tzhash/gogf127/gogf127.go

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2019-07-19 08:59:10 +00:00
// 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 make it work in-place without allocations
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// TODO optimization: no need to perform shift by i every time, cache results
func Mul(a, b, c *GF127) {
r := new(GF127)
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d := new(GF127)
for i := uint(0); i < 64; i++ {
if b[0]&(1<<i) != 0 {
shl(i, a, d)
Add(r, d, r)
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}
}
for i := uint(0); i < 63; i++ {
if b[1]&(1<<i) != 0 {
shl(i+64, a, d)
Add(r, d, r)
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}
}
*c = *r
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}
// 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]
}
}