, 0x62, 0x8b, 0xa7, 0x9b, 0x98, 0x59, 0xf7, 0x41, 0xe0, 0x82, 0x54, 0x2a, 0x38, 0x55, 0x2, 0xf2, 0x5d, 0xbf, 0x55, 0x29, 0x6c, 0x3a, 0x54, 0x5e, 0x38, 0x72, 0x76, 0xa, 0xb7})
	p.y.SetBytes([]byte{0x36, 0x17, 0xde, 0x4a, 0x96, 0x26, 0x2c, 0x6f, 0x5d, 0x9e, 0x98, 0xbf, 0x92, 0x92, 0xdc, 0x29, 0xf8, 0xf4, 0x1d, 0xbd, 0x28, 0x9a, 0x14, 0x7c, 0xe9, 0xda, 0x31, 0x13, 0xb5, 0xf0, 0xb8, 0xc0, 0xa, 0x60, 0xb1, 0xce, 0x1d, 0x7e, 0x81, 0x9d, 0x7a, 0x43, 0x1d, 0x7c, 0x90, 0xea, 0xe, 0x5f})
	p.z.One()
	return p
}

// Set sets p = q and returns p.
func (p *P384Point) Set(q *P384Point) *P384Point {
	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 *P384Point) SetBytes(b []byte) (*P384Point, error) {
	switch {
	// Point at infinity.
	case len(b) == 1 && b[0] == 0:
		return p.Set(NewP384Point()), nil

	// Uncompressed form.
	case len(b) == 1+2*p384ElementLength && b[0] == 4:
		x, err := new(fiat.P384Element).SetBytes(b[1 : 1+p384ElementLength])
		if err != nil {
			return nil, err
		}
		y, err := new(fiat.P384Element).SetBytes(b[1+p384ElementLength:])
		if err != nil {
			return nil, err
		}
		if err := p384CheckOnCurve(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+p384ElementLength && (b[0] == 2 || b[0] == 3):
		x, err := new(fiat.P384Element).SetBytes(b[1:])
		if err != nil {
			return nil, err
		}

		// y² = x³ - 3x + b
		y := p384Polynomial(new(fiat.P384Element), x)
		if !p384Sqrt(y, y) {
			return nil, errors.New("invalid P384 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.P384Element)
		otherRoot.Sub(otherRoot, y)
		cond := y.Bytes()[p384ElementLength-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 P384 point encoding")
	}
}

var _p384B *fiat.P384Element
var _p384BOnce sync.Once

func p384B() *fiat.P384Element {
	_p384BOnce.Do(func() {
		_p384B, _ = new(fiat.P384Element).SetBytes([]byte{0xb3, 0x31, 0x2f, 0xa7, 0xe2, 0x3e, 0xe7, 0xe4, 0x98, 0x8e, 0x5, 0x6b, 0xe3, 0xf8, 0x2d, 0x19, 0x18, 0x1d, 0x9c, 0x6e, 0xfe, 0x81, 0x41, 0x12, 0x3, 0x14, 0x8, 0x8f, 0x50, 0x13, 0x87, 0x5a, 0xc6, 0x56, 0x39, 0x8d, 0x8a, 0x2e, 0xd1, 0x9d, 0x2a, 0x85, 0xc8, 0xed, 0xd3, 0xec, 0x2a, 0xef})
	})
	return _p384B
}

// p384Polynomial sets y2 to x³ - 3x + b, and returns y2.
func p384Polynomial(y2, x *fiat.P384Element) *fiat.P384Element {
	y2.Square(x)
	y2.Mul(y2, x)

	threeX := new(fiat.P384Element).Add(x, x)
	threeX.Add(threeX, x)
	y2.Sub(y2, threeX)

	return y2.Add(y2, p384B())
}

func p384CheckOnCurve(x, y *fiat.P384Element) error {
	// y² = x³ - 3x + b
	rhs := p384Polynomial(new(fiat.P384Element), x)
	lhs := new(fiat.P384Element).Square(y)
	if rhs.Equal(lhs) != 1 {
		return errors.New("P384 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 *P384Point) Bytes() []byte {
	// This function is outlined to make the allocations inline in the caller
	// rather than happen on the heap.
	var out [1 + 2*p384ElementLength]byte
	return p.bytes(&out)
}

func (p *P384Point) bytes(out *[1 + 2*p384ElementLength]byte) []byte {
	if p.z.IsZero() == 1 {
		return append(out[:0], 0)
	}

	zinv := new(fiat.P384Element).Invert(p.z)
	x := new(fiat.P384Element).Mul(p.x, zinv)
	y := new(fiat.P384Element).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 *P384Point) BytesX() ([]byte, error) {
	// This function is outlined to make the allocations inline in the caller
	// rather than happen on the heap.
	var out [p384ElementLength]byte
	return p.bytesX(&out)
}

func (p *P384Point) bytesX(out *[p384ElementLength]byte) ([]byte, error) {
	if p.z.IsZero() == 1 {
		return nil, errors.New("P384 point is the point at infinity")
	}

	zinv := new(fiat.P384Element).Invert(p.z)
	x := new(fiat.P384Element).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 *P384Point) BytesCompressed() []byte {
	// This function is outlined to make the allocations inline in the caller
	// rather than happen on the heap.
	var out [1 + p384ElementLength]byte
	return p.bytesCompressed(&out)
}

func (p *P384Point) bytesCompressed(out *[1 + p384ElementLength]byte) []byte {
	if p.z.IsZero() == 1 {
		return append(out[:0], 0)
	}

	zinv := new(fiat.P384Element).Invert(p.z)
	x := new(fiat.P384Element).Mul(p.x, zinv)
	y := new(fiat.P384Element).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()[p384ElementLength-1] & 1
	buf = append(buf, x.Bytes()...)
	return buf
}

// Add sets q = p1 + p2, and returns q. The points may overlap.
func (q *P384Point) Add(p1, p2 *P384Point) *P384Point {
	// 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.P384Element).Mul(p1.x, p2.x)  // t0 := X1 * X2
	t1 := new(fiat.P384Element).Mul(p1.y, p2.y)  // t1 := Y1 * Y2
	t2 := new(fiat.P384Element).Mul(p1.z, p2.z)  // t2 := Z1 * Z2
	t3 := new(fiat.P384Element).Add(p1.x, p1.y)  // t3 := X1 + Y1
	t4 := new(fiat.P384Element).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.P384Element).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.P384Element).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.P384Element).Mul(p384B(), 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(p384B(), 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 *P384Point) Double(p *P384Point) *P384Point {
	// 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.P384Element).Square(p.x)      // t0 := X ^ 2
	t1 := new(fiat.P384Element).Square(p.y)      // t1 := Y ^ 2
	t2 := new(fiat.P384Element).Square(p.z)      // t2 := Z ^ 2
	t3 := new(fiat.P384Element).Mul(p.x, p.y)    // t3 := X * Y
	t3.Add(t3, t3)                               // t3 := t3 + t3
	z3 := new(fiat.P384Element).Mul(p.x, p.z)    // Z3 := X * Z
	z3.Add(z3, z3)                               // Z3 := Z3 + Z3
	y3 := new(fiat.P384Element).Mul(p384B(), t2) // Y3 := b * t2
	y3.Sub(y3, z3)                               // Y3 := Y3 - Z3
	x3 := new(fiat.P384Element).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(p384B(), 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 *P384Point) Select(p1, p2 *P384Point, cond int) *P384Point {
	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 p384Table 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 p384Table [15]*P384Point

// 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 *p384Table) Select(p *P384Point, n uint8) {
	if n >= 16 {
		panic("nistec: internal error: p384Table called with out-of-bounds value")
	}
	p.Set(NewP384Point())
	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 *P384Point) ScalarMult(q *P384Point, scalar []byte) (*P384Point, error) {
	// Compute a p384Table for the base point q. The explicit NewP384Point
	// calls get inlined, letting the allocations live on the stack.
	var table = p384Table{NewP384Point(), NewP384Point(), NewP384Point(),
		NewP384Point(), NewP384Point(), NewP384Point(), NewP384Point(),
		NewP384Point(), NewP384Point(), NewP384Point(), NewP384Point(),
		NewP384Point(), NewP384Point(), NewP384Point(), NewP384Point()}
	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 := NewP384Point()
	p.Set(NewP384Point())
	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 p384GeneratorTable *[p384ElementLength * 2]p384Table
var p384GeneratorTableOnce sync.Once

// generatorTable returns a sequence of p384Tables. The first table contains
// multiples of G. Each successive table is the previous table doubled four
// times.
func (p *P384Point) generatorTable() *[p384ElementLength * 2]p384Table {
	p384GeneratorTableOnce.Do(func() {
		p384GeneratorTable = new([p384ElementLength * 2]p384Table)
		base := NewP384Point().SetGenerator()
		for i := 0; i < p384ElementLength*2; i++ {
			p384GeneratorTable[i][0] = NewP384Point().Set(base)
			for j := 1; j < 15; j++ {
				p384GeneratorTable[i][j] = NewP384Point().Add(p384GeneratorTable[i][j-1], base)
			}
			base.Double(base)
			base.Double(base)
			base.Double(base)
			base.Double(base)
		}
	})
	return p384GeneratorTable
}

// ScalarBaseMult sets p = scalar * B, where B is the canonical generator, and
// returns p.
func (p *P384Point) ScalarBaseMult(scalar []byte) (*P384Point, error) {
	if len(scalar) != p384ElementLength {
		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 := NewP384Point()
	p.Set(NewP384Point())
	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
}

// p384Sqrt sets e to a square root of x. If x is not a square, p384Sqrt returns
// false and e is unchanged. e and x can overlap.
func p384Sqrt(e, x *fiat.P384Element) (isSquare bool) {
	candidate := new(fiat.P384Element)
	p384SqrtCandidate(candidate, x)
	square := new(fiat.P384Element).Square(candidate)
	if square.Equal(x) != 1 {
		return false
	}
	e.Set(candidate)
	return true
}

// p384SqrtCandidate sets z to a square root candidate for x. z and x must not overlap.
func p384SqrtCandidate(z, x *fiat.P384Element) {
	// Since p = 3 mod 4, exponentiation by (p + 1) / 4 yields a square root candidate.
	//
	// The sequence of 14 multiplications and 381 squarings is derived from the
	// following addition chain generated with github.com/mmcloughlin/addchain v0.4.0.
	//
	//	_10      = 2*1
	//	_11      = 1 + _10
	//	_110     = 2*_11
	//	_111     = 1 + _110
	//	_111000  = _111 << 3
	//	_111111  = _111 + _111000
	//	_1111110 = 2*_111111
	//	_1111111 = 1 + _1111110
	//	x12      = _1111110 << 5 + _111111
	//	x24      = x12 << 12 + x12
	//	x31      = x24 << 7 + _1111111
	//	x32      = 2*x31 + 1
	//	x63      = x32 << 31 + x31
	//	x126     = x63 << 63 + x63
	//	x252     = x126 << 126 + x126
	//	x255     = x252 << 3 + _111
	//	return     ((x255 << 33 + x32) << 64 + 1) << 30
	//
	var t0 = new(fiat.P384Element)
	var t1 = new(fiat.P384Element)
	var t2 = new(fiat.P384Element)

	z.Square(x)
	z.Mul(x, z)
	z.Square(z)
	t0.Mul(x, z)
	z.Square(t0)
	for s := 1; s < 3; s++ {
		z.Square(z)
	}
	t1.Mul(t0, z)
	t2.Square(t1)
	z.Mul(x, t2)
	for s := 0; s < 5; s++ {
		t2.Square(t2)
	}
	t1.Mul(t1, t2)
	t2.Square(t1)
	for s := 1; s < 12; s++ {
		t2.Square(t2)
	}
	t1.Mul(t1, t2)
	for s := 0; s < 7; s++ {
		t1.Square(t1)
	}
	t1.Mul(z, t1)
	z.Square(t1)
	z.Mul(x, z)
	t2.Square(z)
	for s := 1; s < 31; s++ {
		t2.Square(t2)
	}
	t1.Mul(t1, t2)
	t2.Square(t1)
	for s := 1; s < 63; s++ {
		t2.Square(t2)
	}
	t1.Mul(t1, t2)
	t2.Square(t1)
	for s := 1; s < 126; s++ {
		t2.Square(t2)
	}
	t1.Mul(t1, t2)
	for s := 0; s < 3; s++ {
		t1.Square(t1)
	}
	t0.Mul(t0, t1)
	for s := 0; s < 33; s++ {
		t0.Square(t0)
	}
	z.Mul(z, t0)
	for s := 0; s < 64; s++ {
		z.Square(z)
	}
	z.Mul(x, z)
	for s := 0; s < 30; s++ {
		z.Square(z)
	}
}

The pages are generated with Golds v0.7.3-preview. (GOOS=linux GOARCH=amd64)
Golds is a Go 101 project developed by Tapir Liu.
PR and bug reports are welcome and can be submitted to the issue list.
Please follow @zigo_101 (reachable from the left QR code) to get the latest news of Golds.