sm2ec_asm.go

//go:build (amd64 && !purego) || (arm64 && !purego)
// +build amd64,!purego arm64,!purego

package sm2curve

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 Point 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
}

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

// SetGenerator sets p to the canonical generator and returns p.
func (p *Point) SetGenerator() *Point {
    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 *Point) Set(q *Point) *Point {
    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

// toElementArray, convert slice of bytes to pointer to [32]byte.
// This function is required for low version of golang, can type cast directly
// since golang 1.17.
func toElementArray(b []byte) *[32]byte {
    tmpPtr := (*unsafe.Pointer)(unsafe.Pointer(&b))
    return (*[32]byte)(*tmpPtr)
}

// 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 *Point) SetBytes(b []byte) (*Point, 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(NewPoint()), nil

    // Uncompressed form.
    case len(b) == p256UncompressedLength && b[0] == 4:
        var r Point
        p256BigToLittle(&r.x, toElementArray(b[1:33]))
        p256BigToLittle(&r.y, toElementArray(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 Point
        p256BigToLittle(&r.x, toElementArray(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)

    p256Sqr(t4, t1, 3)
    p256Mul(t3, t3, t4) // t3.Mul(t3, t4)
    p256Sqr(t3, t3, 5)
    p256Mul(t1, t1, t3) // t1.Mul(t1, t3)
    p256Sqr(t3, t1, 2)
    p256Mul(t2, t2, t3) // t2.Mul(t2, t3)
    p256Sqr(t2, t2, 14)
    p256Mul(t1, t1, t2) // t1.Mul(t1, t2)

    p256Mul(t0, t0, t1) // t0.Mul(t0, t1)
    p256Sqr(t0, t0, 4)
    p256Sqr(t1, t0, 31)
    p256Mul(t0, t0, t1) //t0.Mul(t0, t1)
    p256Sqr(t1, t1, 32)

    p256Mul(t1, t0, t1) //t1.Mul(t0, t1)
    p256Sqr(t1, t1, 62)
    p256Mul(t0, t0, t1) //t0.Mul(t0, t1)
    p256Mul(z, z, t0)   //z.Mul(z, t0)
    p256Sqr(z, z, 32)
    p256Mul(z, z, x) // z.Mul(x, 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 sm2ec_asm_*.s
var supportBMI2 = 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 *Point, 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)

// lookupTable 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 lookupTable [32]Point

// 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 *Point, table *lookupTable, 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 lookupTable, 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 sm2ec_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 *Point, 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 *Point) int

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

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

// 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 *Point) Add(r1, r2 *Point) *Point {
    var sum, double Point
    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 *Point) Double(p *Point) *Point {
    var double Point
    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 *Point) ScalarBaseMult(scalar []byte) (*Point, error) {
    if len(scalar) != 32 {
        return nil, errors.New("invalid scalar length")
    }
    scalarReversed := new(p256OrdElement)
    p256OrdBigToLittle(scalarReversed, toElementArray(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 *Point) ScalarMult(q *Point, scalar []byte) (*Point, error) {
    if len(scalar) != 32 {
        return nil, errors.New("invalid scalar length")
    }
    scalarReversed := new(p256OrdElement)
    p256OrdBigToLittle(scalarReversed, toElementArray(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 *Point) 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 *Point) 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 *Point) 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(toElementArray(out[1:33]), x)
    p256LittleToBig(toElementArray(out[33:65]), y)

    return out[:]
}

// affineFromMont sets (x, y) to the affine coordinates of p, converted out of the
// Montgomery domain.
func (p *Point) 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 *Point) 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 *Point) 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(toElementArray(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 *Point) 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 *Point) 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(toElementArray(out[1:33]), x)

    return out[:]
}

// Select sets q to p1 if cond == 1, and to p2 if cond == 0.
func (q *Point) Select(p1, p2 *Point, cond int) *Point {
    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 *Point) 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, NewPoint(), zero)
}

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

    // 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)
}