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
32 #include <sys/types.h>
38 #include <openssl/evp.h>
39 #include <openssl/bn.h>
50 #include "openbsd-compat/openssl-compat.h"
52 /* #define SCHNORR_DEBUG */ /* Privacy-violating debugging */
53 /* #define SCHNORR_MAIN */ /* Include main() selftest */
56 # define SCHNORR_DEBUG_BN(a)
57 # define SCHNORR_DEBUG_BUF(a)
59 # define SCHNORR_DEBUG_BN(a) debug3_bn a
60 # define SCHNORR_DEBUG_BUF(a) debug3_buf a
61 #endif /* SCHNORR_DEBUG */
64 * Calculate hash component of Schnorr signature H(g || g^v || g^x || id)
65 * using the hash function defined by "hash_alg". Returns signature as
66 * bignum or NULL on error.
69 schnorr_hash(const BIGNUM *p, const BIGNUM *q, const BIGNUM *g,
70 int hash_alg, const BIGNUM *g_v, const BIGNUM *g_x,
71 const u_char *id, u_int idlen)
79 if ((h = BN_new()) == NULL) {
80 error("%s: BN_new", __func__);
86 /* h = H(g || p || q || g^v || g^x || id) */
87 buffer_put_bignum2(&b, g);
88 buffer_put_bignum2(&b, p);
89 buffer_put_bignum2(&b, q);
90 buffer_put_bignum2(&b, g_v);
91 buffer_put_bignum2(&b, g_x);
92 buffer_put_string(&b, id, idlen);
94 SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
95 "%s: hashblob", __func__));
96 if (hash_buffer(buffer_ptr(&b), buffer_len(&b), hash_alg,
97 &digest, &digest_len) != 0) {
98 error("%s: hash_buffer", __func__);
101 if (BN_bin2bn(digest, (int)digest_len, h) == NULL) {
102 error("%s: BN_bin2bn", __func__);
106 SCHNORR_DEBUG_BN((h, "%s: h = ", __func__));
109 bzero(digest, digest_len);
119 * Generate Schnorr signature to prove knowledge of private value 'x' used
120 * in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
121 * using the hash function "hash_alg".
122 * 'idlen' bytes from 'id' will be included in the signature hash as an anti-
125 * On success, 0 is returned. The signature values are returned as *e_p
126 * (g^v mod p) and *r_p (v - xh mod q). The caller must free these values.
127 * On failure, -1 is returned.
130 schnorr_sign(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
131 int hash_alg, const BIGNUM *x, const BIGNUM *g_x,
132 const u_char *id, u_int idlen, BIGNUM **r_p, BIGNUM **e_p)
135 BIGNUM *h, *tmp, *v, *g_v, *r;
138 SCHNORR_DEBUG_BN((x, "%s: x = ", __func__));
139 SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));
141 /* Avoid degenerate cases: g^0 yields a spoofable signature */
142 if (BN_cmp(g_x, BN_value_one()) <= 0) {
143 error("%s: g_x < 1", __func__);
146 if (BN_cmp(g_x, grp_p) >= 0) {
147 error("%s: g_x > g", __func__);
151 h = g_v = r = tmp = v = NULL;
152 if ((bn_ctx = BN_CTX_new()) == NULL) {
153 error("%s: BN_CTX_new", __func__);
156 if ((g_v = BN_new()) == NULL ||
157 (r = BN_new()) == NULL ||
158 (tmp = BN_new()) == NULL) {
159 error("%s: BN_new", __func__);
164 * v must be a random element of Zq, so 1 <= v < q
165 * we also exclude v = 1, since g^1 looks dangerous
167 if ((v = bn_rand_range_gt_one(grp_p)) == NULL) {
168 error("%s: bn_rand_range2", __func__);
171 SCHNORR_DEBUG_BN((v, "%s: v = ", __func__));
173 /* g_v = g^v mod p */
174 if (BN_mod_exp(g_v, grp_g, v, grp_p, bn_ctx) == -1) {
175 error("%s: BN_mod_exp (g^v mod p)", __func__);
178 SCHNORR_DEBUG_BN((g_v, "%s: g_v = ", __func__));
180 /* h = H(g || g^v || g^x || id) */
181 if ((h = schnorr_hash(grp_p, grp_q, grp_g, hash_alg, g_v, g_x,
182 id, idlen)) == NULL) {
183 error("%s: schnorr_hash failed", __func__);
187 /* r = v - xh mod q */
188 if (BN_mod_mul(tmp, x, h, grp_q, bn_ctx) == -1) {
189 error("%s: BN_mod_mul (tmp = xv mod q)", __func__);
192 if (BN_mod_sub(r, v, tmp, grp_q, bn_ctx) == -1) {
193 error("%s: BN_mod_mul (r = v - tmp)", __func__);
196 SCHNORR_DEBUG_BN((g_v, "%s: e = ", __func__));
197 SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));
215 * Generate Schnorr signature to prove knowledge of private value 'x' used
216 * in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
217 * using a SHA256 hash.
218 * 'idlen' bytes from 'id' will be included in the signature hash as an anti-
220 * On success, 0 is returned and *siglen bytes of signature are returned in
221 * *sig (caller to free). Returns -1 on failure.
224 schnorr_sign_buf(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
225 const BIGNUM *x, const BIGNUM *g_x, const u_char *id, u_int idlen,
226 u_char **sig, u_int *siglen)
231 if (schnorr_sign(grp_p, grp_q, grp_g, SSH_DIGEST_SHA256,
232 x, g_x, id, idlen, &r, &e) != 0)
235 /* Signature is (e, r) */
237 /* XXX sigtype-hash as string? */
238 buffer_put_bignum2(&b, e);
239 buffer_put_bignum2(&b, r);
240 *siglen = buffer_len(&b);
241 *sig = xmalloc(*siglen);
242 memcpy(*sig, buffer_ptr(&b), *siglen);
243 SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
244 "%s: sigblob", __func__));
254 * Verify Schnorr signature { r (v - xh mod q), e (g^v mod p) } against
255 * public exponent g_x (g^x) under group defined by 'grp_p', 'grp_q' and
256 * 'grp_g' using hash "hash_alg".
257 * Signature hash will be salted with 'idlen' bytes from 'id'.
258 * Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
261 schnorr_verify(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
262 int hash_alg, const BIGNUM *g_x, const u_char *id, u_int idlen,
263 const BIGNUM *r, const BIGNUM *e)
266 BIGNUM *h = NULL, *g_xh = NULL, *g_r = NULL, *gx_q = NULL;
267 BIGNUM *expected = NULL;
270 SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));
272 /* Avoid degenerate cases: g^0 yields a spoofable signature */
273 if (BN_cmp(g_x, BN_value_one()) <= 0) {
274 error("%s: g_x <= 1", __func__);
277 if (BN_cmp(g_x, grp_p) >= 0) {
278 error("%s: g_x >= p", __func__);
282 h = g_xh = g_r = expected = NULL;
283 if ((bn_ctx = BN_CTX_new()) == NULL) {
284 error("%s: BN_CTX_new", __func__);
287 if ((g_xh = BN_new()) == NULL ||
288 (g_r = BN_new()) == NULL ||
289 (gx_q = BN_new()) == NULL ||
290 (expected = BN_new()) == NULL) {
291 error("%s: BN_new", __func__);
295 SCHNORR_DEBUG_BN((e, "%s: e = ", __func__));
296 SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));
298 /* gx_q = (g^x)^q must === 1 mod p */
299 if (BN_mod_exp(gx_q, g_x, grp_q, grp_p, bn_ctx) == -1) {
300 error("%s: BN_mod_exp (g_x^q mod p)", __func__);
303 if (BN_cmp(gx_q, BN_value_one()) != 0) {
304 error("%s: Invalid signature (g^x)^q != 1 mod p", __func__);
308 SCHNORR_DEBUG_BN((g_xh, "%s: g_xh = ", __func__));
309 /* h = H(g || g^v || g^x || id) */
310 if ((h = schnorr_hash(grp_p, grp_q, grp_g, hash_alg, e, g_x,
311 id, idlen)) == NULL) {
312 error("%s: schnorr_hash failed", __func__);
317 if (BN_mod_exp(g_xh, g_x, h, grp_p, bn_ctx) == -1) {
318 error("%s: BN_mod_exp (g_x^h mod p)", __func__);
321 SCHNORR_DEBUG_BN((g_xh, "%s: g_xh = ", __func__));
324 if (BN_mod_exp(g_r, grp_g, r, grp_p, bn_ctx) == -1) {
325 error("%s: BN_mod_exp (g_x^h mod p)", __func__);
328 SCHNORR_DEBUG_BN((g_r, "%s: g_r = ", __func__));
330 /* expected = g^r * g_xh */
331 if (BN_mod_mul(expected, g_r, g_xh, grp_p, bn_ctx) == -1) {
332 error("%s: BN_mod_mul (expected = g_r mod p)", __func__);
335 SCHNORR_DEBUG_BN((expected, "%s: expected = ", __func__));
337 /* Check e == expected */
338 success = BN_cmp(expected, e) == 0;
349 if (expected != NULL)
350 BN_clear_free(expected);
355 * Verify Schnorr signature 'sig' of length 'siglen' against public exponent
356 * g_x (g^x) under group defined by 'grp_p', 'grp_q' and 'grp_g' using a
358 * Signature hash will be salted with 'idlen' bytes from 'id'.
359 * Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
362 schnorr_verify_buf(const BIGNUM *grp_p, const BIGNUM *grp_q,
364 const BIGNUM *g_x, const u_char *id, u_int idlen,
365 const u_char *sig, u_int siglen)
373 if ((e = BN_new()) == NULL ||
374 (r = BN_new()) == NULL) {
375 error("%s: BN_new", __func__);
379 /* Extract g^v and r from signature blob */
381 buffer_append(&b, sig, siglen);
382 SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
383 "%s: sigblob", __func__));
384 buffer_get_bignum2(&b, e);
385 buffer_get_bignum2(&b, r);
386 rlen = buffer_len(&b);
389 error("%s: remaining bytes in signature %d", __func__, rlen);
393 ret = schnorr_verify(grp_p, grp_q, grp_g, SSH_DIGEST_SHA256,
394 g_x, id, idlen, r, e);
402 /* Helper functions */
405 * Generate uniformly distributed random number in range (1, high).
406 * Return number on success, NULL on failure.
409 bn_rand_range_gt_one(const BIGNUM *high)
414 if ((tmp = BN_new()) == NULL) {
415 error("%s: BN_new", __func__);
418 if ((r = BN_new()) == NULL) {
419 error("%s: BN_new failed", __func__);
422 if (BN_set_word(tmp, 2) != 1) {
423 error("%s: BN_set_word(tmp, 2)", __func__);
426 if (BN_sub(tmp, high, tmp) == -1) {
427 error("%s: BN_sub failed (tmp = high - 2)", __func__);
430 if (BN_rand_range(r, tmp) == -1) {
431 error("%s: BN_rand_range failed", __func__);
434 if (BN_set_word(tmp, 2) != 1) {
435 error("%s: BN_set_word(tmp, 2)", __func__);
438 if (BN_add(r, r, tmp) == -1) {
439 error("%s: BN_add failed (r = r + 2)", __func__);
451 /* XXX convert all callers of this to use ssh_digest_memory() directly */
453 * Hash contents of buffer 'b' with hash 'md'. Returns 0 on success,
454 * with digest via 'digestp' (caller to free) and length via 'lenp'.
455 * Returns -1 on failure.
458 hash_buffer(const u_char *buf, u_int len, int hash_alg,
459 u_char **digestp, u_int *lenp)
461 u_char digest[SSH_DIGEST_MAX_LENGTH];
462 u_int digest_len = ssh_digest_bytes(hash_alg);
464 if (digest_len == 0) {
465 error("%s: invalid hash", __func__);
468 if (ssh_digest_memory(hash_alg, buf, len, digest, digest_len) != 0) {
469 error("%s: digest_memory failed", __func__);
472 *digestp = xmalloc(digest_len);
474 memcpy(*digestp, digest, *lenp);
475 bzero(digest, sizeof(digest));
480 /* print formatted string followed by bignum */
482 debug3_bn(const BIGNUM *n, const char *fmt, ...)
490 ret = vasprintf(&out, fmt, args);
492 if (ret == -1 || out == NULL)
493 fatal("%s: vasprintf failed", __func__);
496 debug3("%s(null)", out);
499 debug3("%s0x%s", out, h);
505 /* print formatted string followed by buffer contents in hex */
507 debug3_buf(const u_char *buf, u_int len, const char *fmt, ...)
516 ret = vasprintf(&out, fmt, args);
518 if (ret == -1 || out == NULL)
519 fatal("%s: vasprintf failed", __func__);
521 debug3("%s length %u%s", out, len, buf == NULL ? " (null)" : "");
527 for (i = j = 0; i < len; i++) {
528 snprintf(h + j, sizeof(h) - j, "%02x", buf[i]);
530 if (j >= sizeof(h) - 1 || i == len - 1) {
539 * Construct a MODP group from hex strings p (which must be a safe
540 * prime) and g, automatically calculating subgroup q as (p / 2)
543 modp_group_from_g_and_safe_p(const char *grp_g, const char *grp_p)
545 struct modp_group *ret;
547 ret = xcalloc(1, sizeof(*ret));
548 ret->p = ret->q = ret->g = NULL;
549 if (BN_hex2bn(&ret->p, grp_p) == 0 ||
550 BN_hex2bn(&ret->g, grp_g) == 0)
551 fatal("%s: BN_hex2bn", __func__);
552 /* Subgroup order is p/2 (p is a safe prime) */
553 if ((ret->q = BN_new()) == NULL)
554 fatal("%s: BN_new", __func__);
555 if (BN_rshift1(ret->q, ret->p) != 1)
556 fatal("%s: BN_rshift1", __func__);
562 modp_group_free(struct modp_group *grp)
565 BN_clear_free(grp->g);
567 BN_clear_free(grp->p);
569 BN_clear_free(grp->q);
570 bzero(grp, sizeof(*grp));
574 /* main() function for self-test */
578 schnorr_selftest_one(const BIGNUM *grp_p, const BIGNUM *grp_q,
579 const BIGNUM *grp_g, const BIGNUM *x)
586 if ((bn_ctx = BN_CTX_new()) == NULL)
587 fatal("%s: BN_CTX_new", __func__);
588 if ((g_x = BN_new()) == NULL)
589 fatal("%s: BN_new", __func__);
591 if (BN_mod_exp(g_x, grp_g, x, grp_p, bn_ctx) == -1)
592 fatal("%s: g_x", __func__);
593 if (schnorr_sign_buf(grp_p, grp_q, grp_g, x, g_x, "junk", 4,
595 fatal("%s: schnorr_sign", __func__);
596 if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
598 fatal("%s: verify fail", __func__);
599 if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "JUNK", 4,
601 fatal("%s: verify should have failed (bad ID)", __func__);
603 if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
605 fatal("%s: verify should have failed (bit error)", __func__);
612 schnorr_selftest(void)
615 struct modp_group *grp;
619 grp = jpake_default_group();
620 if ((x = BN_new()) == NULL)
621 fatal("%s: BN_new", __func__);
622 SCHNORR_DEBUG_BN((grp->p, "%s: grp->p = ", __func__));
623 SCHNORR_DEBUG_BN((grp->q, "%s: grp->q = ", __func__));
624 SCHNORR_DEBUG_BN((grp->g, "%s: grp->g = ", __func__));
627 for (i = 1; i < 20; i++) {
628 printf("x = %u\n", i);
630 if (BN_set_word(x, i) != 1)
631 fatal("%s: set x word", __func__);
632 schnorr_selftest_one(grp->p, grp->q, grp->g, x);
635 /* 100 x random [0, p) */
636 for (i = 0; i < 100; i++) {
637 if (BN_rand_range(x, grp->p) != 1)
638 fatal("%s: BN_rand_range", __func__);
640 printf("x = (random) 0x%s\n", hh);
643 schnorr_selftest_one(grp->p, grp->q, grp->g, x);
647 if (BN_set_word(x, 20) != 1)
648 fatal("%s: BN_set_word (x = 20)", __func__);
649 if (BN_sub(x, grp->q, x) != 1)
650 fatal("%s: BN_sub (q - x)", __func__);
651 for (i = 0; i < 19; i++) {
653 printf("x = (q - %d) 0x%s\n", 20 - i, hh);
656 schnorr_selftest_one(grp->p, grp->q, grp->g, x);
657 if (BN_add(x, x, BN_value_one()) != 1)
658 fatal("%s: BN_add (x + 1)", __func__);
664 main(int argc, char **argv)
666 log_init(argv[0], SYSLOG_LEVEL_DEBUG3, SYSLOG_FACILITY_USER, 1);