diff options
Diffstat (limited to 'vendor/filippo.io/edwards25519/field/fe.go')
| -rw-r--r-- | vendor/filippo.io/edwards25519/field/fe.go | 420 |
1 files changed, 420 insertions, 0 deletions
diff --git a/vendor/filippo.io/edwards25519/field/fe.go b/vendor/filippo.io/edwards25519/field/fe.go new file mode 100644 index 0000000..5518ef2 --- /dev/null +++ b/vendor/filippo.io/edwards25519/field/fe.go @@ -0,0 +1,420 @@ +// Copyright (c) 2017 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 field implements fast arithmetic modulo 2^255-19. +package field + +import ( + "crypto/subtle" + "encoding/binary" + "errors" + "math/bits" +) + +// Element represents an element of the field GF(2^255-19). Note that this +// is not a cryptographically secure group, and should only be used to interact +// with edwards25519.Point coordinates. +// +// This type works similarly to math/big.Int, and all arguments and receivers +// are allowed to alias. +// +// The zero value is a valid zero element. +type Element struct { + // An element t represents the integer + // t.l0 + t.l1*2^51 + t.l2*2^102 + t.l3*2^153 + t.l4*2^204 + // + // Between operations, all limbs are expected to be lower than 2^52. + l0 uint64 + l1 uint64 + l2 uint64 + l3 uint64 + l4 uint64 +} + +const maskLow51Bits uint64 = (1 << 51) - 1 + +var feZero = &Element{0, 0, 0, 0, 0} + +// Zero sets v = 0, and returns v. +func (v *Element) Zero() *Element { + *v = *feZero + return v +} + +var feOne = &Element{1, 0, 0, 0, 0} + +// One sets v = 1, and returns v. +func (v *Element) One() *Element { + *v = *feOne + return v +} + +// reduce reduces v modulo 2^255 - 19 and returns it. +func (v *Element) reduce() *Element { + v.carryPropagate() + + // After the light reduction we now have a field element representation + // v < 2^255 + 2^13 * 19, but need v < 2^255 - 19. + + // If v >= 2^255 - 19, then v + 19 >= 2^255, which would overflow 2^255 - 1, + // generating a carry. That is, c will be 0 if v < 2^255 - 19, and 1 otherwise. + c := (v.l0 + 19) >> 51 + c = (v.l1 + c) >> 51 + c = (v.l2 + c) >> 51 + c = (v.l3 + c) >> 51 + c = (v.l4 + c) >> 51 + + // If v < 2^255 - 19 and c = 0, this will be a no-op. Otherwise, it's + // effectively applying the reduction identity to the carry. + v.l0 += 19 * c + + v.l1 += v.l0 >> 51 + v.l0 = v.l0 & maskLow51Bits + v.l2 += v.l1 >> 51 + v.l1 = v.l1 & maskLow51Bits + v.l3 += v.l2 >> 51 + v.l2 = v.l2 & maskLow51Bits + v.l4 += v.l3 >> 51 + v.l3 = v.l3 & maskLow51Bits + // no additional carry + v.l4 = v.l4 & maskLow51Bits + + return v +} + +// Add sets v = a + b, and returns v. +func (v *Element) Add(a, b *Element) *Element { + v.l0 = a.l0 + b.l0 + v.l1 = a.l1 + b.l1 + v.l2 = a.l2 + b.l2 + v.l3 = a.l3 + b.l3 + v.l4 = a.l4 + b.l4 + // Using the generic implementation here is actually faster than the + // assembly. Probably because the body of this function is so simple that + // the compiler can figure out better optimizations by inlining the carry + // propagation. + return v.carryPropagateGeneric() +} + +// Subtract sets v = a - b, and returns v. +func (v *Element) Subtract(a, b *Element) *Element { + // We first add 2 * p, to guarantee the subtraction won't underflow, and + // then subtract b (which can be up to 2^255 + 2^13 * 19). + v.l0 = (a.l0 + 0xFFFFFFFFFFFDA) - b.l0 + v.l1 = (a.l1 + 0xFFFFFFFFFFFFE) - b.l1 + v.l2 = (a.l2 + 0xFFFFFFFFFFFFE) - b.l2 + v.l3 = (a.l3 + 0xFFFFFFFFFFFFE) - b.l3 + v.l4 = (a.l4 + 0xFFFFFFFFFFFFE) - b.l4 + return v.carryPropagate() +} + +// Negate sets v = -a, and returns v. +func (v *Element) Negate(a *Element) *Element { + return v.Subtract(feZero, a) +} + +// Invert sets v = 1/z mod p, and returns v. +// +// If z == 0, Invert returns v = 0. +func (v *Element) Invert(z *Element) *Element { + // Inversion is implemented as exponentiation with exponent p − 2. It uses the + // same sequence of 255 squarings and 11 multiplications as [Curve25519]. + var z2, z9, z11, z2_5_0, z2_10_0, z2_20_0, z2_50_0, z2_100_0, t Element + + z2.Square(z) // 2 + t.Square(&z2) // 4 + t.Square(&t) // 8 + z9.Multiply(&t, z) // 9 + z11.Multiply(&z9, &z2) // 11 + t.Square(&z11) // 22 + z2_5_0.Multiply(&t, &z9) // 31 = 2^5 - 2^0 + + t.Square(&z2_5_0) // 2^6 - 2^1 + for i := 0; i < 4; i++ { + t.Square(&t) // 2^10 - 2^5 + } + z2_10_0.Multiply(&t, &z2_5_0) // 2^10 - 2^0 + + t.Square(&z2_10_0) // 2^11 - 2^1 + for i := 0; i < 9; i++ { + t.Square(&t) // 2^20 - 2^10 + } + z2_20_0.Multiply(&t, &z2_10_0) // 2^20 - 2^0 + + t.Square(&z2_20_0) // 2^21 - 2^1 + for i := 0; i < 19; i++ { + t.Square(&t) // 2^40 - 2^20 + } + t.Multiply(&t, &z2_20_0) // 2^40 - 2^0 + + t.Square(&t) // 2^41 - 2^1 + for i := 0; i < 9; i++ { + t.Square(&t) // 2^50 - 2^10 + } + z2_50_0.Multiply(&t, &z2_10_0) // 2^50 - 2^0 + + t.Square(&z2_50_0) // 2^51 - 2^1 + for i := 0; i < 49; i++ { + t.Square(&t) // 2^100 - 2^50 + } + z2_100_0.Multiply(&t, &z2_50_0) // 2^100 - 2^0 + + t.Square(&z2_100_0) // 2^101 - 2^1 + for i := 0; i < 99; i++ { + t.Square(&t) // 2^200 - 2^100 + } + t.Multiply(&t, &z2_100_0) // 2^200 - 2^0 + + t.Square(&t) // 2^201 - 2^1 + for i := 0; i < 49; i++ { + t.Square(&t) // 2^250 - 2^50 + } + t.Multiply(&t, &z2_50_0) // 2^250 - 2^0 + + t.Square(&t) // 2^251 - 2^1 + t.Square(&t) // 2^252 - 2^2 + t.Square(&t) // 2^253 - 2^3 + t.Square(&t) // 2^254 - 2^4 + t.Square(&t) // 2^255 - 2^5 + + return v.Multiply(&t, &z11) // 2^255 - 21 +} + +// Set sets v = a, and returns v. +func (v *Element) Set(a *Element) *Element { + *v = *a + return v +} + +// SetBytes sets v to x, where x is a 32-byte little-endian encoding. If x is +// not of the right length, SetBytes returns nil and an error, and the +// receiver is unchanged. +// +// Consistent with RFC 7748, the most significant bit (the high bit of the +// last byte) is ignored, and non-canonical values (2^255-19 through 2^255-1) +// are accepted. Note that this is laxer than specified by RFC 8032, but +// consistent with most Ed25519 implementations. +func (v *Element) SetBytes(x []byte) (*Element, error) { + if len(x) != 32 { + return nil, errors.New("edwards25519: invalid field element input size") + } + + // Bits 0:51 (bytes 0:8, bits 0:64, shift 0, mask 51). + v.l0 = binary.LittleEndian.Uint64(x[0:8]) + v.l0 &= maskLow51Bits + // Bits 51:102 (bytes 6:14, bits 48:112, shift 3, mask 51). + v.l1 = binary.LittleEndian.Uint64(x[6:14]) >> 3 + v.l1 &= maskLow51Bits + // Bits 102:153 (bytes 12:20, bits 96:160, shift 6, mask 51). + v.l2 = binary.LittleEndian.Uint64(x[12:20]) >> 6 + v.l2 &= maskLow51Bits + // Bits 153:204 (bytes 19:27, bits 152:216, shift 1, mask 51). + v.l3 = binary.LittleEndian.Uint64(x[19:27]) >> 1 + v.l3 &= maskLow51Bits + // Bits 204:255 (bytes 24:32, bits 192:256, shift 12, mask 51). + // Note: not bytes 25:33, shift 4, to avoid overread. + v.l4 = binary.LittleEndian.Uint64(x[24:32]) >> 12 + v.l4 &= maskLow51Bits + + return v, nil +} + +// Bytes returns the canonical 32-byte little-endian encoding of v. +func (v *Element) Bytes() []byte { + // This function is outlined to make the allocations inline in the caller + // rather than happen on the heap. + var out [32]byte + return v.bytes(&out) +} + +func (v *Element) bytes(out *[32]byte) []byte { + t := *v + t.reduce() + + var buf [8]byte + for i, l := range [5]uint64{t.l0, t.l1, t.l2, t.l3, t.l4} { + bitsOffset := i * 51 + binary.LittleEndian.PutUint64(buf[:], l<<uint(bitsOffset%8)) + for i, bb := range buf { + off := bitsOffset/8 + i + if off >= len(out) { + break + } + out[off] |= bb + } + } + + return out[:] +} + +// Equal returns 1 if v and u are equal, and 0 otherwise. +func (v *Element) Equal(u *Element) int { + sa, sv := u.Bytes(), v.Bytes() + return subtle.ConstantTimeCompare(sa, sv) +} + +// mask64Bits returns 0xffffffff if cond is 1, and 0 otherwise. +func mask64Bits(cond int) uint64 { return ^(uint64(cond) - 1) } + +// Select sets v to a if cond == 1, and to b if cond == 0. +func (v *Element) Select(a, b *Element, cond int) *Element { + m := mask64Bits(cond) + v.l0 = (m & a.l0) | (^m & b.l0) + v.l1 = (m & a.l1) | (^m & b.l1) + v.l2 = (m & a.l2) | (^m & b.l2) + v.l3 = (m & a.l3) | (^m & b.l3) + v.l4 = (m & a.l4) | (^m & b.l4) + return v +} + +// Swap swaps v and u if cond == 1 or leaves them unchanged if cond == 0, and returns v. +func (v *Element) Swap(u *Element, cond int) { + m := mask64Bits(cond) + t := m & (v.l0 ^ u.l0) + v.l0 ^= t + u.l0 ^= t + t = m & (v.l1 ^ u.l1) + v.l1 ^= t + u.l1 ^= t + t = m & (v.l2 ^ u.l2) + v.l2 ^= t + u.l2 ^= t + t = m & (v.l3 ^ u.l3) + v.l3 ^= t + u.l3 ^= t + t = m & (v.l4 ^ u.l4) + v.l4 ^= t + u.l4 ^= t +} + +// IsNegative returns 1 if v is negative, and 0 otherwise. +func (v *Element) IsNegative() int { + return int(v.Bytes()[0] & 1) +} + +// Absolute sets v to |u|, and returns v. +func (v *Element) Absolute(u *Element) *Element { + return v.Select(new(Element).Negate(u), u, u.IsNegative()) +} + +// Multiply sets v = x * y, and returns v. +func (v *Element) Multiply(x, y *Element) *Element { + feMul(v, x, y) + return v +} + +// Square sets v = x * x, and returns v. +func (v *Element) Square(x *Element) *Element { + feSquare(v, x) + return v +} + +// Mult32 sets v = x * y, and returns v. +func (v *Element) Mult32(x *Element, y uint32) *Element { + x0lo, x0hi := mul51(x.l0, y) + x1lo, x1hi := mul51(x.l1, y) + x2lo, x2hi := mul51(x.l2, y) + x3lo, x3hi := mul51(x.l3, y) + x4lo, x4hi := mul51(x.l4, y) + v.l0 = x0lo + 19*x4hi // carried over per the reduction identity + v.l1 = x1lo + x0hi + v.l2 = x2lo + x1hi + v.l3 = x3lo + x2hi + v.l4 = x4lo + x3hi + // The hi portions are going to be only 32 bits, plus any previous excess, + // so we can skip the carry propagation. + return v +} + +// mul51 returns lo + hi * 2⁵¹ = a * b. +func mul51(a uint64, b uint32) (lo uint64, hi uint64) { + mh, ml := bits.Mul64(a, uint64(b)) + lo = ml & maskLow51Bits + hi = (mh << 13) | (ml >> 51) + return +} + +// Pow22523 set v = x^((p-5)/8), and returns v. (p-5)/8 is 2^252-3. +func (v *Element) Pow22523(x *Element) *Element { + var t0, t1, t2 Element + + t0.Square(x) // x^2 + t1.Square(&t0) // x^4 + t1.Square(&t1) // x^8 + t1.Multiply(x, &t1) // x^9 + t0.Multiply(&t0, &t1) // x^11 + t0.Square(&t0) // x^22 + t0.Multiply(&t1, &t0) // x^31 + t1.Square(&t0) // x^62 + for i := 1; i < 5; i++ { // x^992 + t1.Square(&t1) + } + t0.Multiply(&t1, &t0) // x^1023 -> 1023 = 2^10 - 1 + t1.Square(&t0) // 2^11 - 2 + for i := 1; i < 10; i++ { // 2^20 - 2^10 + t1.Square(&t1) + } + t1.Multiply(&t1, &t0) // 2^20 - 1 + t2.Square(&t1) // 2^21 - 2 + for i := 1; i < 20; i++ { // 2^40 - 2^20 + t2.Square(&t2) + } + t1.Multiply(&t2, &t1) // 2^40 - 1 + t1.Square(&t1) // 2^41 - 2 + for i := 1; i < 10; i++ { // 2^50 - 2^10 + t1.Square(&t1) + } + t0.Multiply(&t1, &t0) // 2^50 - 1 + t1.Square(&t0) // 2^51 - 2 + for i := 1; i < 50; i++ { // 2^100 - 2^50 + t1.Square(&t1) + } + t1.Multiply(&t1, &t0) // 2^100 - 1 + t2.Square(&t1) // 2^101 - 2 + for i := 1; i < 100; i++ { // 2^200 - 2^100 + t2.Square(&t2) + } + t1.Multiply(&t2, &t1) // 2^200 - 1 + t1.Square(&t1) // 2^201 - 2 + for i := 1; i < 50; i++ { // 2^250 - 2^50 + t1.Square(&t1) + } + t0.Multiply(&t1, &t0) // 2^250 - 1 + t0.Square(&t0) // 2^251 - 2 + t0.Square(&t0) // 2^252 - 4 + return v.Multiply(&t0, x) // 2^252 - 3 -> x^(2^252-3) +} + +// sqrtM1 is 2^((p-1)/4), which squared is equal to -1 by Euler's Criterion. +var sqrtM1 = &Element{1718705420411056, 234908883556509, + 2233514472574048, 2117202627021982, 765476049583133} + +// SqrtRatio sets r to the non-negative square root of the ratio of u and v. +// +// If u/v is square, SqrtRatio returns r and 1. If u/v is not square, SqrtRatio +// sets r according to Section 4.3 of draft-irtf-cfrg-ristretto255-decaf448-00, +// and returns r and 0. +func (r *Element) SqrtRatio(u, v *Element) (R *Element, wasSquare int) { + t0 := new(Element) + + // r = (u * v3) * (u * v7)^((p-5)/8) + v2 := new(Element).Square(v) + uv3 := new(Element).Multiply(u, t0.Multiply(v2, v)) + uv7 := new(Element).Multiply(uv3, t0.Square(v2)) + rr := new(Element).Multiply(uv3, t0.Pow22523(uv7)) + + check := new(Element).Multiply(v, t0.Square(rr)) // check = v * r^2 + + uNeg := new(Element).Negate(u) + correctSignSqrt := check.Equal(u) + flippedSignSqrt := check.Equal(uNeg) + flippedSignSqrtI := check.Equal(t0.Multiply(uNeg, sqrtM1)) + + rPrime := new(Element).Multiply(rr, sqrtM1) // r_prime = SQRT_M1 * r + // r = CT_SELECT(r_prime IF flipped_sign_sqrt | flipped_sign_sqrt_i ELSE r) + rr.Select(rPrime, rr, flippedSignSqrt|flippedSignSqrtI) + + r.Absolute(rr) // Choose the nonnegative square root. + return r, correctSignSqrt | flippedSignSqrt +} |
