| // 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. |
| |
| //go:build (!amd64 && !arm64 && !ppc64le && !s390x) || purego |
| |
| package nistec |
| |
| import ( |
| "crypto/internal/constanttime" |
| "crypto/internal/fips140/nistec/fiat" |
| "crypto/internal/fips140deps/byteorder" |
| "crypto/internal/fips140deps/cpu" |
| "errors" |
| "math/bits" |
| "sync" |
| "unsafe" |
| ) |
| |
| // P256Point is a P-256 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. Infinity is (0:1:0). |
| // |
| // fiat.P256Element is a base field element in [0, P-1] in the Montgomery |
| // domain (with R 2²⁵⁶ and P 2²⁵⁶ - 2²²⁴ + 2¹⁹² + 2⁹⁶ - 1) as four limbs in |
| // little-endian order value. |
| x, y, z fiat.P256Element |
| } |
| |
| // NewP256Point returns a new P256Point representing the point at infinity point. |
| func NewP256Point() *P256Point { |
| p := &P256Point{} |
| p.y.One() |
| return p |
| } |
| |
| // 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 |
| } |
| |
| 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 *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) == p256UncompressedLength && 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) == p256CompressedLength && (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 [p256UncompressedLength]byte |
| return p.bytes(&out) |
| } |
| |
| func (p *P256Point) bytes(out *[p256UncompressedLength]byte) []byte { |
| // The SEC 1 representation of the point at infinity is a single zero byte, |
| // and only infinity has z = 0. |
| 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 [p256CompressedLength]byte |
| return p.bytesCompressed(&out) |
| } |
| |
| func (p *P256Point) bytesCompressed(out *[p256CompressedLength]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 |
| } |
| |
| // 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 fiat.P256Element |
| } |
| |
| func (p *p256AffinePoint) Projective() *P256Point { |
| pp := &P256Point{x: p.x, y: p.y} |
| pp.z.One() |
| return pp |
| } |
| |
| // AddAffine sets q = p1 + p2, if infinity == 0, and to p1 if infinity == 1. |
| // p2 can't be the point at infinity as it can't be represented in affine |
| // coordinates, instead callers can set p2 to an arbitrary point and set |
| // infinity to 1. |
| func (q *P256Point) AddAffine(p1 *P256Point, p2 *p256AffinePoint, infinity int) *P256Point { |
| // Complete mixed addition formula for a = -3 from "Complete addition |
| // formulas for prime order elliptic curves" |
| // (https://eprint.iacr.org/2015/1060), Algorithm 5. |
| |
| t0 := new(fiat.P256Element).Mul(&p1.x, &p2.x) // t0 ← X1 · X2 |
| t1 := new(fiat.P256Element).Mul(&p1.y, &p2.y) // t1 ← Y1 · Y2 |
| t3 := new(fiat.P256Element).Add(&p2.x, &p2.y) // t3 ← X2 + Y2 |
| t4 := new(fiat.P256Element).Add(&p1.x, &p1.y) // t4 ← X1 + Y1 |
| t3.Mul(t3, t4) // t3 ← t3 · t4 |
| t4.Add(t0, t1) // t4 ← t0 + t1 |
| t3.Sub(t3, t4) // t3 ← t3 − t4 |
| t4.Mul(&p2.y, &p1.z) // t4 ← Y2 · Z1 |
| t4.Add(t4, &p1.y) // t4 ← t4 + Y1 |
| y3 := new(fiat.P256Element).Mul(&p2.x, &p1.z) // Y3 ← X2 · Z1 |
| y3.Add(y3, &p1.x) // Y3 ← Y3 + X1 |
| z3 := new(fiat.P256Element).Mul(p256B(), &p1.z) // Z3 ← b · Z1 |
| x3 := new(fiat.P256Element).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(&p1.z, &p1.z) // t1 ← Z1 + Z1 |
| t2 := new(fiat.P256Element).Add(t1, &p1.z) // t2 ← t1 + Z1 |
| 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.Select(&p1.x, x3, infinity) |
| q.y.Select(&p1.y, y3, infinity) |
| q.z.Select(&p1.z, z3, infinity) |
| 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 |
| } |
| |
| // 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 |
| |
| // SetBytes sets s to the big-endian value of x, reducing it as necessary. |
| func (s *p256OrdElement) SetBytes(x []byte) (*p256OrdElement, error) { |
| if len(x) != 32 { |
| return nil, errors.New("invalid scalar length") |
| } |
| |
| s[0] = byteorder.BEUint64(x[24:]) |
| s[1] = byteorder.BEUint64(x[16:]) |
| s[2] = byteorder.BEUint64(x[8:]) |
| s[3] = byteorder.BEUint64(x[:]) |
| |
| // Ensure s is in the range [0, ord(G)-1]. Since 2 * ord(G) > 2²⁵⁶, we can |
| // just conditionally subtract ord(G), keeping the result if it doesn't |
| // underflow. |
| t0, b := bits.Sub64(s[0], 0xf3b9cac2fc632551, 0) |
| t1, b := bits.Sub64(s[1], 0xbce6faada7179e84, b) |
| t2, b := bits.Sub64(s[2], 0xffffffffffffffff, b) |
| t3, b := bits.Sub64(s[3], 0xffffffff00000000, b) |
| tMask := b - 1 // zero if subtraction underflowed |
| s[0] ^= (t0 ^ s[0]) & tMask |
| s[1] ^= (t1 ^ s[1]) & tMask |
| s[2] ^= (t2 ^ s[2]) & tMask |
| s[3] ^= (t3 ^ s[3]) & tMask |
| |
| return s, nil |
| } |
| |
| func (s *p256OrdElement) Bytes() []byte { |
| var out [32]byte |
| byteorder.BEPutUint64(out[24:], s[0]) |
| byteorder.BEPutUint64(out[16:], s[1]) |
| byteorder.BEPutUint64(out[8:], s[2]) |
| byteorder.BEPutUint64(out[:], s[3]) |
| return out[:] |
| } |
| |
| // Rsh returns the 64 least significant bits of x >> n. n must be lower |
| // than 256. The value of n leaks through timing side-channels. |
| func (s *p256OrdElement) Rsh(n int) uint64 { |
| i := n / 64 |
| n = n % 64 |
| res := s[i] >> n |
| // Shift in the more significant limb, if present. |
| if i := i + 1; i < len(s) { |
| res |= s[i] << (64 - n) |
| } |
| return res |
| } |
| |
| // 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 [16]P256Point |
| |
| // Select selects the n-th multiple of the table base point into p. It works in |
| // constant time. n must be in [0, 16]. If n is 0, p is set to the identity point. |
| 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 := constanttime.ByteEq(i, n) |
| p.Select(&table[i-1], p, cond) |
| } |
| } |
| |
| // Compute populates the table to the first 16 multiples of q. |
| func (table *p256Table) Compute(q *P256Point) *p256Table { |
| table[0].Set(q) |
| for i := 1; i < 16; i += 2 { |
| table[i].Double(&table[i/2]) |
| if i+1 < 16 { |
| table[i+1].Add(&table[i], q) |
| } |
| } |
| return table |
| } |
| |
| func boothW5(in uint64) (uint8, int) { |
| s := ^((in >> 5) - 1) |
| d := (1 << 6) - in - 1 |
| d = (d & s) | (in & (^s)) |
| d = (d >> 1) + (d & 1) |
| return uint8(d), int(s & 1) |
| } |
| |
| // ScalarMult sets r = scalar * q, where scalar is a 32-byte big endian value, |
| // and returns r. If scalar is not 32 bytes long, ScalarMult returns an error |
| // and the receiver is unchanged. |
| func (p *P256Point) ScalarMult(q *P256Point, scalar []byte) (*P256Point, error) { |
| s, err := new(p256OrdElement).SetBytes(scalar) |
| if err != nil { |
| return nil, err |
| } |
| |
| // Start scanning the window from the most significant bits. We move by |
| // 5 bits at a time and need to finish at -1, so -1 + 5 * 51 = 254. |
| index := 254 |
| |
| sel, sign := boothW5(s.Rsh(index)) |
| // sign is always zero because the boothW5 input here is at |
| // most two bits long, so the top bit is never set. |
| _ = sign |
| |
| // Neither Select nor Add have exceptions for the point at infinity / |
| // selector zero, so we don't need to check for it here or in the loop. |
| table := new(p256Table).Compute(q) |
| table.Select(p, sel) |
| |
| t := NewP256Point() |
| for index >= 4 { |
| index -= 5 |
| |
| p.Double(p) |
| p.Double(p) |
| p.Double(p) |
| p.Double(p) |
| p.Double(p) |
| |
| if index >= 0 { |
| sel, sign = boothW5(s.Rsh(index) & 0b111111) |
| } else { |
| // Booth encoding considers a virtual zero bit at index -1, |
| // so we shift left the least significant limb. |
| wvalue := (s[0] << 1) & 0b111111 |
| sel, sign = boothW5(wvalue) |
| } |
| |
| table.Select(t, sel) |
| t.Negate(sign) |
| p.Add(p, t) |
| } |
| |
| return p, nil |
| } |
| |
| // Negate sets p to -p, if cond == 1, and to p if cond == 0. |
| func (p *P256Point) Negate(cond int) *P256Point { |
| negY := new(fiat.P256Element) |
| negY.Sub(negY, &p.y) |
| p.y.Select(negY, &p.y, cond) |
| return p |
| } |
| |
| // 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 |
| |
| // Select selects the n-th multiple of the table base point into p. It works in |
| // constant time. n can be in [0, 32], but (unlike p256Table.Select) if n is 0, |
| // p is set to an undefined value. |
| func (table *p256AffineTable) Select(p *p256AffinePoint, n uint8) { |
| if n > 32 { |
| panic("nistec: internal error: p256AffineTable.Select called with out-of-bounds value") |
| } |
| for i := uint8(1); i <= 32; i++ { |
| cond := constanttime.ByteEq(i, n) |
| p.x.Select(&table[i-1].x, &p.x, cond) |
| p.y.Select(&table[i-1].y, &p.y, cond) |
| } |
| } |
| |
| // p256GeneratorTables 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 ScalarBaseMult, and having each table already |
| // pre-doubled lets us avoid the doublings between windows entirely. This table |
| // aliases into p256PrecomputedEmbed. |
| var p256GeneratorTables *[43]p256AffineTable |
| |
| func init() { |
| p256GeneratorTablesPtr := unsafe.Pointer(&p256PrecomputedEmbed) |
| if cpu.BigEndian { |
| var newTable [43 * 32 * 2 * 4]uint64 |
| for i, x := range (*[43 * 32 * 2 * 4][8]byte)(p256GeneratorTablesPtr) { |
| newTable[i] = byteorder.LEUint64(x[:]) |
| } |
| p256GeneratorTablesPtr = unsafe.Pointer(&newTable) |
| } |
| p256GeneratorTables = (*[43]p256AffineTable)(p256GeneratorTablesPtr) |
| } |
| |
| func boothW6(in uint64) (uint8, int) { |
| s := ^((in >> 6) - 1) |
| d := (1 << 7) - in - 1 |
| d = (d & s) | (in & (^s)) |
| d = (d >> 1) + (d & 1) |
| return uint8(d), int(s & 1) |
| } |
| |
| // ScalarBaseMult sets p = 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 (p *P256Point) ScalarBaseMult(scalar []byte) (*P256Point, error) { |
| // This function works like ScalarMult above, but the table is fixed and |
| // "pre-doubled" for each iteration, so instead of doubling we move to the |
| // next table at each iteration. |
| |
| s, err := new(p256OrdElement).SetBytes(scalar) |
| if err != nil { |
| return nil, err |
| } |
| |
| // Start scanning the window from the most significant bits. We move by |
| // 6 bits at a time and need to finish at -1, so -1 + 6 * 42 = 251. |
| index := 251 |
| |
| sel, sign := boothW6(s.Rsh(index)) |
| // sign is always zero because the boothW6 input here is at |
| // most five bits long, so the top bit is never set. |
| _ = sign |
| |
| t := &p256AffinePoint{} |
| table := &p256GeneratorTables[(index+1)/6] |
| table.Select(t, sel) |
| |
| // Select's output is undefined if the selector is zero, when it should be |
| // the point at infinity (because infinity can't be represented in affine |
| // coordinates). Here we conditionally set p to the infinity if sel is zero. |
| // In the loop, that's handled by AddAffine. |
| selIsZero := constanttime.ByteEq(sel, 0) |
| p.Select(NewP256Point(), t.Projective(), selIsZero) |
| |
| for index >= 5 { |
| index -= 6 |
| |
| if index >= 0 { |
| sel, sign = boothW6(s.Rsh(index) & 0b1111111) |
| } else { |
| // Booth encoding considers a virtual zero bit at index -1, |
| // so we shift left the least significant limb. |
| wvalue := (s[0] << 1) & 0b1111111 |
| sel, sign = boothW6(wvalue) |
| } |
| |
| table := &p256GeneratorTables[(index+1)/6] |
| table.Select(t, sel) |
| t.Negate(sign) |
| selIsZero := constanttime.ByteEq(sel, 0) |
| p.AddAffine(p, t, selIsZero) |
| } |
| |
| return p, nil |
| } |
| |
| // Negate sets p to -p, if cond == 1, and to p if cond == 0. |
| func (p *p256AffinePoint) Negate(cond int) *p256AffinePoint { |
| negY := new(fiat.P256Element) |
| negY.Sub(negY, &p.y) |
| p.y.Select(negY, &p.y, cond) |
| return p |
| } |
| |
| // 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) { |
| t0, t1 := new(fiat.P256Element), new(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 |
| // |
| p256Square(t0, x, 1) |
| t0.Mul(x, t0) |
| p256Square(t1, t0, 2) |
| t0.Mul(t0, t1) |
| p256Square(t1, t0, 4) |
| t0.Mul(t0, t1) |
| p256Square(t1, t0, 8) |
| t0.Mul(t0, t1) |
| p256Square(t1, t0, 16) |
| t0.Mul(t0, t1) |
| p256Square(t0, t0, 32) |
| t0.Mul(x, t0) |
| p256Square(t0, t0, 96) |
| t0.Mul(x, t0) |
| p256Square(t0, t0, 94) |
| |
| // Check if the candidate t0 is indeed a square root of x. |
| t1.Square(t0) |
| if t1.Equal(x) != 1 { |
| return false |
| } |
| e.Set(t0) |
| return true |
| } |
| |
| // p256Square sets e to the square of x, repeated n times > 1. |
| func p256Square(e, x *fiat.P256Element, n int) { |
| e.Square(x) |
| for i := 1; i < n; i++ { |
| e.Square(e) |
| } |
| } |
