OpenBSD-4.6/sbin/isakmpd/math_2n.c
/* $OpenBSD: math_2n.c,v 1.26 2007/04/16 13:01:39 moritz Exp $ */
/* $EOM: math_2n.c,v 1.15 1999/04/20 09:23:30 niklas Exp $ */
/*
* Copyright (c) 1998 Niels Provos. All rights reserved.
* Copyright (c) 1999 Niklas Hallqvist. All rights reserved.
*
* 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 ``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 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 code was written under funding by Ericsson Radio Systems.
*/
/*
* B2N is a module for doing arithmetic on the Field GF(2**n) which is
* isomorph to ring of polynomials GF(2)[x]/p(x) where p(x) is an
* irreducible polynomial over GF(2)[x] with grade n.
*
* First we need functions which operate on GF(2)[x], operation
* on GF(2)[x]/p(x) can be done as for Z_p then.
*/
#include <stdlib.h>
#include <string.h>
#include <stdio.h>
#include "math_2n.h"
#include "util.h"
static u_int8_t hex2int(char);
CHUNK_TYPE b2n_mask[CHUNK_BITS] = {
0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, 0x80,
#if CHUNK_BITS > 8
0x0100, 0x0200, 0x0400, 0x0800, 0x1000, 0x2000, 0x4000, 0x8000,
#if CHUNK_BITS > 16
0x00010000, 0x00020000, 0x00040000, 0x00080000,
0x00100000, 0x00200000, 0x00400000, 0x00800000,
0x01000000, 0x02000000, 0x04000000, 0x08000000,
0x10000000, 0x20000000, 0x40000000, 0x80000000,
#endif
#endif
};
/* Convert a hex character to its integer value. */
static u_int8_t
hex2int(char c)
{
if (c <= '9')
return c - '0';
if (c <= 'f')
return 10 + c - 'a';
return 0;
}
int
b2n_random(b2n_ptr n, u_int32_t bits)
{
if (b2n_resize(n, (CHUNK_MASK + bits) >> CHUNK_SHIFTS))
return -1;
getrandom((u_int8_t *) n->limp, CHUNK_BYTES * n->chunks);
/* Get the number of significant bits right */
if (bits & CHUNK_MASK) {
CHUNK_TYPE m =
(((1 << ((bits & CHUNK_MASK) - 1)) - 1) << 1) | 1;
n->limp[n->chunks - 1] &= m;
}
n->dirty = 1;
return 0;
}
/* b2n management functions */
void
b2n_init(b2n_ptr n)
{
n->chunks = 0;
n->limp = 0;
}
void
b2n_clear(b2n_ptr n)
{
free(n->limp);
}
int
b2n_resize(b2n_ptr n, unsigned int chunks)
{
size_t old = n->chunks;
size_t size;
CHUNK_TYPE *new;
if (chunks == 0)
chunks = 1;
if (chunks == old)
return 0;
size = CHUNK_BYTES * chunks;
new = realloc(n->limp, size);
if (!new)
return -1;
n->limp = new;
n->chunks = chunks;
n->bits = chunks << CHUNK_SHIFTS;
n->dirty = 1;
if (chunks > old)
bzero(n->limp + old, size - CHUNK_BYTES * old);
return 0;
}
/* Simple assignment functions. */
int
b2n_set(b2n_ptr d, b2n_ptr s)
{
if (d == s)
return 0;
b2n_sigbit(s);
if (b2n_resize(d, (CHUNK_MASK + s->bits) >> CHUNK_SHIFTS))
return -1;
memcpy(d->limp, s->limp, CHUNK_BYTES * d->chunks);
d->bits = s->bits;
d->dirty = s->dirty;
return 0;
}
int
b2n_set_null(b2n_ptr n)
{
if (b2n_resize(n, 1))
return -1;
n->limp[0] = n->bits = n->dirty = 0;
return 0;
}
int
b2n_set_ui(b2n_ptr n, unsigned int val)
{
#if CHUNK_BITS < 32
int i, chunks;
chunks = (CHUNK_BYTES - 1 + sizeof(val)) / CHUNK_BYTES;
if (b2n_resize(n, chunks))
return -1;
for (i = 0; i < chunks; i++) {
n->limp[i] = val & CHUNK_BMASK;
val >>= CHUNK_BITS;
}
#else
if (b2n_resize(n, 1))
return -1;
n->limp[0] = val;
#endif
n->dirty = 1;
return 0;
}
/* XXX This one only takes hex at the moment. */
int
b2n_set_str(b2n_ptr n, char *str)
{
int i, j, w, len, chunks;
CHUNK_TYPE tmp;
if (strncasecmp(str, "0x", 2))
return -1;
/* Make the hex string even lengthed */
len = strlen(str) - 2;
if (len & 1) {
len++;
str++;
} else
str += 2;
len /= 2;
chunks = (CHUNK_BYTES - 1 + len) / CHUNK_BYTES;
if (b2n_resize(n, chunks))
return -1;
bzero(n->limp, CHUNK_BYTES * n->chunks);
for (w = 0, i = 0; i < chunks; i++) {
tmp = 0;
for (j = (i == 0 ?
((len - 1) % CHUNK_BYTES) + 1 : CHUNK_BYTES);
j > 0; j--) {
tmp <<= 8;
tmp |= (hex2int(str[w]) << 4) | hex2int(str[w + 1]);
w += 2;
}
n->limp[chunks - 1 - i] = tmp;
}
n->dirty = 1;
return 0;
}
/* Arithmetic functions. */
u_int32_t
b2n_sigbit(b2n_ptr n)
{
int i, j;
if (!n->dirty)
return n->bits;
for (i = n->chunks - 1; i > 0; i--)
if (n->limp[i])
break;
if (!n->limp[i])
return 0;
for (j = CHUNK_MASK; j > 0; j--)
if (n->limp[i] & b2n_mask[j])
break;
n->bits = (i << CHUNK_SHIFTS) + j + 1;
n->dirty = 0;
return n->bits;
}
/* Addition on GF(2)[x] is nice, its just an XOR. */
int
b2n_add(b2n_ptr d, b2n_ptr a, b2n_ptr b)
{
int i;
b2n_ptr bmin, bmax;
if (!b2n_cmp_null(a))
return b2n_set(d, b);
if (!b2n_cmp_null(b))
return b2n_set(d, a);
bmin = B2N_MIN(a, b);
bmax = B2N_MAX(a, b);
if (b2n_resize(d, bmax->chunks))
return -1;
for (i = 0; i < bmin->chunks; i++)
d->limp[i] = bmax->limp[i] ^ bmin->limp[i];
/*
* If d is not bmax, we have to copy the rest of the bytes, and also
* need to adjust to number of relevant bits.
*/
if (d != bmax) {
for (; i < bmax->chunks; i++)
d->limp[i] = bmax->limp[i];
d->bits = bmax->bits;
}
/*
* Help to converse memory. When the result of the addition is zero
* truncate the used amount of memory.
*/
if (d != bmax && !b2n_cmp_null(d))
return b2n_set_null(d);
else
d->dirty = 1;
return 0;
}
/* Compare two polynomials. */
int
b2n_cmp(b2n_ptr n, b2n_ptr m)
{
int sn, sm;
int i;
sn = b2n_sigbit(n);
sm = b2n_sigbit(m);
if (sn > sm)
return 1;
if (sn < sm)
return -1;
for (i = n->chunks - 1; i >= 0; i--)
if (n->limp[i] > m->limp[i])
return 1;
else if (n->limp[i] < m->limp[i])
return -1;
return 0;
}
int
b2n_cmp_null(b2n_ptr a)
{
int i = 0;
do {
if (a->limp[i])
return 1;
} while (++i < a->chunks);
return 0;
}
/* Left shift, needed for polynomial multiplication. */
int
b2n_lshift(b2n_ptr d, b2n_ptr n, unsigned int s)
{
int i, maj, min, chunks;
u_int16_t bits = b2n_sigbit(n), add;
CHUNK_TYPE *p, *op;
if (!s)
return b2n_set(d, n);
maj = s >> CHUNK_SHIFTS;
min = s & CHUNK_MASK;
add = (!(bits & CHUNK_MASK) ||
((bits & CHUNK_MASK) + min) > CHUNK_MASK) ? 1 : 0;
chunks = n->chunks;
if (b2n_resize(d, chunks + maj + add))
return -1;
memmove(d->limp + maj, n->limp, CHUNK_BYTES * chunks);
if (maj)
bzero(d->limp, CHUNK_BYTES * maj);
if (add)
d->limp[d->chunks - 1] = 0;
/* If !min there are no bit shifts, we are done */
if (!min)
return 0;
op = p = &d->limp[d->chunks - 1];
for (i = d->chunks - 2; i >= maj; i--) {
op--;
*p = (*p << min) | (*op >> (CHUNK_BITS - min));
p--;
}
*p <<= min;
d->dirty = 0;
d->bits = bits + (maj << CHUNK_SHIFTS) + min;
return 0;
}
/* Right shift, needed for polynomial division. */
int
b2n_rshift(b2n_ptr d, b2n_ptr n, unsigned int s)
{
int maj, min, size = n->chunks, newsize;
b2n_ptr tmp;
if (!s)
return b2n_set(d, n);
maj = s >> CHUNK_SHIFTS;
newsize = size - maj;
if (size < maj)
return b2n_set_null(d);
min = (CHUNK_BITS - (s & CHUNK_MASK)) & CHUNK_MASK;
if (min) {
if ((b2n_sigbit(n) & CHUNK_MASK) > (u_int32_t) min)
newsize++;
if (b2n_lshift(d, n, min))
return -1;
tmp = d;
} else
tmp = n;
memmove(d->limp, tmp->limp + maj + (min ? 1 : 0),
CHUNK_BYTES * newsize);
if (b2n_resize(d, newsize))
return -1;
d->bits = tmp->bits - ((maj + (min ? 1 : 0)) << CHUNK_SHIFTS);
return 0;
}
/* Normal polynomial multiplication. */
int
b2n_mul(b2n_ptr d, b2n_ptr n, b2n_ptr m)
{
int i, j;
b2n_t tmp, tmp2;
if (!b2n_cmp_null(m) || !b2n_cmp_null(n))
return b2n_set_null(d);
if (b2n_sigbit(m) == 1)
return b2n_set(d, n);
if (b2n_sigbit(n) == 1)
return b2n_set(d, m);
b2n_init(tmp);
b2n_init(tmp2);
if (b2n_set(tmp, B2N_MAX(n, m)))
goto fail;
if (b2n_set(tmp2, B2N_MIN(n, m)))
goto fail;
if (b2n_set_null(d))
goto fail;
for (i = 0; i < tmp2->chunks; i++)
if (tmp2->limp[i])
for (j = 0; j < CHUNK_BITS; j++) {
if (tmp2->limp[i] & b2n_mask[j])
if (b2n_add(d, d, tmp))
goto fail;
if (b2n_lshift(tmp, tmp, 1))
goto fail;
}
else if (b2n_lshift(tmp, tmp, CHUNK_BITS))
goto fail;
b2n_clear(tmp);
b2n_clear(tmp2);
return 0;
fail:
b2n_clear(tmp);
b2n_clear(tmp2);
return -1;
}
/*
* Squaring in this polynomial ring is more efficient than normal
* multiplication.
*/
int
b2n_square(b2n_ptr d, b2n_ptr n)
{
int i, j, maj, min, bits, chunk;
b2n_t t;
maj = b2n_sigbit(n);
min = maj & CHUNK_MASK;
maj = (maj + CHUNK_MASK) >> CHUNK_SHIFTS;
b2n_init(t);
if (b2n_resize(t,
2 * maj + ((CHUNK_MASK + 2 * min) >> CHUNK_SHIFTS))) {
b2n_clear(t);
return -1;
}
chunk = 0;
bits = 0;
for (i = 0; i < maj; i++)
if (n->limp[i])
for (j = 0; j < CHUNK_BITS; j++) {
if (n->limp[i] & b2n_mask[j])
t->limp[chunk] ^= b2n_mask[bits];
bits += 2;
if (bits >= CHUNK_BITS) {
chunk++;
bits &= CHUNK_MASK;
}
}
else
chunk += 2;
t->dirty = 1;
B2N_SWAP(d, t);
b2n_clear(t);
return 0;
}
/*
* Normal polynomial division.
* These functions are far from optimal in speed.
*/
int
b2n_div_r(b2n_ptr r, b2n_ptr n, b2n_ptr m)
{
b2n_t q;
int rv;
b2n_init(q);
rv = b2n_div(q, r, n, m);
b2n_clear(q);
return rv;
}
int
b2n_div(b2n_ptr q, b2n_ptr r, b2n_ptr n, b2n_ptr m)
{
int i, j, len, bits;
u_int32_t sm, sn;
b2n_t nenn, div, shift, mask;
/* If Teiler > Zaehler, the result is 0 */
if ((sm = b2n_sigbit(m)) > (sn = b2n_sigbit(n))) {
if (b2n_set_null(q))
return -1;
return b2n_set(r, n);
}
if (sm == 0)
/* Division by Zero */
return -1;
else if (sm == 1) {
/* Division by the One-Element */
if (b2n_set(q, n))
return -1;
return b2n_set_null(r);
}
b2n_init(nenn);
b2n_init(div);
b2n_init(shift);
b2n_init(mask);
if (b2n_set(nenn, n))
goto fail;
if (b2n_set(div, m))
goto fail;
if (b2n_set(shift, m))
goto fail;
if (b2n_set_ui(mask, 1))
goto fail;
if (b2n_resize(q, (sn - sm + CHUNK_MASK) >> CHUNK_SHIFTS))
goto fail;
bzero(q->limp, CHUNK_BYTES * q->chunks);
if (b2n_lshift(shift, shift, sn - sm))
goto fail;
if (b2n_lshift(mask, mask, sn - sm))
goto fail;
/* Number of significant octets */
len = (sn - 1) >> CHUNK_SHIFTS;
/* The first iteration is done over the relevant bits */
bits = (CHUNK_MASK + sn) & CHUNK_MASK;
for (i = len; i >= 0 && b2n_sigbit(nenn) >= sm; i--)
for (j = (i == len ? bits : CHUNK_MASK); j >= 0 &&
b2n_sigbit(nenn) >= sm; j--) {
if (nenn->limp[i] & b2n_mask[j]) {
if (b2n_sub(nenn, nenn, shift))
goto fail;
if (b2n_add(q, q, mask))
goto fail;
}
if (b2n_rshift(shift, shift, 1))
goto fail;
if (b2n_rshift(mask, mask, 1))
goto fail;
}
B2N_SWAP(r, nenn);
b2n_clear(nenn);
b2n_clear(div);
b2n_clear(shift);
b2n_clear(mask);
return 0;
fail:
b2n_clear(nenn);
b2n_clear(div);
b2n_clear(shift);
b2n_clear(mask);
return -1;
}
/* Functions for Operation on GF(2**n) ~= GF(2)[x]/p(x). */
int
b2n_mod(b2n_ptr m, b2n_ptr n, b2n_ptr p)
{
int bits, size;
if (b2n_div_r(m, n, p))
return -1;
bits = b2n_sigbit(m);
size = ((CHUNK_MASK + bits) >> CHUNK_SHIFTS);
if (size == 0)
size = 1;
if (m->chunks > size)
if (b2n_resize(m, size))
return -1;
m->bits = bits;
m->dirty = 0;
return 0;
}
int
b2n_mul_inv(b2n_ptr ga, b2n_ptr be, b2n_ptr p)
{
b2n_t a;
b2n_init(a);
if (b2n_set_ui(a, 1))
goto fail;
if (b2n_div_mod(ga, a, be, p))
goto fail;
b2n_clear(a);
return 0;
fail:
b2n_clear(a);
return -1;
}
int
b2n_div_mod(b2n_ptr ga, b2n_ptr a, b2n_ptr be, b2n_ptr p)
{
b2n_t s0, s1, s2, q, r0, r1;
/* There is no multiplicative inverse to Null. */
if (!b2n_cmp_null(be))
return b2n_set_null(ga);
b2n_init(s0);
b2n_init(s1);
b2n_init(s2);
b2n_init(r0);
b2n_init(r1);
b2n_init(q);
if (b2n_set(r0, p))
goto fail;
if (b2n_set(r1, be))
goto fail;
if (b2n_set_null(s0))
goto fail;
if (b2n_set(s1, a))
goto fail;
while (b2n_cmp_null(r1)) {
if (b2n_div(q, r0, r0, r1))
goto fail;
B2N_SWAP(r0, r1);
if (b2n_mul(s2, q, s1))
goto fail;
if (b2n_mod(s2, s2, p))
goto fail;
if (b2n_sub(s2, s0, s2))
goto fail;
B2N_SWAP(s0, s1);
B2N_SWAP(s1, s2);
}
B2N_SWAP(ga, s0);
b2n_clear(s0);
b2n_clear(s1);
b2n_clear(s2);
b2n_clear(r0);
b2n_clear(r1);
b2n_clear(q);
return 0;
fail:
b2n_clear(s0);
b2n_clear(s1);
b2n_clear(s2);
b2n_clear(r0);
b2n_clear(r1);
b2n_clear(q);
return -1;
}
/*
* The halftrace yields the square root if the degree of the
* irreducible polynomial is odd.
*/
int
b2n_halftrace(b2n_ptr ho, b2n_ptr a, b2n_ptr p)
{
int i, m = b2n_sigbit(p) - 1;
b2n_t h;
b2n_init(h);
if (b2n_set(h, a))
goto fail;
for (i = 0; i < (m - 1) / 2; i++) {
if (b2n_square(h, h))
goto fail;
if (b2n_mod(h, h, p))
goto fail;
if (b2n_square(h, h))
goto fail;
if (b2n_mod(h, h, p))
goto fail;
if (b2n_add(h, h, a))
goto fail;
}
B2N_SWAP(ho, h);
b2n_clear(h);
return 0;
fail:
b2n_clear(h);
return -1;
}
/*
* Solving the equation: y**2 + y = b in GF(2**m) where ip is the
* irreducible polynomial. If m is odd, use the half trace.
*/
int
b2n_sqrt(b2n_ptr zo, b2n_ptr b, b2n_ptr ip)
{
int i, m = b2n_sigbit(ip) - 1;
b2n_t w, p, temp, z;
if (!b2n_cmp_null(b))
return b2n_set_null(z);
if (m & 1)
return b2n_halftrace(zo, b, ip);
b2n_init(z);
b2n_init(w);
b2n_init(p);
b2n_init(temp);
do {
if (b2n_random(p, m))
goto fail;
if (b2n_set_null(z))
goto fail;
if (b2n_set(w, p))
goto fail;
for (i = 1; i < m; i++) {
if (b2n_square(z, z)) /* z**2 */
goto fail;
if (b2n_mod(z, z, ip))
goto fail;
if (b2n_square(w, w)) /* w**2 */
goto fail;
if (b2n_mod(w, w, ip))
goto fail;
if (b2n_mul(temp, w, b)) /* w**2 * b */
goto fail;
if (b2n_mod(temp, temp, ip))
goto fail;
if (b2n_add(z, z, temp)) /* z**2 + w**2 + b */
goto fail;
if (b2n_add(w, w, p)) /* w**2 + p */
goto fail;
}
} while (!b2n_cmp_null(w));
B2N_SWAP(zo, z);
b2n_clear(w);
b2n_clear(p);
b2n_clear(temp);
b2n_clear(z);
return 0;
fail:
b2n_clear(w);
b2n_clear(p);
b2n_clear(temp);
b2n_clear(z);
return -1;
}
/*
* Low-level function to speed up scalar multiplication with
* elliptic curves.
* Multiplies a normal number by 3.
*/
/* Normal addition behaves as Z_{2**n} and not F_{2**n}. */
int
b2n_nadd(b2n_ptr d0, b2n_ptr a0, b2n_ptr b0)
{
int i, carry;
b2n_ptr a, b;
b2n_t d;
if (!b2n_cmp_null(a0))
return b2n_set(d0, b0);
if (!b2n_cmp_null(b0))
return b2n_set(d0, a0);
b2n_init(d);
a = B2N_MAX(a0, b0);
b = B2N_MIN(a0, b0);
if (b2n_resize(d, a->chunks + 1)) {
b2n_clear(d);
return -1;
}
for (carry = i = 0; i < b->chunks; i++) {
d->limp[i] = a->limp[i] + b->limp[i] + carry;
carry = (d->limp[i] < a->limp[i] ? 1 : 0);
}
for (; i < a->chunks && carry; i++) {
d->limp[i] = a->limp[i] + carry;
carry = (d->limp[i] < a->limp[i] ? 1 : 0);
}
if (i < a->chunks)
memcpy(d->limp + i, a->limp + i,
CHUNK_BYTES * (a->chunks - i));
d->dirty = 1;
B2N_SWAP(d0, d);
b2n_clear(d);
return 0;
}
int
b2n_3mul(b2n_ptr d0, b2n_ptr e)
{
b2n_t d;
b2n_init(d);
if (b2n_lshift(d, e, 1))
goto fail;
if (b2n_nadd(d0, d, e))
goto fail;
b2n_clear(d);
return 0;
fail:
b2n_clear(d);
return -1;
}