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mbin_xor.c
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mbin_xor.c
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/*-
* Copyright (c) 2013-2018 Hans Petter Selasky
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
/*
* This file implements helper functions for XOR multiplication,
* division, exponent, logarithm and so on.
*/
#include <stdio.h>
#include <stdint.h>
#include <string.h>
#include "math_bin.h"
/*
* ========================================================================
* BASE-2 XOR
* ========================================================================
*/
#define MBIN_XOR2_MOD_64(p) ((1ULL << (p)) + 1ULL)
#define MBIN_XOR2_BASE_64(p) 3ULL
uint64_t
mbin_xor2_rol_mod_64(uint64_t val, uint8_t shift, uint8_t p)
{
val = (val << shift) | (val >> (p - shift));
val &= (1ULL << p) - 1ULL;
return (val);
}
uint64_t
mbin_xor2_ror_mod_64(uint64_t val, uint8_t shift, uint8_t p)
{
val = (val >> shift) | (val << (p - shift));
val &= (1ULL << p) - 1ULL;
return (val);
}
uint64_t
mbin_xor2_square_mod_64(uint64_t x, uint8_t p)
{
uint64_t r = 0;
uint8_t n;
uint8_t q;
for (q = n = 0; n != p; n++) {
if (x & (1ULL << n))
r ^= 1ULL << q;
q += 2;
if (q >= p)
q -= p;
}
return (r);
}
uint64_t
mbin_xor2_multi_square_mod_64(uint64_t x, uint8_t y, uint8_t p)
{
uint64_t r = 0;
uint8_t q = 0;
do {
if (x & 1)
r ^= (1ULL << q);
q += y;
if (q >= p)
q -= p;
} while ((x /= 2));
return (r);
}
uint64_t
mbin_xor2_root_mod_64(uint64_t x, uint8_t p)
{
uint64_t r = 0;
uint8_t n;
uint8_t q;
for (q = n = 0; n != p; n++) {
if (x & (1ULL << q))
r ^= 1ULL << n;
q += 2;
if (q >= p)
q -= p;
}
return (r);
}
uint64_t
mbin_xor2_exp3_mod_64(uint64_t x, uint64_t y, uint8_t p)
{
uint8_t n;
uint8_t r;
for (r = 1, n = 0; n != p; r *= 2, n++) {
if (r >= p)
r -= p;
if (y & (1ULL << n))
x = x ^ mbin_xor2_rol_mod_64(x, r, p);
}
return (x);
}
uint64_t
mbin_xor2_crc2bin_64(uint64_t z, uint8_t p)
{
uint64_t r;
uint8_t x;
uint8_t y;
r = (z & 1);
for (y = 1, x = 1; x != p; y *= 2, x++) {
if (y >= p)
y -= p;
if (z & (1ULL << y))
r |= (1ULL << x);
}
return (r);
}
uint64_t
mbin_xor2_bin2crc_64(uint64_t z, uint8_t p)
{
uint64_t r;
uint8_t x;
uint8_t y;
r = (z & 1);
for (y = 1, x = 1; x != p; y *= 2, x++) {
if (y >= p)
y -= p;
if (z & (1ULL << x))
r |= (1ULL << y);
}
return (r);
}
void
mbin_xor2_compute_inverse_table_64(struct mbin_xor2_pair_64 src,
struct mbin_xor2_pair_64 dst, uint64_t *ptr, uint8_t lmax)
{
uint64_t table[lmax][2];
uint64_t temp[2][lmax];
uint64_t m;
uint64_t u;
uint64_t v;
uint8_t x;
uint8_t y;
for (u = v = 1, x = 0; x != lmax; x++) {
temp[0][x] = u;
temp[1][x] = v;
u = mbin_xor2_mul_mod_any_64(u, src.val, src.mod);
v = mbin_xor2_mul_mod_any_64(v, dst.val, dst.mod);
}
memset(table, 0, sizeof(table));
memset(ptr, 0, sizeof(ptr[0]) * lmax);
for (x = 0; x != lmax; x++) {
for (y = 0; y != lmax; y++) {
table[y][0] |= ((temp[0][x] >> y) & 1) << x;
table[y][1] |= ((temp[1][x] >> y) & 1) << x;
}
}
for (x = 0; x != lmax; x++) {
if (table[x][0] == 0)
continue;
m = mbin_msb64(table[x][0]);
for (y = 0; y != lmax; y++) {
if (y == x)
continue;
if (table[y][0] & m) {
table[y][0] ^= table[x][0];
table[y][1] ^= table[x][1];
}
}
}
for (x = 0; x != lmax; x++) {
if (table[x][0] == 0)
continue;
y = mbin_sumbits64(mbin_msb64(table[x][0]) - 1ULL);
ptr[y] = table[x][1];
}
}
uint64_t
mbin_xor2_compute_inverse_64(uint64_t x, const uint64_t *ptr, uint8_t lmax)
{
uint64_t r;
uint8_t y;
for (r = 0, y = 0; y != lmax; y++) {
if (mbin_sumbits64(x & ptr[y]) & 1)
r ^= (1ULL << y);
}
return (r);
}
uint64_t
mbin_xor2_compute_div_64(uint64_t rem, uint64_t div,
uint64_t mod, uint64_t *pmod)
{
uint8_t lmax = mbin_sumbits64(mbin_msb64(mod) - 1ULL);
uint64_t table[lmax];
struct mbin_xor2_pair_64 src = {div, mod};
struct mbin_xor2_pair_64 dst = {2, mod};
mbin_xor2_compute_inverse_table_64(src, dst, table, lmax);
if (pmod != NULL) {
/* compute modulus */
uint64_t temp = div;
temp = mbin_xor2_exp_mod_any_64(temp, lmax, mod);
temp = mbin_xor2_compute_inverse_64(temp, table, mod) ^
(1ULL << lmax);
*pmod = temp;
}
return (mbin_xor2_compute_inverse_64(rem, table, mod));
}
uint64_t
mbin_xor2_compute_exp_64(uint64_t base, uint64_t exp, uint64_t mod)
{
uint8_t lmax = mbin_sumbits64(mbin_msb64(mod) - 1ULL);
uint8_t value[lmax];
uint8_t a;
uint8_t x;
uint8_t y;
uint64_t func[lmax];
uint64_t table[lmax];
uint64_t m;
uint64_t u;
/*
* Compute modulus for exponent
*/
for (u = 1, x = 0; x != lmax; x++) {
func[x] = u;
u = mbin_xor2_mul_mod_any_64(u, base, mod);
}
memset(table, 0, sizeof(table));
for (x = 0; x != lmax; x++) {
for (y = 0; y != lmax; y++) {
table[y] |= ((func[x] >> y) & 1ULL) << x;
}
value[x] = (u >> x) & 1;
}
for (x = 0; x != lmax; x++) {
m = mbin_lsb64(table[x]);
for (y = 0; y != lmax; y++) {
if (y == x)
continue;
if (table[y] & m) {
table[y] ^= table[x];
value[y] ^= value[x];
}
}
}
for (u = (1ULL << lmax), x = 0; x != lmax; x++) {
if (table[x] && value[x]) {
y = mbin_sumbits64(table[x] - 1ULL);
u |= 1ULL << y;
}
}
/* compute exponent */
u = mbin_xor2_exp_mod_any_64(2, exp, u);
for (m = 0, a = 0; a != lmax; a++) {
if ((u >> a) & 1)
m ^= func[a];
}
return (m);
}
uint64_t
mbin_xor2_mul_table_64(uint64_t a, uint64_t b, const uint64_t *table,
uint64_t n)
{
uint64_t x, y, r;
r = 0;
for (x = 0; x != n; x++) {
if (!((a >> x) & 1))
continue;
for (y = 0; y != n; y++) {
if (!((b >> y) & 1))
continue;
r ^= table[x + (y * n)];
}
}
return (r);
}
int
mbin_xor2_mul_table_init_64(uint64_t n, const uint64_t *input /* 2*n */ ,
uint64_t *output /* n**2 */ )
{
const uint64_t words = ((n * n) + 63) / 64;
uint64_t bitmap[n * n][words];
uint64_t value[n * n];
uint64_t x, y, u, t, m;
memset(bitmap, 0, sizeof(bitmap));
memset(value, 0, sizeof(value));
memset(output, 0, sizeof(output[0]) * n * n);
for (x = 0; x != n; x++) {
for (y = 0; y != n; y++) {
value[x + (y * n)] = input[x + y];
for (t = 0; t != n; t++) {
for (u = 0; u != n; u++) {
if ((input[x] >> t) & (input[y] >> u) & 1) {
uint64_t off = t + (u * n);
bitmap[x + (y * n)][off / 64] |= (1ULL << (off % 64));
}
}
}
}
}
for (x = 0; x != (n * n); x++) {
for (y = 0; y != words; y++) {
if (bitmap[x][y] == 0)
continue;
m = mbin_lsb64(bitmap[x][y]);
for (u = 0; u != (n * n); u++) {
if (u == x)
continue;
if (bitmap[u][y] & m) {
for (t = 0; t != words; t++)
bitmap[u][t] ^= bitmap[x][t];
value[u] ^= value[x];
}
}
break;
}
if (y == words)
return (-1);
}
for (x = 0; x != (n * n); x++) {
for (u = y = 0; y != words; y++) {
u += mbin_sumbits64(bitmap[x][y]);
}
if (u != 1) {
return (-1);
}
for (y = 0; y != words; y++) {
if (bitmap[x][y] == 0)
continue;
m = mbin_sumbits64(mbin_lsb64(bitmap[x][y]) - 1ULL) + (y * 64);
output[m] = value[x];
break;
}
}
return (0);
}
uint64_t
mbin_xor2_step_fwd_64(uint64_t x, uint64_t mask)
{
uint64_t m = (mbin_msb64(mask) << 1) - 1ULL;
return (((x << 1) | (mbin_sumbits64(x & mask) & 1)) & m);
}
/*
* Example:
* mod=0xa41825, base=2, len=2520
*
* Properties:
* f(a,b) = f(b,a)
*/
uint64_t
mbin_xor2_multiply_plain_64(uint64_t a, uint64_t b, uint64_t mod)
{
uint8_t lmax = mbin_sumbits64(mbin_msb64(mod) - 1ULL);
uint8_t x;
uint64_t r;
uint64_t s;
for (r = 0, s = 1, x = 0; x != lmax; x++) {
if ((a >> x) & 1)
r ^= s;
s = mbin_xor2_mul_mod_any_64(s, b, mod);
}
return (r);
}
/*
* Symmetric version of function above
*/
uint64_t
mbin_xor2_mul_plain_symmetric_64(uint64_t x, uint64_t y, uint64_t mod)
{
uint64_t retval = 0;
uint8_t a;
uint8_t b;
uint8_t max;
max = mbin_sumbits64(mbin_msb64(mod) - 1);
for (a = 0; a != max; a++) {
if (((x >> a) & 1) == 0)
continue;
for (b = 0; b != max; b++) {
if (((y >> b) & 1) == 0)
continue;
retval ^=
mbin_xor2_exp_mod_any_64(2, a * b, mod);
}
}
return (retval);
}
/*
* Square function, derived from function above:
*/
uint64_t
mbin_xor2_square_plain_64(uint64_t x, uint64_t mod)
{
uint64_t retval = 0;
uint8_t a;
uint8_t max;
max = mbin_sumbits64(mbin_msb64(mod) - 1);
for (a = 0; a != max; a++) {
if ((x >> a) & 1) {
retval ^=
mbin_xor2_exp_mod_any_64(2, a * a, mod);
}
}
return (retval);
}
/*
* Negate function which works with polynoms that are mirrored.
*/
uint64_t
mbin_xor2_negate_plain_64(uint64_t x, uint64_t mod)
{
uint8_t max;
#if 0
if (mbin_xor2_is_mirror_64(mod) == 0)
return (0);
#endif
/* compute the maximum power of the polyom we are using */
max = mbin_sumbits64(mbin_msb64(mod) - 1);
/* multiply bias into result before negation */
x = mbin_xor2_mul_mod_any_64(x, (1ULL << (max - 1)), mod);
/* negation value */
x = mbin_bitrev64(x << (64 - max));
return (x);
}
/*
* Polynom generator
*/
void
mbin_xor2_generate_plain_64(uint8_t size, uint64_t *poly, uint64_t *lin_mod)
{
uint64_t x = 3;
uint64_t y = 1;
uint16_t a;
uint16_t b;
for (a = 3;; a += 2) {
/* figure out next prime */
if (a != 3) {
for (b = 3; b != a; b += 2) {
if ((a % b) == 0)
break;
}
if (b != a)
continue;
}
/* see if end */
if (!size--) {
*poly = x;
*lin_mod = y;
break;
}
/* accumulate */
x = mbin_xor2_mul_64(x, (1ULL << a) - 1ULL);
y *= a;
}
}
/* Returns zero if the polynom is reciprocal */
uint64_t
mbin_xor2_is_reciprocal_plain_64(uint64_t mod)
{
uint64_t rem = 0;
uint8_t x;
for (x = 0; x != 64; x++) {
uint64_t y;
if (!(mod & (1ULL << x)))
continue;
y = mbin_xor2_exp_mod_any_64(2, x, mod);
y = mbin_xor2_negate_plain_64(y, mod);
rem ^= y;
}
return (rem);
}
uint64_t
mbin_xor2_divide_plain_64(uint64_t rem, uint64_t div,
uint64_t mod, uint8_t *psolved)
{
uint8_t lmax = mbin_sumbits64(mbin_msb64(mod) - 1ULL);
uint8_t value[lmax];
uint8_t x;
uint8_t y;
uint64_t func[lmax];
uint64_t table[lmax];
uint64_t m;
uint64_t u;
for (u = 1, x = 0; x != lmax; x++) {
func[x] = u;
u = mbin_xor2_mul_mod_any_64(u, div, mod);
}
memset(table, 0, sizeof(table));
for (x = 0; x != lmax; x++) {
for (y = 0; y != lmax; y++) {
table[y] |= ((func[x] >> y) & 1ULL) << x;
}
value[x] = (rem >> x) & 1;
}
top:
for (x = 0; x != lmax; x++) {
m = mbin_lsb64(table[x]);
for (y = 0; y != lmax; y++) {
if (y == x)
continue;
if (table[y] & m) {
table[y] ^= table[x];
value[y] ^= value[x];
}
}
}
for (x = 0; x != lmax; x++) {
m = mbin_lsb64(table[x]);
if (table[x] != m) {
for (y = 0; y != lmax; y++) {
table[y] &= ~m;
}
goto top;
}
}
if (psolved)
*psolved = 1;
/* gather solution */
for (u = 0, x = 0; x != lmax; x++) {
if (value[x]) {
if (table[x]) {
/* store solution bit */
y = mbin_sumbits64(table[x] - 1ULL);
u |= 1ULL << y;
} else {
/* equation set has no solution */
if (psolved)
*psolved = 0;
}
}
}
return (u);
}
uint8_t
mbin_xor2_factor_fast_64(uint64_t rem, uint8_t p)
{
int16_t stats[p];
uint8_t x;
uint8_t y;
memset(stats, 0, sizeof(stats));
/* collect statistics */
for (x = 0; x != p; x++) {
if (((rem >> x) & 1) == 0)
continue;
for (y = x + 1; y != p; y++) {
stats[y - x] += 2 * ((rem >> y) & 1) - 1;
}
}
/* find most hits */
for (x = y = 1; x < p; x++) {
if ((stats[x] + stats[p - x]) > (stats[y] + stats[p - y]))
y = x;
}
return (y);
}
uint64_t
mbin_xor2_log3_mod_64(uint64_t x, uint8_t p)
{
uint64_t mask;
uint64_t d2;
uint64_t z;
uint8_t pm;
uint8_t n;
uint8_t r;
uint8_t sbx;
uint8_t ntable[p];
/* Check for NUL */
if (x <= 1)
return (0);
/* setup MOD table */
pm = (p - 1) / 2;
mask = p * ((1ULL << pm) - 1ULL);
d2 = ((1ULL << pm) - 1ULL);
/* there might be unused entries */
memset(ntable, 0, sizeof(ntable));
for (r = 1, n = 0; n != p; r *= 2, n++) {
if (r >= p)
r -= p;
ntable[r] = n;
}
z = 0;
/* Iterate until a solution is found */
while (1) {
sbx = mbin_sumbits64(x);
if (sbx >= (p - 2) || sbx <= 2)
break;
/* factor found - update */
r = mbin_xor2_factor_fast_64(x, p);
x = x ^ mbin_xor2_rol_mod_64(x, r, p);
z += (1ULL << ntable[r]);
}
if (sbx >= (p - 2))
x ^= (1ULL << p) - 1ULL;
sbx = mbin_sumbits64(x);
/* sanity check */
if (sbx == 0)
return (0);
/* shift down */
while (!(x & 1ULL)) {
z += d2;
x /= 2ULL;
}
if (sbx == 1)
return ((mask - (z % mask)) % mask);
if (sbx != 2)
return (0);
/* locate second bit */
r = mbin_sumbits64(x - 2ULL);
/* properly MOD the "z" variable */
z = (mask + (1ULL << ntable[r]) - (z % mask)) % mask;
return (z);
}
uint64_t
mbin_xor2_log3_mod_64_alt1(uint64_t x, uint8_t p)
{
uint64_t mask;
uint64_t d2;
uint64_t y;
uint64_t z;
uint8_t pm;
uint8_t n;
uint8_t r;
uint8_t sbx;
uint8_t sby;
uint8_t ntable[p];
uint8_t to = p;
/* Check for NUL */
if (x <= 1)
return (0);
/* setup MOD table */
pm = (p - 1) / 2;
mask = p * ((1ULL << pm) - 1ULL);
d2 = ((1ULL << pm) - 1ULL);
/* there might be unused entries */
memset(ntable, 0, sizeof(ntable));
for (r = 1, n = 0; n != p; r *= 2, n++) {
if (r >= p)
r -= p;
ntable[r] = n;
}
z = 0;
sbx = mbin_sumbits64(x);
/* Iterate until a solution is found */
while (to--) {
if (sbx == 2)
break;
for (r = 0; r != p; r++) {
if (ntable[r] == 0)
continue;
y = x ^ mbin_xor2_rol_mod_64(x, r, p);
sby = mbin_sumbits64(y);
if (sby <= sbx) {
z += (1ULL << ntable[r]);
x = y;
sbx = sby;
break;
}
}
/* invert value */
if (r == p) {
y = x ^ ((1ULL << p) - 1ULL);
sby = mbin_sumbits64(y);
x = y;
sbx = sby;
}
}
if (to == (uint8_t)-1)
return (0);
/* shift down */
while (!(x & 1ULL)) {
z += d2;
x /= 2ULL;
}
/* locate the second bit */
for (r = 1; r != p; r++) {
if (x & (1ULL << r))
break;
}
/* properly MOD the "z" variable */
if (ntable[r])
z = (mask + (1ULL << ntable[r]) - (z % mask)) % mask;
else
z = (mask - (z % mask)) % mask;
return (z);
}
uint64_t
mbin_xor2_is_div_by_3_64(uint64_t x)
{
return (mbin_sumbits64(x) & 1);
}
uint64_t
mbin_xor2_sliced_exp_64(uint64_t x, uint64_t y,
uint64_t base, uint64_t mod)
{
uint64_t r;
uint64_t bb;
uint8_t msb;
uint8_t a;
uint8_t b;
msb = mbin_sumbits64(mbin_msb64(mod) - 1ULL);
for (a = 0; a != msb; a++) {
if (y & (1ULL << a)) {
bb = base;
r = 0;
for (b = 0; b != msb; b++) {
r |= (mbin_sumbits64(x & bb) & 1) << b;
bb = mbin_xor2_mul_mod_any_64(2, bb, mod);
}
x = r;
}
base = mbin_xor2_mul_mod_any_64(base, base, mod);
}
return (x);
}
uint8_t
mbin_xor2_is_mirror_64(uint64_t x)
{
uint8_t end;
uint64_t rev;
uint64_t mask;
uint8_t found;
found = 0;
while (x > 1) {
mask = mbin_lsb64(x) - 1ULL;
end = mbin_sumbits64(mask);
x >>= end;
mask = mbin_msb64(x) - 1ULL;
end = mbin_sumbits64(mask);
rev = mbin_bitrev64(x << (63 - end));
if (rev != x)
break;
found++;
x >>= ((end + 1) / 2);
}
return (found);
}
uint64_t
mbin_xor2_reduce_64(uint64_t x, uint8_t p)
{
if (x & (1ULL << (p - 1)))
x ^= (1ULL << p) - 1ULL;
return (x);
}
uint64_t
mbin_xor2_exp_mod_64(uint64_t x, uint64_t y, uint8_t p)
{
uint64_t r = 1;
uint8_t n = 1;
do {
if (y & 1) {
uint64_t z;
z = mbin_xor2_multi_square_mod_64(x, n, p);
if (r == 1)
r = z;
else
r = mbin_xor2_mul_mod_64(r, z, p);
}
n *= 2;
if (n >= p)
n -= p;
} while ((y /= 2));
return (r);
}
uint64_t
mbin_xor2_exp_64(uint64_t x, uint64_t y)
{
uint64_t r = 1;
while (y) {
if (y & 1)
r = mbin_xor2_mul_64(r, x);
x = mbin_xor2_mul_64(x, x);
y /= 2;
}
return (r);
}
uint64_t
mbin_xor2_exp_mod_any_64(uint64_t x, uint64_t y, uint64_t p)
{
uint64_t r = 1;
while (y) {
if (y & 1)
r = mbin_xor2_mul_mod_any_64(r, x, p);
x = mbin_xor2_mul_mod_any_64(x, x, p);
y /= 2;
}
return (r);
}
uint32_t
mbin_xor2_exp_mod_any_32(uint32_t x, uint32_t y, uint32_t p)
{
uint32_t r = 1;
while (y) {
if (y & 1)
r = mbin_xor2_mul_mod_any_32(r, x, p);
x = mbin_xor2_mul_mod_any_32(x, x, p);
y /= 2;
}
return (r);
}
uint64_t
mbin_xor2_neg_mod_64(uint64_t x, uint8_t p)
{
uint8_t n = p - 2;
uint64_t r = 1;
while (n--) {
x = mbin_xor2_mul_mod_64(x, x, p);
r = mbin_xor2_mul_mod_64(r, x, p);
}
return (r);
}
uint64_t
mbin_xor2_log_mod_64(uint64_t x, uint8_t p)
{
uint64_t mask = (1ULL << p) - 1ULL;
uint64_t r = mask ^ 1ULL;
uint64_t y = 0;
/* Check for NUL */
if (x == 0)
return (0);
if (x & 1)
x = ~x & mask;
else
x = x & mask;
while (x != r) {
r = mbin_xor2_mul_mod_64(r, p, p);
if (r & 1)
r = ~r & mask;
if (y > mask)
return (0);
y++;
}
return (y);
}
uint64_t
mbin_xor2_mul_64(uint64_t x, uint64_t y)
{
uint64_t temp[4];
uint64_t r = 0;
uint8_t n;
/* optimise */
temp[0] = 0;
temp[1] = x;
temp[2] = 2 * x;
temp[3] = x ^ (2 * x);
for (n = 0; n != 64; n += 2) {
r ^= temp[y & 3] << n;