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// Copyright 2010 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 gf256 implements arithmetic over the Galois Field GF(256).
package gf256
import "strconv"
// A Field represents an instance of GF(256) defined by a specific polynomial.
type Field struct {
log [256]byte // log[0] is unused
exp [510]byte
}
// NewField returns a new field corresponding to the polynomial poly
// and generator α. The Reed-Solomon encoding in QR codes uses
// polynomial 0x11d with generator 2.
//
// The choice of generator α only affects the Exp and Log operations.
func NewField(poly, α int) *Field {
if poly < 0x100 || poly >= 0x200 || reducible(poly) {
panic("gf256: invalid polynomial: " + strconv.Itoa(poly))
}
var f Field
x := 1
for i := 0; i < 255; i++ {
if x == 1 && i != 0 {
panic("gf256: invalid generator " + strconv.Itoa(α) +
" for polynomial " + strconv.Itoa(poly))
}
f.exp[i] = byte(x)
f.exp[i+255] = byte(x)
f.log[x] = byte(i)
x = mul(x, α, poly)
}
f.log[0] = 255
for i := 0; i < 255; i++ {
if f.log[f.exp[i]] != byte(i) {
panic("bad log")
}
if f.log[f.exp[i+255]] != byte(i) {
panic("bad log")
}
}
for i := 1; i < 256; i++ {
if f.exp[f.log[i]] != byte(i) {
panic("bad log")
}
}
return &f
}
// nbit returns the number of significant in p.
func nbit(p int) uint {
n := uint(0)
for ; p > 0; p >>= 1 {
n++
}
return n
}
// polyDiv divides the polynomial p by q and returns the remainder.
func polyDiv(p, q int) int {
np := nbit(p)
nq := nbit(q)
for ; np >= nq; np-- {
if p&(1<<(np-1)) != 0 {
p ^= q << (np - nq)
}
}
return p
}
// mul returns the product x*y mod poly, a GF(256) multiplication.
func mul(x, y, poly int) int {
z := 0
for x > 0 {
if x&1 != 0 {
z ^= y
}
x >>= 1
y <<= 1
if y&0x100 != 0 {
y ^= poly
}
}
return z
}
// reducible reports whether p is reducible.
func reducible(p int) bool {
// Multiplying n-bit * n-bit produces (2n-1)-bit,
// so if p is reducible, one of its factors must be
// of np/2+1 bits or fewer.
np := nbit(p)
for q := 2; q < 1<<(np/2+1); q++ {
if polyDiv(p, q) == 0 {
return true
}
}
return false
}
// Add returns the sum of x and y in the field.
func (f *Field) Add(x, y byte) byte {
return x ^ y
}
// Exp returns the the base-α exponential of e in the field.
// If e < 0, Exp returns 0.
func (f *Field) Exp(e int) byte {
if e < 0 {
return 0
}
return f.exp[e%255]
}
// Log returns the base-α logarithm of x in the field.
// If x == 0, Log returns -1.
func (f *Field) Log(x byte) int {
if x == 0 {
return -1
}
return int(f.log[x])
}
// Inv returns the multiplicative inverse of x in the field.
// If x == 0, Inv returns 0.
func (f *Field) Inv(x byte) byte {
if x == 0 {
return 0
}
return f.exp[255-f.log[x]]
}
// Mul returns the product of x and y in the field.
func (f *Field) Mul(x, y byte) byte {
if x == 0 || y == 0 {
return 0
}
return f.exp[int(f.log[x])+int(f.log[y])]
}
// An RSEncoder implements Reed-Solomon encoding
// over a given field using a given number of error correction bytes.
type RSEncoder struct {
f *Field
c int
gen []byte
lgen []byte
p []byte
}
func (f *Field) gen(e int) (gen, lgen []byte) {
// p = 1
p := make([]byte, e+1)
p[e] = 1
for i := 0; i < e; i++ {
// p *= (x + Exp(i))
// p[j] = p[j]*Exp(i) + p[j+1].
c := f.Exp(i)
for j := 0; j < e; j++ {
p[j] = f.Mul(p[j], c) ^ p[j+1]
}
p[e] = f.Mul(p[e], c)
}
// lp = log p.
lp := make([]byte, e+1)
for i, c := range p {
if c == 0 {
lp[i] = 255
} else {
lp[i] = byte(f.Log(c))
}
}
return p, lp
}
// NewRSEncoder returns a new Reed-Solomon encoder
// over the given field and number of error correction bytes.
func NewRSEncoder(f *Field, c int) *RSEncoder {
gen, lgen := f.gen(c)
return &RSEncoder{f: f, c: c, gen: gen, lgen: lgen}
}
// ECC writes to check the error correcting code bytes
// for data using the given Reed-Solomon parameters.
func (rs *RSEncoder) ECC(data []byte, check []byte) {
if len(check) < rs.c {
panic("gf256: invalid check byte length")
}
if rs.c == 0 {
return
}
// The check bytes are the remainder after dividing
// data padded with c zeros by the generator polynomial.
// p = data padded with c zeros.
var p []byte
n := len(data) + rs.c
if len(rs.p) >= n {
p = rs.p
} else {
p = make([]byte, n)
}
copy(p, data)
for i := len(data); i < len(p); i++ {
p[i] = 0
}
// Divide p by gen, leaving the remainder in p[len(data):].
// p[0] is the most significant term in p, and
// gen[0] is the most significant term in the generator,
// which is always 1.
// To avoid repeated work, we store various values as
// lv, not v, where lv = log[v].
f := rs.f
lgen := rs.lgen[1:]
for i := 0; i < len(data); i++ {
c := p[i]
if c == 0 {
continue
}
q := p[i+1:]
exp := f.exp[f.log[c]:]
for j, lg := range lgen {
if lg != 255 { // lgen uses 255 for log 0
q[j] ^= exp[lg]
}
}
}
copy(check, p[len(data):])
rs.p = p
}
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