1 /* $OpenBSD: schnorr.c,v 1.9 2014/01/09 23:20:00 djm Exp $ */
4 * Copyright (c) 2008 Damien Miller. All rights reserved.
6 * Permission to use, copy, modify, and distribute this software for any
7 * purpose with or without fee is hereby granted, provided that the above
8 * copyright notice and this permission notice appear in all copies.
10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
20 * Implementation of Schnorr signatures / zero-knowledge proofs, based on
23 * F. Hao, P. Ryan, "Password Authenticated Key Exchange by Juggling",
24 * 16th Workshop on Security Protocols, Cambridge, April 2008
26 * http://grouper.ieee.org/groups/1363/Research/contributions/hao-ryan-2008.pdf
31 #include <sys/types.h>
37 #include <openssl/evp.h>
38 #include <openssl/bn.h>
49 #include "openbsd-compat/openssl-compat.h"
51 /* #define SCHNORR_DEBUG */ /* Privacy-violating debugging */
52 /* #define SCHNORR_MAIN */ /* Include main() selftest */
55 # define SCHNORR_DEBUG_BN(a)
56 # define SCHNORR_DEBUG_BUF(a)
58 # define SCHNORR_DEBUG_BN(a) debug3_bn a
59 # define SCHNORR_DEBUG_BUF(a) debug3_buf a
60 #endif /* SCHNORR_DEBUG */
63 * Calculate hash component of Schnorr signature H(g || g^v || g^x || id)
64 * using the hash function defined by "hash_alg". Returns signature as
65 * bignum or NULL on error.
68 schnorr_hash(const BIGNUM *p, const BIGNUM *q, const BIGNUM *g,
69 int hash_alg, const BIGNUM *g_v, const BIGNUM *g_x,
70 const u_char *id, u_int idlen)
78 if ((h = BN_new()) == NULL) {
79 error("%s: BN_new", __func__);
85 /* h = H(g || p || q || g^v || g^x || id) */
86 buffer_put_bignum2(&b, g);
87 buffer_put_bignum2(&b, p);
88 buffer_put_bignum2(&b, q);
89 buffer_put_bignum2(&b, g_v);
90 buffer_put_bignum2(&b, g_x);
91 buffer_put_string(&b, id, idlen);
93 SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
94 "%s: hashblob", __func__));
95 if (hash_buffer(buffer_ptr(&b), buffer_len(&b), hash_alg,
96 &digest, &digest_len) != 0) {
97 error("%s: hash_buffer", __func__);
100 if (BN_bin2bn(digest, (int)digest_len, h) == NULL) {
101 error("%s: BN_bin2bn", __func__);
105 SCHNORR_DEBUG_BN((h, "%s: h = ", __func__));
108 bzero(digest, digest_len);
118 * Generate Schnorr signature to prove knowledge of private value 'x' used
119 * in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
120 * using the hash function "hash_alg".
121 * 'idlen' bytes from 'id' will be included in the signature hash as an anti-
124 * On success, 0 is returned. The signature values are returned as *e_p
125 * (g^v mod p) and *r_p (v - xh mod q). The caller must free these values.
126 * On failure, -1 is returned.
129 schnorr_sign(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
130 int hash_alg, const BIGNUM *x, const BIGNUM *g_x,
131 const u_char *id, u_int idlen, BIGNUM **r_p, BIGNUM **e_p)
134 BIGNUM *h, *tmp, *v, *g_v, *r;
137 SCHNORR_DEBUG_BN((x, "%s: x = ", __func__));
138 SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));
140 /* Avoid degenerate cases: g^0 yields a spoofable signature */
141 if (BN_cmp(g_x, BN_value_one()) <= 0) {
142 error("%s: g_x < 1", __func__);
145 if (BN_cmp(g_x, grp_p) >= 0) {
146 error("%s: g_x > g", __func__);
150 h = g_v = r = tmp = v = NULL;
151 if ((bn_ctx = BN_CTX_new()) == NULL) {
152 error("%s: BN_CTX_new", __func__);
155 if ((g_v = BN_new()) == NULL ||
156 (r = BN_new()) == NULL ||
157 (tmp = BN_new()) == NULL) {
158 error("%s: BN_new", __func__);
163 * v must be a random element of Zq, so 1 <= v < q
164 * we also exclude v = 1, since g^1 looks dangerous
166 if ((v = bn_rand_range_gt_one(grp_p)) == NULL) {
167 error("%s: bn_rand_range2", __func__);
170 SCHNORR_DEBUG_BN((v, "%s: v = ", __func__));
172 /* g_v = g^v mod p */
173 if (BN_mod_exp(g_v, grp_g, v, grp_p, bn_ctx) == -1) {
174 error("%s: BN_mod_exp (g^v mod p)", __func__);
177 SCHNORR_DEBUG_BN((g_v, "%s: g_v = ", __func__));
179 /* h = H(g || g^v || g^x || id) */
180 if ((h = schnorr_hash(grp_p, grp_q, grp_g, hash_alg, g_v, g_x,
181 id, idlen)) == NULL) {
182 error("%s: schnorr_hash failed", __func__);
186 /* r = v - xh mod q */
187 if (BN_mod_mul(tmp, x, h, grp_q, bn_ctx) == -1) {
188 error("%s: BN_mod_mul (tmp = xv mod q)", __func__);
191 if (BN_mod_sub(r, v, tmp, grp_q, bn_ctx) == -1) {
192 error("%s: BN_mod_mul (r = v - tmp)", __func__);
195 SCHNORR_DEBUG_BN((g_v, "%s: e = ", __func__));
196 SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));
214 * Generate Schnorr signature to prove knowledge of private value 'x' used
215 * in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
216 * using a SHA256 hash.
217 * 'idlen' bytes from 'id' will be included in the signature hash as an anti-
219 * On success, 0 is returned and *siglen bytes of signature are returned in
220 * *sig (caller to free). Returns -1 on failure.
223 schnorr_sign_buf(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
224 const BIGNUM *x, const BIGNUM *g_x, const u_char *id, u_int idlen,
225 u_char **sig, u_int *siglen)
230 if (schnorr_sign(grp_p, grp_q, grp_g, SSH_DIGEST_SHA256,
231 x, g_x, id, idlen, &r, &e) != 0)
234 /* Signature is (e, r) */
236 /* XXX sigtype-hash as string? */
237 buffer_put_bignum2(&b, e);
238 buffer_put_bignum2(&b, r);
239 *siglen = buffer_len(&b);
240 *sig = xmalloc(*siglen);
241 memcpy(*sig, buffer_ptr(&b), *siglen);
242 SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
243 "%s: sigblob", __func__));
253 * Verify Schnorr signature { r (v - xh mod q), e (g^v mod p) } against
254 * public exponent g_x (g^x) under group defined by 'grp_p', 'grp_q' and
255 * 'grp_g' using hash "hash_alg".
256 * Signature hash will be salted with 'idlen' bytes from 'id'.
257 * Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
260 schnorr_verify(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
261 int hash_alg, const BIGNUM *g_x, const u_char *id, u_int idlen,
262 const BIGNUM *r, const BIGNUM *e)
265 BIGNUM *h = NULL, *g_xh = NULL, *g_r = NULL, *gx_q = NULL;
266 BIGNUM *expected = NULL;
269 SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));
271 /* Avoid degenerate cases: g^0 yields a spoofable signature */
272 if (BN_cmp(g_x, BN_value_one()) <= 0) {
273 error("%s: g_x <= 1", __func__);
276 if (BN_cmp(g_x, grp_p) >= 0) {
277 error("%s: g_x >= p", __func__);
281 h = g_xh = g_r = expected = NULL;
282 if ((bn_ctx = BN_CTX_new()) == NULL) {
283 error("%s: BN_CTX_new", __func__);
286 if ((g_xh = BN_new()) == NULL ||
287 (g_r = BN_new()) == NULL ||
288 (gx_q = BN_new()) == NULL ||
289 (expected = BN_new()) == NULL) {
290 error("%s: BN_new", __func__);
294 SCHNORR_DEBUG_BN((e, "%s: e = ", __func__));
295 SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));
297 /* gx_q = (g^x)^q must === 1 mod p */
298 if (BN_mod_exp(gx_q, g_x, grp_q, grp_p, bn_ctx) == -1) {
299 error("%s: BN_mod_exp (g_x^q mod p)", __func__);
302 if (BN_cmp(gx_q, BN_value_one()) != 0) {
303 error("%s: Invalid signature (g^x)^q != 1 mod p", __func__);
307 SCHNORR_DEBUG_BN((g_xh, "%s: g_xh = ", __func__));
308 /* h = H(g || g^v || g^x || id) */
309 if ((h = schnorr_hash(grp_p, grp_q, grp_g, hash_alg, e, g_x,
310 id, idlen)) == NULL) {
311 error("%s: schnorr_hash failed", __func__);
316 if (BN_mod_exp(g_xh, g_x, h, grp_p, bn_ctx) == -1) {
317 error("%s: BN_mod_exp (g_x^h mod p)", __func__);
320 SCHNORR_DEBUG_BN((g_xh, "%s: g_xh = ", __func__));
323 if (BN_mod_exp(g_r, grp_g, r, grp_p, bn_ctx) == -1) {
324 error("%s: BN_mod_exp (g_x^h mod p)", __func__);
327 SCHNORR_DEBUG_BN((g_r, "%s: g_r = ", __func__));
329 /* expected = g^r * g_xh */
330 if (BN_mod_mul(expected, g_r, g_xh, grp_p, bn_ctx) == -1) {
331 error("%s: BN_mod_mul (expected = g_r mod p)", __func__);
334 SCHNORR_DEBUG_BN((expected, "%s: expected = ", __func__));
336 /* Check e == expected */
337 success = BN_cmp(expected, e) == 0;
348 if (expected != NULL)
349 BN_clear_free(expected);
354 * Verify Schnorr signature 'sig' of length 'siglen' against public exponent
355 * g_x (g^x) under group defined by 'grp_p', 'grp_q' and 'grp_g' using a
357 * Signature hash will be salted with 'idlen' bytes from 'id'.
358 * Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
361 schnorr_verify_buf(const BIGNUM *grp_p, const BIGNUM *grp_q,
363 const BIGNUM *g_x, const u_char *id, u_int idlen,
364 const u_char *sig, u_int siglen)
372 if ((e = BN_new()) == NULL ||
373 (r = BN_new()) == NULL) {
374 error("%s: BN_new", __func__);
378 /* Extract g^v and r from signature blob */
380 buffer_append(&b, sig, siglen);
381 SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
382 "%s: sigblob", __func__));
383 buffer_get_bignum2(&b, e);
384 buffer_get_bignum2(&b, r);
385 rlen = buffer_len(&b);
388 error("%s: remaining bytes in signature %d", __func__, rlen);
392 ret = schnorr_verify(grp_p, grp_q, grp_g, SSH_DIGEST_SHA256,
393 g_x, id, idlen, r, e);
401 /* Helper functions */
404 * Generate uniformly distributed random number in range (1, high).
405 * Return number on success, NULL on failure.
408 bn_rand_range_gt_one(const BIGNUM *high)
413 if ((tmp = BN_new()) == NULL) {
414 error("%s: BN_new", __func__);
417 if ((r = BN_new()) == NULL) {
418 error("%s: BN_new failed", __func__);
421 if (BN_set_word(tmp, 2) != 1) {
422 error("%s: BN_set_word(tmp, 2)", __func__);
425 if (BN_sub(tmp, high, tmp) == -1) {
426 error("%s: BN_sub failed (tmp = high - 2)", __func__);
429 if (BN_rand_range(r, tmp) == -1) {
430 error("%s: BN_rand_range failed", __func__);
433 if (BN_set_word(tmp, 2) != 1) {
434 error("%s: BN_set_word(tmp, 2)", __func__);
437 if (BN_add(r, r, tmp) == -1) {
438 error("%s: BN_add failed (r = r + 2)", __func__);
450 /* XXX convert all callers of this to use ssh_digest_memory() directly */
452 * Hash contents of buffer 'b' with hash 'md'. Returns 0 on success,
453 * with digest via 'digestp' (caller to free) and length via 'lenp'.
454 * Returns -1 on failure.
457 hash_buffer(const u_char *buf, u_int len, int hash_alg,
458 u_char **digestp, u_int *lenp)
460 u_char digest[SSH_DIGEST_MAX_LENGTH];
461 u_int digest_len = ssh_digest_bytes(hash_alg);
463 if (digest_len == 0) {
464 error("%s: invalid hash", __func__);
467 if (ssh_digest_memory(hash_alg, buf, len, digest, digest_len) != 0) {
468 error("%s: digest_memory failed", __func__);
471 *digestp = xmalloc(digest_len);
473 memcpy(*digestp, digest, *lenp);
474 bzero(digest, sizeof(digest));
479 /* print formatted string followed by bignum */
481 debug3_bn(const BIGNUM *n, const char *fmt, ...)
489 ret = vasprintf(&out, fmt, args);
491 if (ret == -1 || out == NULL)
492 fatal("%s: vasprintf failed", __func__);
495 debug3("%s(null)", out);
498 debug3("%s0x%s", out, h);
504 /* print formatted string followed by buffer contents in hex */
506 debug3_buf(const u_char *buf, u_int len, const char *fmt, ...)
515 ret = vasprintf(&out, fmt, args);
517 if (ret == -1 || out == NULL)
518 fatal("%s: vasprintf failed", __func__);
520 debug3("%s length %u%s", out, len, buf == NULL ? " (null)" : "");
526 for (i = j = 0; i < len; i++) {
527 snprintf(h + j, sizeof(h) - j, "%02x", buf[i]);
529 if (j >= sizeof(h) - 1 || i == len - 1) {
538 * Construct a MODP group from hex strings p (which must be a safe
539 * prime) and g, automatically calculating subgroup q as (p / 2)
542 modp_group_from_g_and_safe_p(const char *grp_g, const char *grp_p)
544 struct modp_group *ret;
546 ret = xcalloc(1, sizeof(*ret));
547 ret->p = ret->q = ret->g = NULL;
548 if (BN_hex2bn(&ret->p, grp_p) == 0 ||
549 BN_hex2bn(&ret->g, grp_g) == 0)
550 fatal("%s: BN_hex2bn", __func__);
551 /* Subgroup order is p/2 (p is a safe prime) */
552 if ((ret->q = BN_new()) == NULL)
553 fatal("%s: BN_new", __func__);
554 if (BN_rshift1(ret->q, ret->p) != 1)
555 fatal("%s: BN_rshift1", __func__);
561 modp_group_free(struct modp_group *grp)
564 BN_clear_free(grp->g);
566 BN_clear_free(grp->p);
568 BN_clear_free(grp->q);
569 bzero(grp, sizeof(*grp));
573 /* main() function for self-test */
577 schnorr_selftest_one(const BIGNUM *grp_p, const BIGNUM *grp_q,
578 const BIGNUM *grp_g, const BIGNUM *x)
585 if ((bn_ctx = BN_CTX_new()) == NULL)
586 fatal("%s: BN_CTX_new", __func__);
587 if ((g_x = BN_new()) == NULL)
588 fatal("%s: BN_new", __func__);
590 if (BN_mod_exp(g_x, grp_g, x, grp_p, bn_ctx) == -1)
591 fatal("%s: g_x", __func__);
592 if (schnorr_sign_buf(grp_p, grp_q, grp_g, x, g_x, "junk", 4,
594 fatal("%s: schnorr_sign", __func__);
595 if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
597 fatal("%s: verify fail", __func__);
598 if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "JUNK", 4,
600 fatal("%s: verify should have failed (bad ID)", __func__);
602 if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
604 fatal("%s: verify should have failed (bit error)", __func__);
611 schnorr_selftest(void)
614 struct modp_group *grp;
618 grp = jpake_default_group();
619 if ((x = BN_new()) == NULL)
620 fatal("%s: BN_new", __func__);
621 SCHNORR_DEBUG_BN((grp->p, "%s: grp->p = ", __func__));
622 SCHNORR_DEBUG_BN((grp->q, "%s: grp->q = ", __func__));
623 SCHNORR_DEBUG_BN((grp->g, "%s: grp->g = ", __func__));
626 for (i = 1; i < 20; i++) {
627 printf("x = %u\n", i);
629 if (BN_set_word(x, i) != 1)
630 fatal("%s: set x word", __func__);
631 schnorr_selftest_one(grp->p, grp->q, grp->g, x);
634 /* 100 x random [0, p) */
635 for (i = 0; i < 100; i++) {
636 if (BN_rand_range(x, grp->p) != 1)
637 fatal("%s: BN_rand_range", __func__);
639 printf("x = (random) 0x%s\n", hh);
642 schnorr_selftest_one(grp->p, grp->q, grp->g, x);
646 if (BN_set_word(x, 20) != 1)
647 fatal("%s: BN_set_word (x = 20)", __func__);
648 if (BN_sub(x, grp->q, x) != 1)
649 fatal("%s: BN_sub (q - x)", __func__);
650 for (i = 0; i < 19; i++) {
652 printf("x = (q - %d) 0x%s\n", 20 - i, hh);
655 schnorr_selftest_one(grp->p, grp->q, grp->g, x);
656 if (BN_add(x, x, BN_value_one()) != 1)
657 fatal("%s: BN_add (x + 1)", __func__);
663 main(int argc, char **argv)
665 log_init(argv[0], SYSLOG_LEVEL_DEBUG3, SYSLOG_FACILITY_USER, 1);