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Diffstat (limited to 'src/crypto/elliptic/internal/nistec/p521.go')
| -rw-r--r-- | src/crypto/elliptic/internal/nistec/p521.go | 310 |
1 files changed, 310 insertions, 0 deletions
diff --git a/src/crypto/elliptic/internal/nistec/p521.go b/src/crypto/elliptic/internal/nistec/p521.go new file mode 100644 index 0000000..cdbd195 --- /dev/null +++ b/src/crypto/elliptic/internal/nistec/p521.go @@ -0,0 +1,310 @@ +// Copyright 2021 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. + +// Package nistec implements the NIST P elliptic curves from FIPS 186-4. +// +// This package uses fiat-crypto for its backend field arithmetic (not math/big) +// and exposes constant-time, heap allocation-free, byte slice-based safe APIs. +// Group operations use modern and safe complete addition formulas. The point at +// infinity is handled and encoded according to SEC 1, Version 2.0, and invalid +// curve points can't be represented. +package nistec + +import ( + "crypto/elliptic/internal/fiat" + "crypto/subtle" + "errors" +) + +var p521B, _ = new(fiat.P521Element).SetBytes([]byte{ + 0x00, 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, 0x09, 0xe1, 0x56, 0x19, + 0x39, 0x51, 0xec, 0x7e, 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1, + 0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c, 0x34, 0xf1, 0xef, 0x45, + 0x1f, 0xd4, 0x6b, 0x50, 0x3f, 0x00}) + +var p521G, _ = NewP521Point().SetBytes([]byte{0x04, + 0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04, 0xe9, 0xcd, 0x9e, 0x3e, + 0xcb, 0x66, 0x23, 0x95, 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 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, 0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, + 0x9a, 0x3b, 0xc0, 0x04, 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, 0x01, 0x3f, 0xad, 0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, + 0xa2, 0x72, 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1, 0x66, 0x50}) + +const p521ElementLength = 66 + +// P521Point is a P-521 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] == 0: + return nil, errors.New("unimplemented") // TODO(filippo) + + default: + return nil, errors.New("invalid P521 point encoding") + } +} + +func p521CheckOnCurve(x, y *fiat.P521Element) error { + // x³ - 3x + b. + x3 := new(fiat.P521Element).Square(x) + x3.Mul(x3, x) + + threeX := new(fiat.P521Element).Add(x, x) + threeX.Add(threeX, x) + + x3.Sub(x3, threeX) + x3.Add(x3, p521B) + + // y² = x³ - 3x + b + y2 := new(fiat.P521Element).Square(y) + + if x3.Equal(y2) != 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 [133]byte + return p.bytes(&out) +} + +func (p *P521Point) bytes(out *[133]byte) []byte { + if p.z.IsZero() == 1 { + return append(out[:0], 0) + } + + zinv := new(fiat.P521Element).Invert(p.z) + xx := new(fiat.P521Element).Mul(p.x, zinv) + yy := new(fiat.P521Element).Mul(p.y, zinv) + + buf := append(out[:0], 4) + buf = append(buf, xx.Bytes()...) + buf = append(buf, yy.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 +} + +// ScalarMult sets p = scalar * q, and returns p. +func (p *P521Point) ScalarMult(q *P521Point, scalar []byte) *P521Point { + // table holds the first 16 multiples of q. The explicit newP521Point calls + // get inlined, letting the allocations live on the stack. + var table = [16]*P521Point{ + NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(), + NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(), + NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(), + NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(), + } + for i := 1; i < 16; i++ { + table[i].Add(table[i-1], 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 _, byte := range scalar { + p.Double(p) + p.Double(p) + p.Double(p) + p.Double(p) + + for i := uint8(0); i < 16; i++ { + cond := subtle.ConstantTimeByteEq(byte>>4, i) + t.Select(table[i], t, cond) + } + p.Add(p, t) + + p.Double(p) + p.Double(p) + p.Double(p) + p.Double(p) + + for i := uint8(0); i < 16; i++ { + cond := subtle.ConstantTimeByteEq(byte&0b1111, i) + t.Select(table[i], t, cond) + } + p.Add(p, t) + } + + return p +} |