sm2p256_asm.go

// It is by standing on the shoulders of giants.

// This file contains the Go wrapper for the constant-time, 64-bit assembly
// implementation of P256. The optimizations performed here are described in
// detail in:
// S.Gueron and V.Krasnov, "Fast prime field elliptic-curve cryptography with
//                          256-bit primes"
// https://link.springer.com/article/10.1007%2Fs13389-014-0090-x
// https://eprint.iacr.org/2013/816.pdf
//go:build (amd64 || arm64) && !purego

package sm2ec

import (
	_ "embed"
	"errors"
	"math/bits"
	"unsafe"

	"golang.org/x/sys/cpu"
)

// p256Element is a P-256 base field element in [0, P-1] in the Montgomery
// domain (with R 2²⁵⁶) as four limbs in little-endian order value.
type p256Element [4]uint64

// p256One is one in the Montgomery domain.
var p256One = p256Element{0x0000000000000001, 0x00000000ffffffff, 0x0000000000000000, 0x0000000100000000}

var p256Zero = p256Element{}

// p256P is 2^256 - 2^224 - 2^96 + 2^64 - 1.
var p256P = p256Element{0xffffffffffffffff, 0xffffffff00000000,
	0xffffffffffffffff, 0xfffffffeffffffff}

// P256Point is a P-256 point. The zero value should not be assumed to be valid
// (although it is in this implementation).
type SM2P256Point struct {
	// (X:Y:Z) are Jacobian coordinates where x = X/Z² and y = Y/Z³. The point
	// at infinity can be represented by any set of coordinates with Z = 0.
	x, y, z p256Element
}

// NewSM2P256Point returns a new SM2P256Point representing the point at infinity.
func NewSM2P256Point() *SM2P256Point {
	return &SM2P256Point{
		x: p256One, y: p256One, z: p256Zero,
	}
}

// SetGenerator sets p to the canonical generator and returns p.
func (p *SM2P256Point) SetGenerator() *SM2P256Point {
	p.x = p256Element{0x61328990f418029e, 0x3e7981eddca6c050,
		0xd6a1ed99ac24c3c3, 0x91167a5ee1c13b05}
	p.y = p256Element{0xc1354e593c2d0ddd, 0xc1f5e5788d3295fa,
		0x8d4cfb066e2a48f8, 0x63cd65d481d735bd}
	p.z = p256One
	return p
}

// Set sets p = q and returns p.
func (p *SM2P256Point) Set(q *SM2P256Point) *SM2P256Point {
	p.x, p.y, p.z = q.x, q.y, q.z
	return p
}

const p256ElementLength = 32
const p256UncompressedLength = 1 + 2*p256ElementLength
const p256CompressedLength = 1 + p256ElementLength

// 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 *SM2P256Point) SetBytes(b []byte) (*SM2P256Point, error) {
	// p256Mul operates in the Montgomery domain with R = 2²⁵⁶ mod p. Thus rr
	// here is R in the Montgomery domain, or R×R mod p. See comment in
	// P256OrdInverse about how this is used.
	rr := p256Element{0x0000000200000003, 0x00000002ffffffff,
		0x0000000100000001, 0x0000000400000002}

	switch {
	// Point at infinity.
	case len(b) == 1 && b[0] == 0:
		return p.Set(NewSM2P256Point()), nil

	// Uncompressed form.
	case len(b) == p256UncompressedLength && b[0] == 4:
		var r SM2P256Point
		p256BigToLittle(&r.x, (*[32]byte)(b[1:33]))
		p256BigToLittle(&r.y, (*[32]byte)(b[33:65]))
		if p256LessThanP(&r.x) == 0 || p256LessThanP(&r.y) == 0 {
			return nil, errors.New("invalid P256 element encoding")
		}
		p256Mul(&r.x, &r.x, &rr)
		p256Mul(&r.y, &r.y, &rr)
		if err := p256CheckOnCurve(&r.x, &r.y); err != nil {
			return nil, err
		}
		r.z = p256One
		return p.Set(&r), nil

	// Compressed form.
	case len(b) == p256CompressedLength && (b[0] == 2 || b[0] == 3):
		var r SM2P256Point
		p256BigToLittle(&r.x, (*[32]byte)(b[1:33]))
		if p256LessThanP(&r.x) == 0 {
			return nil, errors.New("invalid P256 element encoding")
		}
		p256Mul(&r.x, &r.x, &rr)

		// y² = x³ - 3x + b
		p256Polynomial(&r.y, &r.x)
		if !p256Sqrt(&r.y, &r.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.
		yy := new(p256Element)
		p256FromMont(yy, &r.y)
		cond := int(yy[0]&1) ^ int(b[0]&1)
		p256NegCond(&r.y, cond)

		r.z = p256One
		return p.Set(&r), nil

	default:
		return nil, errors.New("invalid P256 point encoding")
	}
}

// p256Polynomial sets y2 to x³ - 3x + b, and returns y2.
func p256Polynomial(y2, x *p256Element) *p256Element {
	x3 := new(p256Element)
	p256Sqr(x3, x, 1)
	p256Mul(x3, x3, x)

	threeX := new(p256Element)
	p256Add(threeX, x, x)
	p256Add(threeX, threeX, x)
	p256NegCond(threeX, 1)

	p256B := &p256Element{0x90d230632bc0dd42, 0x71cf379ae9b537ab,
		0x527981505ea51c3c, 0x240fe188ba20e2c8}

	p256Add(x3, x3, threeX)
	p256Add(x3, x3, p256B)

	*y2 = *x3
	return y2
}

func p256CheckOnCurve(x, y *p256Element) error {
	// y² = x³ - 3x + b
	rhs := p256Polynomial(new(p256Element), x)
	lhs := new(p256Element)
	p256Sqr(lhs, y, 1)
	if p256Equal(lhs, rhs) != 1 {
		return errors.New("point not on SM2 P256 curve")
	}
	return nil
}

// p256LessThanP returns 1 if x < p, and 0 otherwise. Note that a p256Element is
// not allowed to be equal to or greater than p, so if this function returns 0
// then x is invalid.
func p256LessThanP(x *p256Element) int {
	var b uint64
	_, b = bits.Sub64(x[0], p256P[0], b)
	_, b = bits.Sub64(x[1], p256P[1], b)
	_, b = bits.Sub64(x[2], p256P[2], b)
	_, b = bits.Sub64(x[3], p256P[3], b)
	return int(b)
}

// p256Add sets res = x + y.
func p256Add(res, x, y *p256Element) {
	var c, b uint64
	t1 := make([]uint64, 4)
	t1[0], c = bits.Add64(x[0], y[0], 0)
	t1[1], c = bits.Add64(x[1], y[1], c)
	t1[2], c = bits.Add64(x[2], y[2], c)
	t1[3], c = bits.Add64(x[3], y[3], c)
	t2 := make([]uint64, 4)
	t2[0], b = bits.Sub64(t1[0], p256P[0], 0)
	t2[1], b = bits.Sub64(t1[1], p256P[1], b)
	t2[2], b = bits.Sub64(t1[2], p256P[2], b)
	t2[3], b = bits.Sub64(t1[3], p256P[3], b)
	// Three options:
	//   - a+b < p
	//     then c is 0, b is 1, and t1 is correct
	//   - p <= a+b < 2^256
	//     then c is 0, b is 0, and t2 is correct
	//   - 2^256 <= a+b
	//     then c is 1, b is 1, and t2 is correct
	t2Mask := (c ^ b) - 1
	res[0] = (t1[0] & ^t2Mask) | (t2[0] & t2Mask)
	res[1] = (t1[1] & ^t2Mask) | (t2[1] & t2Mask)
	res[2] = (t1[2] & ^t2Mask) | (t2[2] & t2Mask)
	res[3] = (t1[3] & ^t2Mask) | (t2[3] & t2Mask)
}

// 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 *p256Element) (isSquare bool) {
	z, t0, t1, t2, t3, t4 := new(p256Element), new(p256Element), new(p256Element), new(p256Element), new(p256Element), new(p256Element)

	// Since p = 3 mod 4, exponentiation by (p + 1) / 4 yields a square root candidate.
	//
	// The sequence of 13 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
	//	_110     = 2*_11
	//	_111     = 1 + _110
	//	_1110    = 2*_111
	//	_1111    = 1 + _1110
	//	_11110   = 2*_1111
	//	_111100  = 2*_11110
	//	_1111000 = 2*_111100
	//	i19      = (_1111000 << 3 + _111100) << 5 + _1111000
	//	x31      = (i19 << 2 + _11110) << 14 + i19 + _111
	//	i42      = x31 << 4
	//	i73      = i42 << 31
	//	i74      = i42 + i73
	//	i171     = (i73 << 32 + i74) << 62 + i74 + _1111
	//	return     (i171 << 32 + 1) << 62
	//
	p256Sqr(z, x, 1)   // z.Square(x)
	p256Mul(z, x, z)   // z.Mul(x, z)
	p256Sqr(z, z, 1)   // z.Square(z)
	p256Mul(t0, x, z)  // t0.Mul(x, z)
	p256Sqr(z, t0, 1)  // z.Square(t0)
	p256Mul(z, x, z)   // z.Mul(x, z)
	p256Sqr(t2, z, 1)  // t2.Square(z)
	p256Sqr(t3, t2, 1) // t3.Square(t2)
	p256Sqr(t1, t3, 1) // t1.Square(t3)
	// t4.Square(t1)
	//for s := 1; s < 3; s++ {
	//	t4.Square(t4)
	//}
	p256Sqr(t4, t1, 3)
	p256Mul(t3, t3, t4) // t3.Mul(t3, t4)
	//for s := 0; s < 5; s++ {
	//	t3.Square(t3)
	//}
	p256Sqr(t3, t3, 5)
	p256Mul(t1, t1, t3) // t1.Mul(t1, t3)
	//t3.Square(t1)
	//for s := 1; s < 2; s++ {
	//	t3.Square(t3)
	//}
	p256Sqr(t3, t1, 2)
	p256Mul(t2, t2, t3) // t2.Mul(t2, t3)
	//for s := 0; s < 14; s++ {
	//	t2.Square(t2)
	//}
	p256Sqr(t2, t2, 14)
	p256Mul(t1, t1, t2) // t1.Mul(t1, t2)

	p256Mul(t0, t0, t1) // t0.Mul(t0, t1)
	//for s := 0; s < 4; s++ {
	//	t0.Square(t0)
	//}
	p256Sqr(t0, t0, 4)
	//t1.Square(t0)
	//for s := 1; s < 31; s++ {
	//	t1.Square(t1)
	//}
	p256Sqr(t1, t0, 31)
	p256Mul(t0, t0, t1) //t0.Mul(t0, t1)
	//for s := 0; s < 32; s++ {
	//	t1.Square(t1)
	//}
	p256Sqr(t1, t1, 32)

	p256Mul(t1, t0, t1) //t1.Mul(t0, t1)
	//for s := 0; s < 62; s++ {
	//	t1.Square(t1)
	//}
	p256Sqr(t1, t1, 62)
	p256Mul(t0, t0, t1) //t0.Mul(t0, t1)
	p256Mul(z, z, t0)   //z.Mul(z, t0)
	//for s := 0; s < 32; s++ {
	//	e.Square(e)
	//}
	p256Sqr(z, z, 32)
	p256Mul(z, z, x) // z.Mul(x, z)
	//for s := 0; s < 62; s++ {
	//	z.Square(z)
	//}
	p256Sqr(z, z, 62)

	p256Sqr(t1, z, 1)
	if p256Equal(t1, x) != 1 {
		return false
	}
	*e = *z
	return true
}

// The following assembly functions are implemented in p256_asm_*.s

// amd64 assembly uses ADCX/ADOX/MULX
var supportBMI2 = cpu.X86.HasADX && cpu.X86.HasBMI2

var supportAVX2 = cpu.X86.HasAVX2

// Montgomery multiplication. Sets res = in1 * in2 * R⁻¹ mod p.
//
//go:noescape
func p256Mul(res, in1, in2 *p256Element)

// Montgomery square, repeated n times (n >= 1).
//
//go:noescape
func p256Sqr(res, in *p256Element, n int)

// Montgomery multiplication by R⁻¹, or 1 outside the domain.
// Sets res = in * R⁻¹, bringing res out of the Montgomery domain.
//
//go:noescape
func p256FromMont(res, in *p256Element)

// If cond is not 0, sets val = -val mod p.
//
//go:noescape
func p256NegCond(val *p256Element, cond int)

// If cond is 0, sets res = b, otherwise sets res = a.
//
//go:noescape
func p256MovCond(res, a, b *SM2P256Point, cond int)

//go:noescape
func p256BigToLittle(res *p256Element, in *[32]byte)

//go:noescape
func p256LittleToBig(res *[32]byte, in *p256Element)

//go:noescape
func p256OrdBigToLittle(res *p256OrdElement, in *[32]byte)

//go:noescape
func p256OrdLittleToBig(res *[32]byte, in *p256OrdElement)

// p256OrdReduce ensures s is in the range [0, ord(G)-1].
//
//go:noescape
func p256OrdReduce(s *p256OrdElement)

// p256Table is a table of the first 16 multiples of a point. Points are stored
// at an index offset of -1 so [8]P is at index 7, P is at 0, and [16]P is at 15.
// [0]P is the point at infinity and it's not stored.
type p256Table [32]SM2P256Point

// p256Select sets res to the point at index idx in the table.
// idx must be in [0, limit-1]. It executes in constant time.
//
//go:noescape
func p256Select(res *SM2P256Point, table *p256Table, idx, limit int)

// p256AffinePoint is a point in affine coordinates (x, y). x and y are still
// Montgomery domain elements. The point can't be the point at infinity.
type p256AffinePoint struct {
	x, y p256Element
}

// p256AffineTable is a table of the first 32 multiples of a point. Points are
// stored at an index offset of -1 like in p256Table, and [0]P is not stored.
type p256AffineTable [32]p256AffinePoint

// p256Precomputed is a series of precomputed multiples of G, the canonical
// generator. The first p256AffineTable contains multiples of G. The second one
// multiples of [2⁶]G, the third one of [2¹²]G, and so on, where each successive
// table is the previous table doubled six times. Six is the width of the
// sliding window used in p256ScalarMult, and having each table already
// pre-doubled lets us avoid the doublings between windows entirely. This table
// MUST NOT be modified, as it aliases into p256PrecomputedEmbed below.
var p256Precomputed *[43]p256AffineTable

//go:embed p256_asm_table.bin
var p256PrecomputedEmbed string

func init() {
	p256PrecomputedPtr := (*unsafe.Pointer)(unsafe.Pointer(&p256PrecomputedEmbed))
	p256Precomputed = (*[43]p256AffineTable)(*p256PrecomputedPtr)
}

// p256SelectAffine sets res to the point at index idx in the table.
// idx must be in [0, 31]. It executes in constant time.
//
//go:noescape
func p256SelectAffine(res *p256AffinePoint, table *p256AffineTable, idx int)

// Point addition with an affine point and constant time conditions.
// If zero is 0, sets res = in2. If sel is 0, sets res = in1.
// If sign is not 0, sets res = in1 + -in2. Otherwise, sets res = in1 + in2
//
//go:noescape
func p256PointAddAffineAsm(res, in1 *SM2P256Point, in2 *p256AffinePoint, sign, sel, zero int)

// Point addition. Sets res = in1 + in2. Returns one if the two input points
// were equal and zero otherwise. If in1 or in2 are the point at infinity, res
// and the return value are undefined.
//
//go:noescape
func p256PointAddAsm(res, in1, in2 *SM2P256Point) int

// Point doubling. Sets res = in + in. in can be the point at infinity.
//
//go:noescape
func p256PointDoubleAsm(res, in *SM2P256Point)

// Point doubling 6 times. in can be the point at infinity.
//
//go:noescape
func p256PointDouble6TimesAsm(res, in *SM2P256Point)

// p256OrdElement is a P-256 scalar field element in [0, ord(G)-1] in the
// Montgomery domain (with R 2²⁵⁶) as four uint64 limbs in little-endian order.
type p256OrdElement [4]uint64

// Add sets q = p1 + p2, and returns q. The points may overlap.
func (q *SM2P256Point) Add(r1, r2 *SM2P256Point) *SM2P256Point {
	var sum, double SM2P256Point
	r1IsInfinity := r1.isInfinity()
	r2IsInfinity := r2.isInfinity()
	pointsEqual := p256PointAddAsm(&sum, r1, r2)
	p256PointDoubleAsm(&double, r1)
	p256MovCond(&sum, &double, &sum, pointsEqual)
	p256MovCond(&sum, r1, &sum, r2IsInfinity)
	p256MovCond(&sum, r2, &sum, r1IsInfinity)
	return q.Set(&sum)
}

// Double sets q = p + p, and returns q. The points may overlap.
func (q *SM2P256Point) Double(p *SM2P256Point) *SM2P256Point {
	var double SM2P256Point
	p256PointDoubleAsm(&double, p)
	return q.Set(&double)
}

// ScalarBaseMult sets r = scalar * generator, where scalar is a 32-byte big
// endian value, and returns r. If scalar is not 32 bytes long, ScalarBaseMult
// returns an error and the receiver is unchanged.
func (r *SM2P256Point) ScalarBaseMult(scalar []byte) (*SM2P256Point, error) {
	if len(scalar) != 32 {
		return nil, errors.New("invalid scalar length")
	}
	scalarReversed := new(p256OrdElement)
	p256OrdBigToLittle(scalarReversed, (*[32]byte)(scalar))
	p256OrdReduce(scalarReversed)
	r.p256BaseMult(scalarReversed)
	return r, nil
}

// ScalarMult sets r = scalar * q, where scalar is a 32-byte big endian value,
// and returns r. If scalar is not 32 bytes long, ScalarBaseMult returns an
// error and the receiver is unchanged.
func (r *SM2P256Point) ScalarMult(q *SM2P256Point, scalar []byte) (*SM2P256Point, error) {
	if len(scalar) != 32 {
		return nil, errors.New("invalid scalar length")
	}
	scalarReversed := new(p256OrdElement)
	p256OrdBigToLittle(scalarReversed, (*[32]byte)(scalar))
	p256OrdReduce(scalarReversed)
	r.Set(q).p256ScalarMult(scalarReversed)
	return r, nil
}

// uint64IsZero returns 1 if x is zero and zero otherwise.
func uint64IsZero(x uint64) int {
	x = ^x
	x &= x >> 32
	x &= x >> 16
	x &= x >> 8
	x &= x >> 4
	x &= x >> 2
	x &= x >> 1
	return int(x & 1)
}

// p256Equal returns 1 if a and b are equal and 0 otherwise.
func p256Equal(a, b *p256Element) int {
	var acc uint64
	for i := range a {
		acc |= a[i] ^ b[i]
	}
	return uint64IsZero(acc)
}

// isInfinity returns 1 if p is the point at infinity and 0 otherwise.
func (p *SM2P256Point) isInfinity() int {
	return p256Equal(&p.z, &p256Zero)
}

// 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 *SM2P256Point) Bytes() []byte {
	// This function is outlined to make the allocations inline in the caller
	// rather than happen on the heap.
	var out [p256UncompressedLength]byte
	return p.bytes(&out)
}

func (p *SM2P256Point) bytes(out *[p256UncompressedLength]byte) []byte {
	// The proper representation of the point at infinity is a single zero byte.
	if p.isInfinity() == 1 {
		return append(out[:0], 0)
	}

	x, y := new(p256Element), new(p256Element)
	p.affineFromMont(x, y)

	out[0] = 4 // Uncompressed form.
	p256LittleToBig((*[32]byte)(out[1:33]), x)
	p256LittleToBig((*[32]byte)(out[33:65]), y)

	return out[:]
}

// affineFromMont sets (x, y) to the affine coordinates of p, converted out of the
// Montgomery domain.
func (p *SM2P256Point) affineFromMont(x, y *p256Element) {
	p256Inverse(y, &p.z)
	p256Sqr(x, y, 1)
	p256Mul(y, y, x)

	p256Mul(x, &p.x, x)
	p256Mul(y, &p.y, y)

	p256FromMont(x, x)
	p256FromMont(y, y)
}

// 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 *SM2P256Point) 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 *SM2P256Point) bytesX(out *[p256ElementLength]byte) ([]byte, error) {
	if p.isInfinity() == 1 {
		return nil, errors.New("SM2 point is the point at infinity")
	}

	x := new(p256Element)
	p256Inverse(x, &p.z)
	p256Sqr(x, x, 1)
	p256Mul(x, &p.x, x)
	p256FromMont(x, x)
	p256LittleToBig((*[32]byte)(out[:]), x)

	return out[:], 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 *SM2P256Point) BytesCompressed() []byte {
	// This function is outlined to make the allocations inline in the caller
	// rather than happen on the heap.
	var out [p256CompressedLength]byte
	return p.bytesCompressed(&out)
}

func (p *SM2P256Point) bytesCompressed(out *[p256CompressedLength]byte) []byte {
	if p.isInfinity() == 1 {
		return append(out[:0], 0)
	}

	x, y := new(p256Element), new(p256Element)
	p.affineFromMont(x, y)

	out[0] = 2 | byte(y[0]&1)
	p256LittleToBig((*[32]byte)(out[1:33]), x)

	return out[:]
}

// Select sets q to p1 if cond == 1, and to p2 if cond == 0.
func (q *SM2P256Point) Select(p1, p2 *SM2P256Point, cond int) *SM2P256Point {
	p256MovCond(q, p1, p2, cond)
	return q
}

// p256Inverse sets out to in⁻¹ mod p. If in is zero, out will be zero.
func p256Inverse(out, in *p256Element) {
	// Inversion is calculated through exponentiation by p - 2, per Fermat's
	// little theorem.
	//
	// The sequence of 14 multiplications and 255 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
	//      i39      = x31 << 2
	//      i68      = i39 << 29
	//      x62      = x31 + i68
	//      i71      = i68 << 2
	//      x64      = i39 + i71 + _11
	//      i265     = ((i71 << 32 + x64) << 64 + x64) << 94
	//      return     (x62 + i265) << 2 + 1
	// Allocate Temporaries.
	var (
		t0 = new(p256Element)
		t1 = new(p256Element)
		t2 = new(p256Element)
	)
	// Step 1: z = x^0x2
	//z.Sqr(x)
	p256Sqr(out, in, 1)

	// Step 2: t0 = x^0x3
	// t0.Mul(x, z)
	p256Mul(t0, in, out)

	// Step 3: z = x^0x6
	// z.Sqr(t0)
	p256Sqr(out, t0, 1)

	// Step 4: z = x^0x7
	// z.Mul(x, z)
	p256Mul(out, in, out)

	// Step 7: t1 = x^0x38
	//t1.Sqr(z)
	//for s := 1; s < 3; s++ {
	//	t1.Sqr(t1)
	//}
	p256Sqr(t1, out, 3)

	// Step 8: t1 = x^0x3f
	//t1.Mul(z, t1)
	p256Mul(t1, out, t1)

	// Step 9: t2 = x^0x7e
	//t2.Sqr(t1)
	p256Sqr(t2, t1, 1)

	// Step 10: z = x^0x7f
	//z.Mul(x, t2)
	p256Mul(out, in, t2)

	// Step 15: t2 = x^0xfc0
	//for s := 0; s < 5; s++ {
	//	t2.Sqr(t2)
	//}
	p256Sqr(t2, t2, 5)

	// Step 16: t1 = x^0xfff
	//t1.Mul(t1, t2)
	p256Mul(t1, t1, t2)

	// Step 28: t2 = x^0xfff000
	//t2.Sqr(t1)
	//for s := 1; s < 12; s++ {
	//	t2.Sqr(t2)
	//}
	p256Sqr(t2, t1, 12)

	// Step 29: t1 = x^0xffffff
	//t1.Mul(t1, t2)
	p256Mul(t1, t1, t2)

	// Step 36: t1 = x^0x7fffff80
	//for s := 0; s < 7; s++ {
	//	t1.Sqr(t1)
	//}
	p256Sqr(t1, t1, 7)

	// Step 37: z = x^0x7fffffff
	//z.Mul(z, t1)
	p256Mul(out, out, t1)

	// Step 39: t2 = x^0x1fffffffc
	//t2.Sqr(z)
	//for s := 1; s < 2; s++ {
	//	t2.Sqr(t2)
	//}
	p256Sqr(t2, out, 2)

	// Step 68: t1 = x^0x3fffffff80000000
	//t1.Sqr(t2)
	//for s := 1; s < 29; s++ {
	//	t1.Sqr(t1)
	//}
	p256Sqr(t1, t2, 29)

	// Step 69: z = x^0x3fffffffffffffff
	//z.Mul(z, t1)
	p256Mul(out, out, t1)

	// Step 71: t1 = x^0xfffffffe00000000
	//for s := 0; s < 2; s++ {
	//	t1.Sqr(t1)
	//}
	p256Sqr(t1, t1, 2)

	// Step 72: t2 = x^0xfffffffffffffffc
	//t2.Mul(t2, t1)
	p256Mul(t2, t2, t1)

	// Step 73: t0 = x^0xffffffffffffffff
	//t0.Mul(t0, t2)
	p256Mul(t0, t0, t2)

	// Step 105: t1 = x^0xfffffffe0000000000000000
	//for s := 0; s < 32; s++ {
	//	t1.Sqr(t1)
	//}
	p256Sqr(t1, t1, 32)

	// Step 106: t1 = x^0xfffffffeffffffffffffffff
	//t1.Mul(t0, t1)
	p256Mul(t1, t0, t1)

	// Step 170: t1 = x^0xfffffffeffffffffffffffff0000000000000000
	//for s := 0; s < 64; s++ {
	//	t1.Sqr(t1)
	//}
	p256Sqr(t1, t1, 64)

	// Step 171: t0 = x^0xfffffffeffffffffffffffffffffffffffffffff
	//t0.Mul(t0, t1)
	p256Mul(t0, t0, t1)

	// Step 265: t0 = x^0x3fffffffbfffffffffffffffffffffffffffffffc00000000000000000000000
	//for s := 0; s < 94; s++ {
	//	t0.Sqr(t0)
	//}
	p256Sqr(t0, t0, 94)

	// Step 266: z = x^0x3fffffffbfffffffffffffffffffffffffffffffc00000003fffffffffffffff
	//z.Mul(z, t0)
	p256Mul(out, out, t0)

	// Step 268: z = x^0xfffffffeffffffffffffffffffffffffffffffff00000000fffffffffffffffc
	//for s := 0; s < 2; s++ {
	//	z.Sqr(z)
	//}
	p256Sqr(out, out, 2)

	// Step 269: z = x^0xfffffffeffffffffffffffffffffffffffffffff00000000fffffffffffffffd
	//z.Mul(x, z)
	p256Mul(out, in, out)
}

// This function takes those six bits as an integer (0 .. 63), writing the
// recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
// value, in the range 0 .. 16).  Note that this integer essentially provides
// the input bits "shifted to the left" by one position: for example, the input
// to compute the least significant recoded digit, given that there's no bit
// b_-1, has to be b_4 b_3 b_2 b_1 b_0 0.
//
// Reference:
// https://github.com/openssl/openssl/blob/master/crypto/ec/ecp_nistputil.c
// https://github.com/google/boringssl/blob/master/crypto/fipsmodule/ec/util.c
func boothW5(in uint) (int, int) {
	var s uint = ^((in >> 5) - 1)  // sets all bits to MSB(in), 'in' seen as 6-bit value
	var d uint = (1 << 6) - in - 1 // d = 63 - in, or d = ^in & 0x3f
	d = (d & s) | (in & (^s))      // d = in if in < 2^5; otherwise, d = 63 - in
	d = (d >> 1) + (d & 1)         // d = (d + 1) / 2
	return int(d), int(s & 1)
}

func boothW6(in uint) (int, int) {
	var s uint = ^((in >> 6) - 1)
	var d uint = (1 << 7) - in - 1
	d = (d & s) | (in & (^s))
	d = (d >> 1) + (d & 1)
	return int(d), int(s & 1)
}

func (p *SM2P256Point) p256BaseMult(scalar *p256OrdElement) {
	var t0 p256AffinePoint

	wvalue := (scalar[0] << 1) & 0x7f
	sel, sign := boothW6(uint(wvalue))
	p256SelectAffine(&t0, &p256Precomputed[0], sel)
	p.x, p.y, p.z = t0.x, t0.y, p256One
	p256NegCond(&p.y, sign)

	index := uint(5)
	zero := sel

	for i := 1; i < 43; i++ {
		if index >= 192 {
			wvalue = (scalar[3] >> (index & 63)) & 0x7f
		} else if index >= 128 {
			wvalue = ((scalar[2] >> (index & 63)) + (scalar[3] << (64 - (index & 63)))) & 0x7f
		} else if index >= 64 {
			wvalue = ((scalar[1] >> (index & 63)) + (scalar[2] << (64 - (index & 63)))) & 0x7f
		} else {
			wvalue = ((scalar[0] >> (index & 63)) + (scalar[1] << (64 - (index & 63)))) & 0x7f
		}
		index += 6
		sel, sign = boothW6(uint(wvalue))
		p256SelectAffine(&t0, &p256Precomputed[i], sel)
		p256PointAddAffineAsm(p, p, &t0, sign, sel, zero)
		zero |= sel
	}

	// If the whole scalar was zero, set to the point at infinity.
	p256MovCond(p, p, NewSM2P256Point(), zero)
}

func (p *SM2P256Point) p256ScalarMult(scalar *p256OrdElement) {
	// precomp is a table of precomputed points that stores powers of p
	// from p^1 to p^32.
	var precomp p256Table
	var t0, t1 SM2P256Point

	// Prepare the table
	precomp[0] = *p // 1

	p256PointDoubleAsm(&precomp[1], p)             //2
	p256PointAddAsm(&precomp[2], &precomp[1], p)   //3
	p256PointDoubleAsm(&precomp[3], &precomp[1])   //4
	p256PointAddAsm(&precomp[4], &precomp[3], p)   //5
	p256PointDoubleAsm(&precomp[5], &precomp[2])   //6
	p256PointAddAsm(&precomp[6], &precomp[5], p)   //7
	p256PointDoubleAsm(&precomp[7], &precomp[3])   //8
	p256PointAddAsm(&precomp[8], &precomp[7], p)   //9
	p256PointDoubleAsm(&precomp[9], &precomp[4])   //10
	p256PointAddAsm(&precomp[10], &precomp[9], p)  //11
	p256PointDoubleAsm(&precomp[11], &precomp[5])  //12
	p256PointAddAsm(&precomp[12], &precomp[11], p) //13
	p256PointDoubleAsm(&precomp[13], &precomp[6])  //14
	p256PointAddAsm(&precomp[14], &precomp[13], p) //15
	p256PointDoubleAsm(&precomp[15], &precomp[7])  //16

	p256PointAddAsm(&precomp[16], &precomp[15], p) //17
	p256PointDoubleAsm(&precomp[17], &precomp[8])  //18
	p256PointAddAsm(&precomp[18], &precomp[17], p) //19
	p256PointDoubleAsm(&precomp[19], &precomp[9])  //20
	p256PointAddAsm(&precomp[20], &precomp[19], p) //21
	p256PointDoubleAsm(&precomp[21], &precomp[10]) //22
	p256PointAddAsm(&precomp[22], &precomp[21], p) //23
	p256PointDoubleAsm(&precomp[23], &precomp[11]) //24
	p256PointAddAsm(&precomp[24], &precomp[23], p) //25
	p256PointDoubleAsm(&precomp[25], &precomp[12]) //26
	p256PointAddAsm(&precomp[26], &precomp[25], p) //27
	p256PointDoubleAsm(&precomp[27], &precomp[13]) //28
	p256PointAddAsm(&precomp[28], &precomp[27], p) //29
	p256PointDoubleAsm(&precomp[29], &precomp[14]) //30
	p256PointAddAsm(&precomp[30], &precomp[29], p) //31
	p256PointDoubleAsm(&precomp[31], &precomp[15]) //32

	// Start scanning the window from top bit
	index := uint(251)
	var sel, sign int

	wvalue := (scalar[index/64] >> (index % 64)) & 0x7f
	sel, _ = boothW6(uint(wvalue))

	p256Select(p, &precomp, sel, 32)
	zero := sel

	for index > 5 {
		index -= 6

		p256PointDouble6TimesAsm(p, p)

		if index >= 192 {
			wvalue = (scalar[3] >> (index & 63)) & 0x7f
		} else if index >= 128 {
			wvalue = ((scalar[2] >> (index & 63)) + (scalar[3] << (64 - (index & 63)))) & 0x7f
		} else if index >= 64 {
			wvalue = ((scalar[1] >> (index & 63)) + (scalar[2] << (64 - (index & 63)))) & 0x7f
		} else {
			wvalue = ((scalar[0] >> (index & 63)) + (scalar[1] << (64 - (index & 63)))) & 0x7f
		}

		sel, sign = boothW6(uint(wvalue))

		p256Select(&t0, &precomp, sel, 32)
		p256NegCond(&t0.y, sign)
		p256PointAddAsm(&t1, p, &t0)
		p256MovCond(&t1, &t1, p, sel)
		p256MovCond(p, &t1, &t0, zero)
		zero |= sel
	}
	p256PointDouble6TimesAsm(p, p)

	wvalue = (scalar[0] << 1) & 0x7f
	sel, sign = boothW6(uint(wvalue))

	p256Select(&t0, &precomp, sel, 32)
	p256NegCond(&t0.y, sign)
	p256PointAddAsm(&t1, p, &t0)
	p256MovCond(&t1, &t1, p, sel)
	p256MovCond(p, &t1, &t0, zero)
}