Add pure-go GF(2^127) implementation
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Evgenii
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2 changed files with 321 additions and 0 deletions
110
gogf127/gf127_test.go
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110
gogf127/gf127_test.go
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package gogf127
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import (
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"testing"
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"github.com/stretchr/testify/require"
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)
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const maxUint64 = ^uint64(0)
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func TestAdd(t *testing.T) {
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var (
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a = Random()
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b = Random()
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e = &GF127{a[0] ^ b[0], a[1] ^ b[1]}
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c = new(GF127)
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)
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Add(a, b, c)
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require.Equal(t, e, c)
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}
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var testCasesMul = [][3]*GF127{
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// (x+1)*(x^63+x^62+...+1) == x^64+1
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{&GF127{3, 0}, &GF127{maxUint64, 0}, &GF127{1, 1}},
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// x^126 * x^2 == x^128 == x^64 + x
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{&GF127{0, 1 << 62}, &GF127{4, 0}, &GF127{2, 1}},
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// (x^64+x^63+1) * (x^64+x) == x^128+x^65+x^127+x^64+x^64+x == x^65+x^64+x^63+1
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{&GF127{1 + 1<<63, 1}, &GF127{2, 1}, &GF127{0x8000000000000001, 3}},
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}
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func TestMul(t *testing.T) {
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c := new(GF127)
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for _, tc := range testCasesMul {
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Mul(tc[0], tc[1], c)
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require.Equal(t, tc[2], c)
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}
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}
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var testCasesMul10 = [][2]*GF127{
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{&GF127{123, 0}, &GF127{246, 0}},
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{&GF127{maxUint64, 2}, &GF127{maxUint64 - 1, 5}},
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{&GF127{0, maxUint64 >> 1}, &GF127{1 + 1<<63, maxUint64>>1 - 1}},
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}
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func TestMul10(t *testing.T) {
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c := new(GF127)
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for _, tc := range testCasesMul10 {
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Mul10(tc[0], c)
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require.Equal(t, tc[1], c)
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}
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}
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var testCasesMul11 = [][2]*GF127{
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{&GF127{123, 0}, &GF127{141, 0}},
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{&GF127{maxUint64, 2}, &GF127{1, 7}},
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{&GF127{0, maxUint64 >> 1}, &GF127{1 + 1<<63, 1}},
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}
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func TestMul11(t *testing.T) {
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c := new(GF127)
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for _, tc := range testCasesMul11 {
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Mul11(tc[0], c)
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require.Equal(t, tc[1], c)
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}
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}
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var testCasesInv = [][2]*GF127{
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{&GF127{1, 0}, &GF127{1, 0}},
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{&GF127{3, 0}, &GF127{msb64, ^msb64}},
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{&GF127{54321, 12345}, &GF127{8230555108620784737, 3929873967650665114}},
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}
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func TestInv(t *testing.T) {
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var a, b, c = new(GF127), new(GF127), new(GF127)
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for _, tc := range testCasesInv {
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Inv(tc[0], c)
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require.Equal(t, tc[1], c)
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}
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for i := 0; i < 3; i++ {
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// 0 has no inverse
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if a = Random(); a.Equals(&GF127{0, 0}) {
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continue
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}
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Inv(a, b)
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Mul(a, b, c)
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require.Equal(t, &GF127{1, 0}, c)
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}
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}
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func TestGF127_MarshalBinary(t *testing.T) {
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a := New(0xFF, 0xEE)
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data, err := a.MarshalBinary()
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require.NoError(t, err)
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require.Equal(t, data, []byte{0, 0, 0, 0, 0, 0, 0, 0xEE, 0, 0, 0, 0, 0, 0, 0, 0xFF})
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a = Random()
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data, err = a.MarshalBinary()
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require.NoError(t, err)
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b := new(GF127)
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err = b.UnmarshalBinary(data)
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require.NoError(t, err)
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require.Equal(t, a, b)
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err = b.UnmarshalBinary([]byte{0, 1, 2, 3})
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require.Error(t, err)
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}
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211
gogf127/gogf127.go
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211
gogf127/gogf127.go
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// 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 gogf127
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import (
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"encoding/binary"
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"encoding/hex"
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"errors"
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"math/bits"
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"math/rand"
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)
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// GF127 represents element of GF(2^127)
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type GF127 [2]uint64
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const (
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msb64 = uint64(0x8000000000000000)
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byteSize = 16
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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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// Random returns random element from GF(2^127).
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// Is used mostly for testing.
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func Random() *GF127 {
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return &GF127{rand.Uint64(), rand.Uint64() >> 1}
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}
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// String returns hex-encoded representation, starting with MSB.
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func (c *GF127) String() string {
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return hex.EncodeToString(c.ByteArray())
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}
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// Equals checks if two reduced (zero MSB) elements of GF(2^127) are equal
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func (c *GF127) Equals(b *GF127) bool {
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return c[0] == b[0] && c[1] == b[1]
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}
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// ByteArray represents element of GF(2^127) as byte array of length 16.
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func (c *GF127) ByteArray() (buf []byte) {
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buf = make([]byte, 16)
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binary.BigEndian.PutUint64(buf[:8], c[1])
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binary.BigEndian.PutUint64(buf[8:], c[0])
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return
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}
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// MarshalBinary implements encoding.BinaryMarshaler.
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func (c *GF127) MarshalBinary() (data []byte, err error) {
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return c.ByteArray(), nil
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}
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// UnmarshalBinary implements encoding.BinaryUnmarshaler.
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func (c *GF127) UnmarshalBinary(data []byte) error {
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if len(data) != byteSize {
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return errors.New("data must be 16-bytes long")
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}
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c[0] = binary.BigEndian.Uint64(data[8:])
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c[1] = binary.BigEndian.Uint64(data[:8])
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if c[1]&msb64 != 0 {
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return errors.New("MSB must be zero")
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}
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return nil
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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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// Mul sets c to the product a*b and returns c.
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func (c *GF127) Mul(a, b *GF127) *GF127 {
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Mul(a, b, c)
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return c
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}
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// Add sets c to the sum a+b and returns c.
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func (c *GF127) Add(a, b *GF127) *GF127 {
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Add(a, b, c)
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return c
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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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// TODO optimization: no need to perform shift by i every time, cache results
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func Mul(a, b, c *GF127) {
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c[0] = 0
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c[1] = 0
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d := new(GF127)
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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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shl(i, a, d)
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Add(c, d, c)
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}
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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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shl(i+64, a, d)
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Add(c, d, c)
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}
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}
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}
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// shl performs left shift by consecutive multiplications by 2.
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func shl(count uint, a, b *GF127) {
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b[0] = a[0]
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b[1] = a[1]
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for i := uint(0); i < count; i++ {
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Mul10(b, b)
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}
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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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