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-rw-r--r--keyexchange/isakmpd-20041012/math_2n.c1107
1 files changed, 1107 insertions, 0 deletions
diff --git a/keyexchange/isakmpd-20041012/math_2n.c b/keyexchange/isakmpd-20041012/math_2n.c
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+++ b/keyexchange/isakmpd-20041012/math_2n.c
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+/* $OpenBSD: math_2n.c,v 1.16 2004/06/14 09:55:41 ho 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
+ * irreduciable 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 "sysdep.h"
+
+#include "math_2n.h"
+#include "util.h"
+
+static u_int8_t hex2int(char);
+
+static char int2hex[] = "0123456789abcdef";
+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)
+{
+ if (n->limp)
+ 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)
+ memset(n->limp + old, 0, 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;
+ memset(n->limp, 0, 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;
+}
+
+/* Output function, mainly for debugging purposes. */
+void
+b2n_print(b2n_ptr n)
+{
+ int i, j, w, flag = 0;
+ int left;
+ char buffer[2 * CHUNK_BYTES];
+ CHUNK_TYPE tmp;
+
+ left = ((((7 + b2n_sigbit(n)) >> 3) - 1) % CHUNK_BYTES) + 1;
+ printf("0x");
+ for (i = 0; i < n->chunks; i++) {
+ tmp = n->limp[n->chunks - 1 - i];
+ memset(buffer, '0', sizeof(buffer));
+ for (w = 0, j = (i == 0 ? left : CHUNK_BYTES); j > 0; j--) {
+ buffer[w++] = int2hex[(tmp >> 4) & 0xf];
+ buffer[w++] = int2hex[tmp & 0xf];
+ tmp >>= 8;
+ }
+
+ for (j = (i == 0 ? left - 1 : CHUNK_BYTES - 1); j >= 0; j--)
+ if (flag || (i == n->chunks - 1 && j == 0) ||
+ buffer[2 * j] != '0' || buffer[2 * j + 1] != '0') {
+ putchar(buffer[2 * j]);
+ putchar(buffer[2 * j + 1]);
+ flag = 1;
+ }
+ }
+ printf("\n");
+}
+
+int
+b2n_snprint(char *buf, size_t sz, b2n_ptr n)
+{
+ int i, j, w, flag = 0;
+ size_t k;
+ int left;
+ char buffer[2 * CHUNK_BYTES];
+ CHUNK_TYPE tmp;
+
+ left = ((((7 + b2n_sigbit(n)) >> 3) - 1) % CHUNK_BYTES) + 1;
+
+ k = strlcpy(buf, "0x", sz);
+ for (i = 0; i < n->chunks && k < sz - 1; i++) {
+ tmp = n->limp[n->chunks - 1 - i];
+ memset(buffer, '0', sizeof(buffer));
+ for (w = 0, j = (i == 0 ? left : CHUNK_BYTES); j > 0; j--) {
+ buffer[w++] = int2hex[(tmp >> 4) & 0xf];
+ buffer[w++] = int2hex[tmp & 0xf];
+ tmp >>= 8;
+ }
+
+ for (j = (i == 0 ? left - 1 : CHUNK_BYTES - 1); j >= 0
+ && k < sz - 3; j--)
+ if (flag || (i == n->chunks - 1 && j == 0) ||
+ buffer[2 * j] != '0' || buffer[2 * j + 1] != '0') {
+ buf[k++] = buffer[2 * j];
+ buf[k++] = buffer[2 * j + 1];
+ flag = 1;
+ }
+ }
+
+ buf[k++] = 0;
+ return k;
+}
+
+/* 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)
+ memset(d->limp, 0, 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_q(b2n_ptr d, b2n_ptr n, b2n_ptr m)
+{
+ b2n_t r;
+ int rv;
+
+ b2n_init(r);
+ rv = b2n_div(d, r, n, m);
+ b2n_clear(r);
+ return rv;
+}
+
+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;
+ memset(q->limp, 0, 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_gcd(b2n_ptr e, b2n_ptr go, b2n_ptr ho)
+{
+ b2n_t g, h;
+
+ b2n_init(g);
+ b2n_init(h);
+ if (b2n_set(g, go))
+ goto fail;
+ if (b2n_set(h, ho))
+ goto fail;
+
+ while (b2n_cmp_null(h)) {
+ if (b2n_mod(g, g, h))
+ goto fail;
+ B2N_SWAP(g, h);
+ }
+
+ B2N_SWAP(e, g);
+
+ b2n_clear(g);
+ b2n_clear(h);
+ return 0;
+
+fail:
+ b2n_clear(g);
+ b2n_clear(h);
+ return -1;
+}
+
+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 trace tells us if there do exist any square roots
+ * for 'a' in GF(2)[x]/p(x). The number of square roots is
+ * 2 - 2*Trace.
+ * If z is a square root, z + 1 is the other.
+ */
+int
+b2n_trace(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; i++) {
+ 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;
+}
+
+/*
+ * The halftrace yields the square root if the degree of the
+ * irreduceable 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
+ * irreduceable 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;
+}
+
+/* Exponentiation modulo a polynomial. */
+int
+b2n_exp_mod(b2n_ptr d, b2n_ptr b0, u_int32_t e, b2n_ptr p)
+{
+ b2n_t u, b;
+
+ b2n_init(u);
+ b2n_init(b);
+ if (b2n_set_ui(u, 1))
+ goto fail;
+ if (b2n_mod(b, b0, p))
+ goto fail;
+
+ while (e) {
+ if (e & 1) {
+ if (b2n_mul(u, u, b))
+ goto fail;
+ if (b2n_mod(u, u, p))
+ goto fail;
+ }
+ if (b2n_square(b, b))
+ goto fail;
+ if (b2n_mod(b, b, p))
+ goto fail;
+ e >>= 1;
+ }
+
+ B2N_SWAP(d, u);
+
+ b2n_clear(u);
+ b2n_clear(b);
+ return 0;
+
+fail:
+ b2n_clear(u);
+ b2n_clear(b);
+ 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;
+}
+
+/* Very special sub, a > b. */
+int
+b2n_nsub(b2n_ptr d0, b2n_ptr a, b2n_ptr b)
+{
+ int i, carry;
+ b2n_t d;
+
+ if (b2n_cmp(a, b) <= 0)
+ return b2n_set_null(d0);
+
+ b2n_init(d);
+ if (b2n_resize(d, a->chunks)) {
+ 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;
+}
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