Current File : //usr/local/go119/src/crypto/internal/edwards25519/field/fe.go
// 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
}
Mini Shell