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