Add pure-go GF(2^127) implementation

gogf127/gf127_test.go

@@ -0,0 +1,110 @@
+package gogf127
+
+import (
+	"testing"
+
+	"github.com/stretchr/testify/require"
+)
+
+const maxUint64 = ^uint64(0)
+
+func TestAdd(t *testing.T) {
+	var (
+		a = Random()
+		b = Random()
+		e = &GF127{a[0] ^ b[0], a[1] ^ b[1]}
+		c = new(GF127)
+	)
+	Add(a, b, c)
+	require.Equal(t, e, c)
+}
+
+var testCasesMul = [][3]*GF127{
+	// (x+1)*(x^63+x^62+...+1) == x^64+1
+	{&GF127{3, 0}, &GF127{maxUint64, 0}, &GF127{1, 1}},
+
+	// x^126 * x^2 == x^128 == x^64 + x
+	{&GF127{0, 1 << 62}, &GF127{4, 0}, &GF127{2, 1}},
+
+	// (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
+	{&GF127{1 + 1<<63, 1}, &GF127{2, 1}, &GF127{0x8000000000000001, 3}},
+}
+
+func TestMul(t *testing.T) {
+	c := new(GF127)
+	for _, tc := range testCasesMul {
+		Mul(tc[0], tc[1], c)
+		require.Equal(t, tc[2], c)
+	}
+}
+
+var testCasesMul10 = [][2]*GF127{
+	{&GF127{123, 0}, &GF127{246, 0}},
+	{&GF127{maxUint64, 2}, &GF127{maxUint64 - 1, 5}},
+	{&GF127{0, maxUint64 >> 1}, &GF127{1 + 1<<63, maxUint64>>1 - 1}},
+}
+
+func TestMul10(t *testing.T) {
+	c := new(GF127)
+	for _, tc := range testCasesMul10 {
+		Mul10(tc[0], c)
+		require.Equal(t, tc[1], c)
+	}
+}
+
+var testCasesMul11 = [][2]*GF127{
+	{&GF127{123, 0}, &GF127{141, 0}},
+	{&GF127{maxUint64, 2}, &GF127{1, 7}},
+	{&GF127{0, maxUint64 >> 1}, &GF127{1 + 1<<63, 1}},
+}
+
+func TestMul11(t *testing.T) {
+	c := new(GF127)
+	for _, tc := range testCasesMul11 {
+		Mul11(tc[0], c)
+		require.Equal(t, tc[1], c)
+	}
+}
+
+var testCasesInv = [][2]*GF127{
+	{&GF127{1, 0}, &GF127{1, 0}},
+	{&GF127{3, 0}, &GF127{msb64, ^msb64}},
+	{&GF127{54321, 12345}, &GF127{8230555108620784737, 3929873967650665114}},
+}
+
+func TestInv(t *testing.T) {
+	var a, b, c = new(GF127), new(GF127), new(GF127)
+	for _, tc := range testCasesInv {
+		Inv(tc[0], c)
+		require.Equal(t, tc[1], c)
+	}
+
+	for i := 0; i < 3; i++ {
+		// 0 has no inverse
+		if a = Random(); a.Equals(&GF127{0, 0}) {
+			continue
+		}
+		Inv(a, b)
+		Mul(a, b, c)
+		require.Equal(t, &GF127{1, 0}, c)
+	}
+}
+
+func TestGF127_MarshalBinary(t *testing.T) {
+	a := New(0xFF, 0xEE)
+	data, err := a.MarshalBinary()
+	require.NoError(t, err)
+	require.Equal(t, data, []byte{0, 0, 0, 0, 0, 0, 0, 0xEE, 0, 0, 0, 0, 0, 0, 0, 0xFF})
+
+	a = Random()
+	data, err = a.MarshalBinary()
+	require.NoError(t, err)
+
+	b := new(GF127)
+	err = b.UnmarshalBinary(data)
+	require.NoError(t, err)
+	require.Equal(t, a, b)
+
+	err = b.UnmarshalBinary([]byte{0, 1, 2, 3})
+	require.Error(t, err)
+}

gogf127/gogf127.go

@@ -0,0 +1,211 @@
+// Copyright 2019 (c) NSPCC
+//
+// Package gf127 implements the GF(2^127) arithmetic
+// modulo reduction polynomial x^127 + x^63 + 1 .
+// Implementation is in pure Go.
+package gogf127
+
+import (
+	"encoding/binary"
+	"encoding/hex"
+	"errors"
+	"math/bits"
+	"math/rand"
+)
+
+// GF127 represents element of GF(2^127)
+type GF127 [2]uint64
+
+const (
+	msb64    = uint64(0x8000000000000000)
+	byteSize = 16
+)
+
+var (
+	// x126x631 is reduction polynomial x^127+x^63+1
+	x127x631 = GF127{msb64 + 1, msb64}
+)
+
+// New constructs new element of GF(2^127) as hi*x^64 + lo.
+// It is assumed that hi has zero MSB.
+func New(lo, hi uint64) *GF127 {
+	return &GF127{lo, hi}
+}
+
+// Random returns random element from GF(2^127).
+// Is used mostly for testing.
+func Random() *GF127 {
+	return &GF127{rand.Uint64(), rand.Uint64() >> 1}
+}
+
+// String returns hex-encoded representation, starting with MSB.
+func (c *GF127) String() string {
+	return hex.EncodeToString(c.ByteArray())
+}
+
+// Equals checks if two reduced (zero MSB) elements of GF(2^127) are equal
+func (c *GF127) Equals(b *GF127) bool {
+	return c[0] == b[0] && c[1] == b[1]
+}
+
+// ByteArray represents element of GF(2^127) as byte array of length 16.
+func (c *GF127) ByteArray() (buf []byte) {
+	buf = make([]byte, 16)
+	binary.BigEndian.PutUint64(buf[:8], c[1])
+	binary.BigEndian.PutUint64(buf[8:], c[0])
+	return
+}
+
+// MarshalBinary implements encoding.BinaryMarshaler.
+func (c *GF127) MarshalBinary() (data []byte, err error) {
+	return c.ByteArray(), nil
+}
+
+// UnmarshalBinary implements encoding.BinaryUnmarshaler.
+func (c *GF127) UnmarshalBinary(data []byte) error {
+	if len(data) != byteSize {
+		return errors.New("data must be 16-bytes long")
+	}
+
+	c[0] = binary.BigEndian.Uint64(data[8:])
+	c[1] = binary.BigEndian.Uint64(data[:8])
+	if c[1]&msb64 != 0 {
+		return errors.New("MSB must be zero")
+	}
+
+	return nil
+}
+
+// Inv sets b to a^-1
+// Algorithm is based on Extended Euclidean Algorithm
+// and is described by Hankerson, Hernandez, Menezes in
+// https://link.springer.com/content/pdf/10.1007/3-540-44499-8_1.pdf
+func Inv(a, b *GF127) {
+	var (
+		v    = x127x631
+		u    = *a
+		c, d = &GF127{1, 0}, &GF127{0, 0}
+		t    = new(GF127)
+		x    *GF127
+	)
+
+	// degree of polynomial is a position of most significant bit
+	for du, dv := msb(&u), msb(&v); du != 0; du, dv = msb(&u), msb(&v) {
+		if du < dv {
+			v, u = u, v
+			dv, du = du, dv
+			d, c = c, d
+		}
+
+		x = xN(du - dv)
+
+		Mul(x, &v, t)
+		Add(&u, t, &u)
+
+		// becasuse mul performs reduction on t, we need
+		// manually reduce u at first step
+		if msb(&u) == 127 {
+			Add(&u, &x127x631, &u)
+		}
+
+		Mul(x, d, t)
+		Add(c, t, c)
+	}
+	*b = *c
+}
+
+func xN(n int) *GF127 {
+	if n < 64 {
+		return &GF127{1 << uint(n), 0}
+	}
+	return &GF127{0, 1 << uint(n-64)}
+}
+
+func msb(a *GF127) (x int) {
+	x = bits.LeadingZeros64(a[1])
+	if x == 64 {
+		x = bits.LeadingZeros64(a[0]) + 64
+	}
+	return 127 - x
+}
+
+// Mul sets c to the product a*b and returns c.
+func (c *GF127) Mul(a, b *GF127) *GF127 {
+	Mul(a, b, c)
+	return c
+}
+
+// Add sets c to the sum a+b and returns c.
+func (c *GF127) Add(a, b *GF127) *GF127 {
+	Add(a, b, c)
+	return c
+}
+
+// Mul1 copies a to b.
+func Mul1(a, b *GF127) {
+	b[0] = a[0]
+	b[1] = a[1]
+}
+
+// And sets c to a & b (bitwise-and).
+func And(a, b, c *GF127) {
+	c[0] = a[0] & b[0]
+	c[1] = a[1] & b[1]
+}
+
+// Add sets c to a+b.
+func Add(a, b, c *GF127) {
+	c[0] = a[0] ^ b[0]
+	c[1] = a[1] ^ b[1]
+}
+
+// Mul sets c to a*b.
+// TODO optimization: no need to perform shift by i every time, cache results
+func Mul(a, b, c *GF127) {
+	c[0] = 0
+	c[1] = 0
+	d := new(GF127)
+	for i := uint(0); i < 64; i++ {
+		if b[0]&(1<<i) != 0 {
+			shl(i, a, d)
+			Add(c, d, c)
+		}
+	}
+	for i := uint(0); i < 63; i++ {
+		if b[1]&(1<<i) != 0 {
+			shl(i+64, a, d)
+			Add(c, d, c)
+		}
+	}
+}
+
+// shl performs left shift by consecutive multiplications by 2.
+func shl(count uint, a, b *GF127) {
+	b[0] = a[0]
+	b[1] = a[1]
+	for i := uint(0); i < count; i++ {
+		Mul10(b, b)
+	}
+}
+
+// Mul10 sets b to a*x.
+func Mul10(a, b *GF127) {
+	c := (a[0] & msb64) >> 63
+	b[0] = a[0] << 1
+	b[1] = (a[1] << 1) ^ c
+	if b[1]&msb64 != 0 {
+		b[0] ^= x127x631[0]
+		b[1] ^= x127x631[1]
+	}
+}
+
+// Mul11 sets b to a*(x+1).
+func Mul11(a, b *GF127) {
+	c := (a[0] & msb64) >> 63
+	b[0] = a[0] ^ (a[0] << 1)
+	b[1] = a[1] ^ (a[1] << 1) ^ c
+	if b[1]&msb64 != 0 {
+		b[0] ^= x127x631[0]
+		b[1] ^= x127x631[1]
+	}
+}