d891a9c591
Provide default implementations in gf127 package and all optimizations in subpackages. This way it will be easier to use from a client.
133 lines
2.4 KiB
Go
133 lines
2.4 KiB
Go
// Copyright 2019 (c) NSPCC
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//
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// Package gf127 implements the GF(2^127) arithmetic
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// modulo reduction polynomial x^127 + x^63 + 1 .
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// Implementation is in pure Go.
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package gf127
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import (
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"math/bits"
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)
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var (
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// x126x631 is reduction polynomial x^127+x^63+1
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x127x631 = GF127{msb64 + 1, msb64}
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)
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// New constructs new element of GF(2^127) as hi*x^64 + lo.
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// It is assumed that hi has zero MSB.
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func New(lo, hi uint64) *GF127 {
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return &GF127{lo, hi}
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}
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// Inv sets b to a^-1
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// Algorithm is based on Extended Euclidean Algorithm
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// and is described by Hankerson, Hernandez, Menezes in
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// https://link.springer.com/content/pdf/10.1007/3-540-44499-8_1.pdf
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func Inv(a, b *GF127) {
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var (
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v = x127x631
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u = *a
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c, d = &GF127{1, 0}, &GF127{0, 0}
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t = new(GF127)
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x *GF127
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)
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// degree of polynomial is a position of most significant bit
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for du, dv := msb(&u), msb(&v); du != 0; du, dv = msb(&u), msb(&v) {
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if du < dv {
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v, u = u, v
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dv, du = du, dv
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d, c = c, d
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}
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x = xN(du - dv)
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Mul(x, &v, t)
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Add(&u, t, &u)
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// becasuse mul performs reduction on t, we need
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// manually reduce u at first step
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if msb(&u) == 127 {
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Add(&u, &x127x631, &u)
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}
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Mul(x, d, t)
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Add(c, t, c)
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}
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*b = *c
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}
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func xN(n int) *GF127 {
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if n < 64 {
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return &GF127{1 << uint(n), 0}
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}
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return &GF127{0, 1 << uint(n-64)}
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}
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func msb(a *GF127) (x int) {
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x = bits.LeadingZeros64(a[1])
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if x == 64 {
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x = bits.LeadingZeros64(a[0]) + 64
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}
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return 127 - x
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}
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// Mul1 copies a to b.
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func Mul1(a, b *GF127) {
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b[0] = a[0]
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b[1] = a[1]
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}
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// And sets c to a & b (bitwise-and).
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func And(a, b, c *GF127) {
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c[0] = a[0] & b[0]
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c[1] = a[1] & b[1]
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}
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// Add sets c to a+b.
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func Add(a, b, c *GF127) {
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c[0] = a[0] ^ b[0]
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c[1] = a[1] ^ b[1]
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}
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// Mul sets c to a*b.
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func Mul(a, b, c *GF127) {
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r := new(GF127)
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d := *a
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for i := uint(0); i < 64; i++ {
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if b[0]&(1<<i) != 0 {
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Add(r, &d, r)
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}
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Mul10(&d, &d)
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}
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for i := uint(0); i < 63; i++ {
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if b[1]&(1<<i) != 0 {
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Add(r, &d, r)
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}
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Mul10(&d, &d)
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}
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*c = *r
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}
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// Mul10 sets b to a*x.
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func Mul10(a, b *GF127) {
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c := (a[0] & msb64) >> 63
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b[0] = a[0] << 1
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b[1] = (a[1] << 1) ^ c
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if b[1]&msb64 != 0 {
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b[0] ^= x127x631[0]
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b[1] ^= x127x631[1]
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}
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}
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// Mul11 sets b to a*(x+1).
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func Mul11(a, b *GF127) {
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c := (a[0] & msb64) >> 63
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b[0] = a[0] ^ (a[0] << 1)
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b[1] = a[1] ^ (a[1] << 1) ^ c
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if b[1]&msb64 != 0 {
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b[0] ^= x127x631[0]
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b[1] ^= x127x631[1]
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}
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}
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