| // Copyright 2022 The Go Authors. All rights reserved. |
| // Use of this source code is governed by a BSD-style |
| // license that can be found in the LICENSE file. |
| |
| // Code generated by generate.go. DO NOT EDIT. |
| |
| package nistec |
| |
| import ( |
| "crypto/internal/nistec/fiat" |
| "crypto/subtle" |
| "errors" |
| "sync" |
| ) |
| |
| var p521B, _ = new(fiat.P521Element).SetBytes([]byte{0x0, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85, 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3, 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1, 0x9, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e, 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1, 0xbf, 0x7, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c, 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50, 0x3f, 0x0}) |
| |
| var p521G, _ = NewP521Point().SetBytes([]byte{0x4, 0x0, 0xc6, 0x85, 0x8e, 0x6, 0xb7, 0x4, 0x4, 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95, 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x5, 0x3f, 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d, 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7, 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff, 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a, 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5, 0xbd, 0x66, 0x1, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, 0xc0, 0x4, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d, 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b, 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e, 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4, 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x1, 0x3f, 0xad, 0x7, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72, 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1, 0x66, 0x50}) |
| |
| // p521ElementLength is the length of an element of the base or scalar field, |
| // which have the same bytes length for all NIST P curves. |
| const p521ElementLength = 66 |
| |
| // P521Point is a P521 point. The zero value is NOT valid. |
| type P521Point struct { |
| // The point is represented in projective coordinates (X:Y:Z), |
| // where x = X/Z and y = Y/Z. |
| x, y, z *fiat.P521Element |
| } |
| |
| // NewP521Point returns a new P521Point representing the point at infinity point. |
| func NewP521Point() *P521Point { |
| return &P521Point{ |
| x: new(fiat.P521Element), |
| y: new(fiat.P521Element).One(), |
| z: new(fiat.P521Element), |
| } |
| } |
| |
| // NewP521Generator returns a new P521Point set to the canonical generator. |
| func NewP521Generator() *P521Point { |
| return (&P521Point{ |
| x: new(fiat.P521Element), |
| y: new(fiat.P521Element), |
| z: new(fiat.P521Element), |
| }).Set(p521G) |
| } |
| |
| // Set sets p = q and returns p. |
| func (p *P521Point) Set(q *P521Point) *P521Point { |
| p.x.Set(q.x) |
| p.y.Set(q.y) |
| p.z.Set(q.z) |
| return p |
| } |
| |
| // SetBytes sets p to the compressed, uncompressed, or infinity value encoded in |
| // b, as specified in SEC 1, Version 2.0, Section 2.3.4. If the point is not on |
| // the curve, it returns nil and an error, and the receiver is unchanged. |
| // Otherwise, it returns p. |
| func (p *P521Point) SetBytes(b []byte) (*P521Point, error) { |
| switch { |
| // Point at infinity. |
| case len(b) == 1 && b[0] == 0: |
| return p.Set(NewP521Point()), nil |
| |
| // Uncompressed form. |
| case len(b) == 1+2*p521ElementLength && b[0] == 4: |
| x, err := new(fiat.P521Element).SetBytes(b[1 : 1+p521ElementLength]) |
| if err != nil { |
| return nil, err |
| } |
| y, err := new(fiat.P521Element).SetBytes(b[1+p521ElementLength:]) |
| if err != nil { |
| return nil, err |
| } |
| if err := p521CheckOnCurve(x, y); err != nil { |
| return nil, err |
| } |
| p.x.Set(x) |
| p.y.Set(y) |
| p.z.One() |
| return p, nil |
| |
| // Compressed form. |
| case len(b) == 1+p521ElementLength && (b[0] == 2 || b[0] == 3): |
| x, err := new(fiat.P521Element).SetBytes(b[1:]) |
| if err != nil { |
| return nil, err |
| } |
| |
| // y² = x³ - 3x + b |
| y := p521Polynomial(new(fiat.P521Element), x) |
| if !p521Sqrt(y, y) { |
| return nil, errors.New("invalid P521 compressed point encoding") |
| } |
| |
| // Select the positive or negative root, as indicated by the least |
| // significant bit, based on the encoding type byte. |
| otherRoot := new(fiat.P521Element) |
| otherRoot.Sub(otherRoot, y) |
| cond := y.Bytes()[p521ElementLength-1]&1 ^ b[0]&1 |
| y.Select(otherRoot, y, int(cond)) |
| |
| p.x.Set(x) |
| p.y.Set(y) |
| p.z.One() |
| return p, nil |
| |
| default: |
| return nil, errors.New("invalid P521 point encoding") |
| } |
| } |
| |
| // p521Polynomial sets y2 to x³ - 3x + b, and returns y2. |
| func p521Polynomial(y2, x *fiat.P521Element) *fiat.P521Element { |
| y2.Square(x) |
| y2.Mul(y2, x) |
| |
| threeX := new(fiat.P521Element).Add(x, x) |
| threeX.Add(threeX, x) |
| |
| y2.Sub(y2, threeX) |
| return y2.Add(y2, p521B) |
| } |
| |
| func p521CheckOnCurve(x, y *fiat.P521Element) error { |
| // y² = x³ - 3x + b |
| rhs := p521Polynomial(new(fiat.P521Element), x) |
| lhs := new(fiat.P521Element).Square(y) |
| if rhs.Equal(lhs) != 1 { |
| return errors.New("P521 point not on curve") |
| } |
| return nil |
| } |
| |
| // Bytes returns the uncompressed or infinity encoding of p, as specified in |
| // SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the point at |
| // infinity is shorter than all other encodings. |
| func (p *P521Point) Bytes() []byte { |
| // This function is outlined to make the allocations inline in the caller |
| // rather than happen on the heap. |
| var out [1 + 2*p521ElementLength]byte |
| return p.bytes(&out) |
| } |
| |
| func (p *P521Point) bytes(out *[1 + 2*p521ElementLength]byte) []byte { |
| if p.z.IsZero() == 1 { |
| return append(out[:0], 0) |
| } |
| |
| zinv := new(fiat.P521Element).Invert(p.z) |
| x := new(fiat.P521Element).Mul(p.x, zinv) |
| y := new(fiat.P521Element).Mul(p.y, zinv) |
| |
| buf := append(out[:0], 4) |
| buf = append(buf, x.Bytes()...) |
| buf = append(buf, y.Bytes()...) |
| return buf |
| } |
| |
| // BytesCompressed returns the compressed or infinity encoding of p, as |
| // specified in SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the |
| // point at infinity is shorter than all other encodings. |
| func (p *P521Point) BytesCompressed() []byte { |
| // This function is outlined to make the allocations inline in the caller |
| // rather than happen on the heap. |
| var out [1 + p521ElementLength]byte |
| return p.bytesCompressed(&out) |
| } |
| |
| func (p *P521Point) bytesCompressed(out *[1 + p521ElementLength]byte) []byte { |
| if p.z.IsZero() == 1 { |
| return append(out[:0], 0) |
| } |
| |
| zinv := new(fiat.P521Element).Invert(p.z) |
| x := new(fiat.P521Element).Mul(p.x, zinv) |
| y := new(fiat.P521Element).Mul(p.y, zinv) |
| |
| // Encode the sign of the y coordinate (indicated by the least significant |
| // bit) as the encoding type (2 or 3). |
| buf := append(out[:0], 2) |
| buf[0] |= y.Bytes()[p521ElementLength-1] & 1 |
| buf = append(buf, x.Bytes()...) |
| return buf |
| } |
| |
| // Add sets q = p1 + p2, and returns q. The points may overlap. |
| func (q *P521Point) Add(p1, p2 *P521Point) *P521Point { |
| // Complete addition formula for a = -3 from "Complete addition formulas for |
| // prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2. |
| |
| t0 := new(fiat.P521Element).Mul(p1.x, p2.x) // t0 := X1 * X2 |
| t1 := new(fiat.P521Element).Mul(p1.y, p2.y) // t1 := Y1 * Y2 |
| t2 := new(fiat.P521Element).Mul(p1.z, p2.z) // t2 := Z1 * Z2 |
| t3 := new(fiat.P521Element).Add(p1.x, p1.y) // t3 := X1 + Y1 |
| t4 := new(fiat.P521Element).Add(p2.x, p2.y) // t4 := X2 + Y2 |
| t3.Mul(t3, t4) // t3 := t3 * t4 |
| t4.Add(t0, t1) // t4 := t0 + t1 |
| t3.Sub(t3, t4) // t3 := t3 - t4 |
| t4.Add(p1.y, p1.z) // t4 := Y1 + Z1 |
| x3 := new(fiat.P521Element).Add(p2.y, p2.z) // X3 := Y2 + Z2 |
| t4.Mul(t4, x3) // t4 := t4 * X3 |
| x3.Add(t1, t2) // X3 := t1 + t2 |
| t4.Sub(t4, x3) // t4 := t4 - X3 |
| x3.Add(p1.x, p1.z) // X3 := X1 + Z1 |
| y3 := new(fiat.P521Element).Add(p2.x, p2.z) // Y3 := X2 + Z2 |
| x3.Mul(x3, y3) // X3 := X3 * Y3 |
| y3.Add(t0, t2) // Y3 := t0 + t2 |
| y3.Sub(x3, y3) // Y3 := X3 - Y3 |
| z3 := new(fiat.P521Element).Mul(p521B, t2) // Z3 := b * t2 |
| x3.Sub(y3, z3) // X3 := Y3 - Z3 |
| z3.Add(x3, x3) // Z3 := X3 + X3 |
| x3.Add(x3, z3) // X3 := X3 + Z3 |
| z3.Sub(t1, x3) // Z3 := t1 - X3 |
| x3.Add(t1, x3) // X3 := t1 + X3 |
| y3.Mul(p521B, y3) // Y3 := b * Y3 |
| t1.Add(t2, t2) // t1 := t2 + t2 |
| t2.Add(t1, t2) // t2 := t1 + t2 |
| y3.Sub(y3, t2) // Y3 := Y3 - t2 |
| y3.Sub(y3, t0) // Y3 := Y3 - t0 |
| t1.Add(y3, y3) // t1 := Y3 + Y3 |
| y3.Add(t1, y3) // Y3 := t1 + Y3 |
| t1.Add(t0, t0) // t1 := t0 + t0 |
| t0.Add(t1, t0) // t0 := t1 + t0 |
| t0.Sub(t0, t2) // t0 := t0 - t2 |
| t1.Mul(t4, y3) // t1 := t4 * Y3 |
| t2.Mul(t0, y3) // t2 := t0 * Y3 |
| y3.Mul(x3, z3) // Y3 := X3 * Z3 |
| y3.Add(y3, t2) // Y3 := Y3 + t2 |
| x3.Mul(t3, x3) // X3 := t3 * X3 |
| x3.Sub(x3, t1) // X3 := X3 - t1 |
| z3.Mul(t4, z3) // Z3 := t4 * Z3 |
| t1.Mul(t3, t0) // t1 := t3 * t0 |
| z3.Add(z3, t1) // Z3 := Z3 + t1 |
| |
| q.x.Set(x3) |
| q.y.Set(y3) |
| q.z.Set(z3) |
| return q |
| } |
| |
| // Double sets q = p + p, and returns q. The points may overlap. |
| func (q *P521Point) Double(p *P521Point) *P521Point { |
| // Complete addition formula for a = -3 from "Complete addition formulas for |
| // prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2. |
| |
| t0 := new(fiat.P521Element).Square(p.x) // t0 := X ^ 2 |
| t1 := new(fiat.P521Element).Square(p.y) // t1 := Y ^ 2 |
| t2 := new(fiat.P521Element).Square(p.z) // t2 := Z ^ 2 |
| t3 := new(fiat.P521Element).Mul(p.x, p.y) // t3 := X * Y |
| t3.Add(t3, t3) // t3 := t3 + t3 |
| z3 := new(fiat.P521Element).Mul(p.x, p.z) // Z3 := X * Z |
| z3.Add(z3, z3) // Z3 := Z3 + Z3 |
| y3 := new(fiat.P521Element).Mul(p521B, t2) // Y3 := b * t2 |
| y3.Sub(y3, z3) // Y3 := Y3 - Z3 |
| x3 := new(fiat.P521Element).Add(y3, y3) // X3 := Y3 + Y3 |
| y3.Add(x3, y3) // Y3 := X3 + Y3 |
| x3.Sub(t1, y3) // X3 := t1 - Y3 |
| y3.Add(t1, y3) // Y3 := t1 + Y3 |
| y3.Mul(x3, y3) // Y3 := X3 * Y3 |
| x3.Mul(x3, t3) // X3 := X3 * t3 |
| t3.Add(t2, t2) // t3 := t2 + t2 |
| t2.Add(t2, t3) // t2 := t2 + t3 |
| z3.Mul(p521B, z3) // Z3 := b * Z3 |
| z3.Sub(z3, t2) // Z3 := Z3 - t2 |
| z3.Sub(z3, t0) // Z3 := Z3 - t0 |
| t3.Add(z3, z3) // t3 := Z3 + Z3 |
| z3.Add(z3, t3) // Z3 := Z3 + t3 |
| t3.Add(t0, t0) // t3 := t0 + t0 |
| t0.Add(t3, t0) // t0 := t3 + t0 |
| t0.Sub(t0, t2) // t0 := t0 - t2 |
| t0.Mul(t0, z3) // t0 := t0 * Z3 |
| y3.Add(y3, t0) // Y3 := Y3 + t0 |
| t0.Mul(p.y, p.z) // t0 := Y * Z |
| t0.Add(t0, t0) // t0 := t0 + t0 |
| z3.Mul(t0, z3) // Z3 := t0 * Z3 |
| x3.Sub(x3, z3) // X3 := X3 - Z3 |
| z3.Mul(t0, t1) // Z3 := t0 * t1 |
| z3.Add(z3, z3) // Z3 := Z3 + Z3 |
| z3.Add(z3, z3) // Z3 := Z3 + Z3 |
| |
| q.x.Set(x3) |
| q.y.Set(y3) |
| q.z.Set(z3) |
| return q |
| } |
| |
| // Select sets q to p1 if cond == 1, and to p2 if cond == 0. |
| func (q *P521Point) Select(p1, p2 *P521Point, cond int) *P521Point { |
| q.x.Select(p1.x, p2.x, cond) |
| q.y.Select(p1.y, p2.y, cond) |
| q.z.Select(p1.z, p2.z, cond) |
| return q |
| } |
| |
| // A p521Table holds the first 15 multiples of a point at offset -1, so [1]P |
| // is at table[0], [15]P is at table[14], and [0]P is implicitly the identity |
| // point. |
| type p521Table [15]*P521Point |
| |
| // Select selects the n-th multiple of the table base point into p. It works in |
| // constant time by iterating over every entry of the table. n must be in [0, 15]. |
| func (table *p521Table) Select(p *P521Point, n uint8) { |
| if n >= 16 { |
| panic("nistec: internal error: p521Table called with out-of-bounds value") |
| } |
| p.Set(NewP521Point()) |
| for i := uint8(1); i < 16; i++ { |
| cond := subtle.ConstantTimeByteEq(i, n) |
| p.Select(table[i-1], p, cond) |
| } |
| } |
| |
| // ScalarMult sets p = scalar * q, and returns p. |
| func (p *P521Point) ScalarMult(q *P521Point, scalar []byte) (*P521Point, error) { |
| // Compute a p521Table for the base point q. The explicit NewP521Point |
| // calls get inlined, letting the allocations live on the stack. |
| var table = p521Table{NewP521Point(), NewP521Point(), NewP521Point(), |
| NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(), |
| NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(), |
| NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point()} |
| table[0].Set(q) |
| for i := 1; i < 15; i += 2 { |
| table[i].Double(table[i/2]) |
| table[i+1].Add(table[i], q) |
| } |
| |
| // Instead of doing the classic double-and-add chain, we do it with a |
| // four-bit window: we double four times, and then add [0-15]P. |
| t := NewP521Point() |
| p.Set(NewP521Point()) |
| for i, byte := range scalar { |
| // No need to double on the first iteration, as p is the identity at |
| // this point, and [N]∞ = ∞. |
| if i != 0 { |
| p.Double(p) |
| p.Double(p) |
| p.Double(p) |
| p.Double(p) |
| } |
| |
| windowValue := byte >> 4 |
| table.Select(t, windowValue) |
| p.Add(p, t) |
| |
| p.Double(p) |
| p.Double(p) |
| p.Double(p) |
| p.Double(p) |
| |
| windowValue = byte & 0b1111 |
| table.Select(t, windowValue) |
| p.Add(p, t) |
| } |
| |
| return p, nil |
| } |
| |
| var p521GeneratorTable *[p521ElementLength * 2]p521Table |
| var p521GeneratorTableOnce sync.Once |
| |
| // generatorTable returns a sequence of p521Tables. The first table contains |
| // multiples of G. Each successive table is the previous table doubled four |
| // times. |
| func (p *P521Point) generatorTable() *[p521ElementLength * 2]p521Table { |
| p521GeneratorTableOnce.Do(func() { |
| p521GeneratorTable = new([p521ElementLength * 2]p521Table) |
| base := NewP521Generator() |
| for i := 0; i < p521ElementLength*2; i++ { |
| p521GeneratorTable[i][0] = NewP521Point().Set(base) |
| for j := 1; j < 15; j++ { |
| p521GeneratorTable[i][j] = NewP521Point().Add(p521GeneratorTable[i][j-1], base) |
| } |
| base.Double(base) |
| base.Double(base) |
| base.Double(base) |
| base.Double(base) |
| } |
| }) |
| return p521GeneratorTable |
| } |
| |
| // ScalarBaseMult sets p = scalar * B, where B is the canonical generator, and |
| // returns p. |
| func (p *P521Point) ScalarBaseMult(scalar []byte) (*P521Point, error) { |
| if len(scalar) != p521ElementLength { |
| return nil, errors.New("invalid scalar length") |
| } |
| tables := p.generatorTable() |
| |
| // This is also a scalar multiplication with a four-bit window like in |
| // ScalarMult, but in this case the doublings are precomputed. The value |
| // [windowValue]G added at iteration k would normally get doubled |
| // (totIterations-k)×4 times, but with a larger precomputation we can |
| // instead add [2^((totIterations-k)×4)][windowValue]G and avoid the |
| // doublings between iterations. |
| t := NewP521Point() |
| p.Set(NewP521Point()) |
| tableIndex := len(tables) - 1 |
| for _, byte := range scalar { |
| windowValue := byte >> 4 |
| tables[tableIndex].Select(t, windowValue) |
| p.Add(p, t) |
| tableIndex-- |
| |
| windowValue = byte & 0b1111 |
| tables[tableIndex].Select(t, windowValue) |
| p.Add(p, t) |
| tableIndex-- |
| } |
| |
| return p, nil |
| } |
| |
| // p521Sqrt sets e to a square root of x. If x is not a square, p521Sqrt returns |
| // false and e is unchanged. e and x can overlap. |
| func p521Sqrt(e, x *fiat.P521Element) (isSquare bool) { |
| candidate := new(fiat.P521Element) |
| p521SqrtCandidate(candidate, x) |
| square := new(fiat.P521Element).Square(candidate) |
| if square.Equal(x) != 1 { |
| return false |
| } |
| e.Set(candidate) |
| return true |
| } |
| |
| // p521SqrtCandidate sets z to a square root candidate for x. z and x must not overlap. |
| func p521SqrtCandidate(z, x *fiat.P521Element) { |
| // Since p = 3 mod 4, exponentiation by (p + 1) / 4 yields a square root candidate. |
| // |
| // The sequence of 0 multiplications and 519 squarings is derived from the |
| // following addition chain generated with github.com/mmcloughlin/addchain v0.4.0. |
| // |
| // return 1 << 519 |
| // |
| |
| z.Square(x) |
| for s := 1; s < 519; s++ { |
| z.Square(z) |
| } |
| } |
