schnorr.c revision 197679
1/* $OpenBSD: schnorr.c,v 1.3 2009/03/05 07:18:19 djm Exp $ */
2/* $FreeBSD: head/crypto/openssh/schnorr.c 197679 2009-10-01 17:12:52Z des $ */
3/*
4 * Copyright (c) 2008 Damien Miller.  All rights reserved.
5 *
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.
9 *
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.
17 */
18
19/*
20 * Implementation of Schnorr signatures / zero-knowledge proofs, based on
21 * description in:
22 *
23 * F. Hao, P. Ryan, "Password Authenticated Key Exchange by Juggling",
24 * 16th Workshop on Security Protocols, Cambridge, April 2008
25 *
26 * http://grouper.ieee.org/groups/1363/Research/contributions/hao-ryan-2008.pdf
27 */
28
29#include "includes.h"
30
31#include <sys/types.h>
32
33#include <string.h>
34#include <stdarg.h>
35#include <stdio.h>
36
37#include <openssl/evp.h>
38#include <openssl/bn.h>
39
40#include "xmalloc.h"
41#include "buffer.h"
42#include "log.h"
43
44#include "schnorr.h"
45
46#ifdef JPAKE
47
48#include "openbsd-compat/openssl-compat.h"
49
50/* #define SCHNORR_DEBUG */		/* Privacy-violating debugging */
51/* #define SCHNORR_MAIN */		/* Include main() selftest */
52
53#ifndef SCHNORR_DEBUG
54# define SCHNORR_DEBUG_BN(a)
55# define SCHNORR_DEBUG_BUF(a)
56#else
57# define SCHNORR_DEBUG_BN(a)	debug3_bn a
58# define SCHNORR_DEBUG_BUF(a)	debug3_buf a
59#endif /* SCHNORR_DEBUG */
60
61/*
62 * Calculate hash component of Schnorr signature H(g || g^v || g^x || id)
63 * using the hash function defined by "evp_md". Returns signature as
64 * bignum or NULL on error.
65 */
66static BIGNUM *
67schnorr_hash(const BIGNUM *p, const BIGNUM *q, const BIGNUM *g,
68    const EVP_MD *evp_md, const BIGNUM *g_v, const BIGNUM *g_x,
69    const u_char *id, u_int idlen)
70{
71	u_char *digest;
72	u_int digest_len;
73	BIGNUM *h;
74	Buffer b;
75	int success = -1;
76
77	if ((h = BN_new()) == NULL) {
78		error("%s: BN_new", __func__);
79		return NULL;
80	}
81
82	buffer_init(&b);
83
84	/* h = H(g || p || q || g^v || g^x || id) */
85	buffer_put_bignum2(&b, g);
86	buffer_put_bignum2(&b, p);
87	buffer_put_bignum2(&b, q);
88	buffer_put_bignum2(&b, g_v);
89	buffer_put_bignum2(&b, g_x);
90	buffer_put_string(&b, id, idlen);
91
92	SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
93	    "%s: hashblob", __func__));
94	if (hash_buffer(buffer_ptr(&b), buffer_len(&b), evp_md,
95	    &digest, &digest_len) != 0) {
96		error("%s: hash_buffer", __func__);
97		goto out;
98	}
99	if (BN_bin2bn(digest, (int)digest_len, h) == NULL) {
100		error("%s: BN_bin2bn", __func__);
101		goto out;
102	}
103	success = 0;
104	SCHNORR_DEBUG_BN((h, "%s: h = ", __func__));
105 out:
106	buffer_free(&b);
107	bzero(digest, digest_len);
108	xfree(digest);
109	digest_len = 0;
110	if (success == 0)
111		return h;
112	BN_clear_free(h);
113	return NULL;
114}
115
116/*
117 * Generate Schnorr signature to prove knowledge of private value 'x' used
118 * in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
119 * using the hash function "evp_md".
120 * 'idlen' bytes from 'id' will be included in the signature hash as an anti-
121 * replay salt.
122 *
123 * On success, 0 is returned. The signature values are returned as *e_p
124 * (g^v mod p) and *r_p (v - xh mod q). The caller must free these values.
125 * On failure, -1 is returned.
126 */
127int
128schnorr_sign(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
129    const EVP_MD *evp_md, const BIGNUM *x, const BIGNUM *g_x,
130    const u_char *id, u_int idlen, BIGNUM **r_p, BIGNUM **e_p)
131{
132	int success = -1;
133	BIGNUM *h, *tmp, *v, *g_v, *r;
134	BN_CTX *bn_ctx;
135
136	SCHNORR_DEBUG_BN((x, "%s: x = ", __func__));
137	SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));
138
139	/* Avoid degenerate cases: g^0 yields a spoofable signature */
140	if (BN_cmp(g_x, BN_value_one()) <= 0) {
141		error("%s: g_x < 1", __func__);
142		return -1;
143	}
144
145	h = g_v = r = tmp = v = NULL;
146	if ((bn_ctx = BN_CTX_new()) == NULL) {
147		error("%s: BN_CTX_new", __func__);
148		goto out;
149	}
150	if ((g_v = BN_new()) == NULL ||
151	    (r = BN_new()) == NULL ||
152	    (tmp = BN_new()) == NULL) {
153		error("%s: BN_new", __func__);
154		goto out;
155	}
156
157	/*
158	 * v must be a random element of Zq, so 1 <= v < q
159	 * we also exclude v = 1, since g^1 looks dangerous
160	 */
161	if ((v = bn_rand_range_gt_one(grp_p)) == NULL) {
162		error("%s: bn_rand_range2", __func__);
163		goto out;
164	}
165	SCHNORR_DEBUG_BN((v, "%s: v = ", __func__));
166
167	/* g_v = g^v mod p */
168	if (BN_mod_exp(g_v, grp_g, v, grp_p, bn_ctx) == -1) {
169		error("%s: BN_mod_exp (g^v mod p)", __func__);
170		goto out;
171	}
172	SCHNORR_DEBUG_BN((g_v, "%s: g_v = ", __func__));
173
174	/* h = H(g || g^v || g^x || id) */
175	if ((h = schnorr_hash(grp_p, grp_q, grp_g, evp_md, g_v, g_x,
176	    id, idlen)) == NULL) {
177		error("%s: schnorr_hash failed", __func__);
178		goto out;
179	}
180
181	/* r = v - xh mod q */
182	if (BN_mod_mul(tmp, x, h, grp_q, bn_ctx) == -1) {
183		error("%s: BN_mod_mul (tmp = xv mod q)", __func__);
184		goto out;
185	}
186	if (BN_mod_sub(r, v, tmp, grp_q, bn_ctx) == -1) {
187		error("%s: BN_mod_mul (r = v - tmp)", __func__);
188		goto out;
189	}
190	SCHNORR_DEBUG_BN((g_v, "%s: e = ", __func__));
191	SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));
192
193	*e_p = g_v;
194	*r_p = r;
195
196	success = 0;
197 out:
198	BN_CTX_free(bn_ctx);
199	if (h != NULL)
200		BN_clear_free(h);
201	if (v != NULL)
202		BN_clear_free(v);
203	BN_clear_free(tmp);
204
205	return success;
206}
207
208/*
209 * Generate Schnorr signature to prove knowledge of private value 'x' used
210 * in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
211 * using a SHA256 hash.
212 * 'idlen' bytes from 'id' will be included in the signature hash as an anti-
213 * replay salt.
214 * On success, 0 is returned and *siglen bytes of signature are returned in
215 * *sig (caller to free). Returns -1 on failure.
216 */
217int
218schnorr_sign_buf(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
219    const BIGNUM *x, const BIGNUM *g_x, const u_char *id, u_int idlen,
220    u_char **sig, u_int *siglen)
221{
222	Buffer b;
223	BIGNUM *r, *e;
224
225	if (schnorr_sign(grp_p, grp_q, grp_g, EVP_sha256(),
226	    x, g_x, id, idlen, &r, &e) != 0)
227		return -1;
228
229	/* Signature is (e, r) */
230	buffer_init(&b);
231	/* XXX sigtype-hash as string? */
232	buffer_put_bignum2(&b, e);
233	buffer_put_bignum2(&b, r);
234	*siglen = buffer_len(&b);
235	*sig = xmalloc(*siglen);
236	memcpy(*sig, buffer_ptr(&b), *siglen);
237	SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
238	    "%s: sigblob", __func__));
239	buffer_free(&b);
240
241	BN_clear_free(r);
242	BN_clear_free(e);
243
244	return 0;
245}
246
247/*
248 * Verify Schnorr signature { r (v - xh mod q), e (g^v mod p) } against
249 * public exponent g_x (g^x) under group defined by 'grp_p', 'grp_q' and
250 * 'grp_g' using hash "evp_md".
251 * Signature hash will be salted with 'idlen' bytes from 'id'.
252 * Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
253 */
254int
255schnorr_verify(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
256    const EVP_MD *evp_md, const BIGNUM *g_x, const u_char *id, u_int idlen,
257    const BIGNUM *r, const BIGNUM *e)
258{
259	int success = -1;
260	BIGNUM *h, *g_xh, *g_r, *expected;
261	BN_CTX *bn_ctx;
262
263	SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));
264
265	/* Avoid degenerate cases: g^0 yields a spoofable signature */
266	if (BN_cmp(g_x, BN_value_one()) <= 0) {
267		error("%s: g_x < 1", __func__);
268		return -1;
269	}
270
271	h = g_xh = g_r = expected = NULL;
272	if ((bn_ctx = BN_CTX_new()) == NULL) {
273		error("%s: BN_CTX_new", __func__);
274		goto out;
275	}
276	if ((g_xh = BN_new()) == NULL ||
277	    (g_r = BN_new()) == NULL ||
278	    (expected = BN_new()) == NULL) {
279		error("%s: BN_new", __func__);
280		goto out;
281	}
282
283	SCHNORR_DEBUG_BN((e, "%s: e = ", __func__));
284	SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));
285
286	/* h = H(g || g^v || g^x || id) */
287	if ((h = schnorr_hash(grp_p, grp_q, grp_g, evp_md, e, g_x,
288	    id, idlen)) == NULL) {
289		error("%s: schnorr_hash failed", __func__);
290		goto out;
291	}
292
293	/* g_xh = (g^x)^h */
294	if (BN_mod_exp(g_xh, g_x, h, grp_p, bn_ctx) == -1) {
295		error("%s: BN_mod_exp (g_x^h mod p)", __func__);
296		goto out;
297	}
298	SCHNORR_DEBUG_BN((g_xh, "%s: g_xh = ", __func__));
299
300	/* g_r = g^r */
301	if (BN_mod_exp(g_r, grp_g, r, grp_p, bn_ctx) == -1) {
302		error("%s: BN_mod_exp (g_x^h mod p)", __func__);
303		goto out;
304	}
305	SCHNORR_DEBUG_BN((g_r, "%s: g_r = ", __func__));
306
307	/* expected = g^r * g_xh */
308	if (BN_mod_mul(expected, g_r, g_xh, grp_p, bn_ctx) == -1) {
309		error("%s: BN_mod_mul (expected = g_r mod p)", __func__);
310		goto out;
311	}
312	SCHNORR_DEBUG_BN((expected, "%s: expected = ", __func__));
313
314	/* Check e == expected */
315	success = BN_cmp(expected, e) == 0;
316 out:
317	BN_CTX_free(bn_ctx);
318	if (h != NULL)
319		BN_clear_free(h);
320	BN_clear_free(g_xh);
321	BN_clear_free(g_r);
322	BN_clear_free(expected);
323	return success;
324}
325
326/*
327 * Verify Schnorr signature 'sig' of length 'siglen' against public exponent
328 * g_x (g^x) under group defined by 'grp_p', 'grp_q' and 'grp_g' using a
329 * SHA256 hash.
330 * Signature hash will be salted with 'idlen' bytes from 'id'.
331 * Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
332 */
333int
334schnorr_verify_buf(const BIGNUM *grp_p, const BIGNUM *grp_q,
335    const BIGNUM *grp_g,
336    const BIGNUM *g_x, const u_char *id, u_int idlen,
337    const u_char *sig, u_int siglen)
338{
339	Buffer b;
340	int ret = -1;
341	u_int rlen;
342	BIGNUM *r, *e;
343
344	e = r = NULL;
345	if ((e = BN_new()) == NULL ||
346	    (r = BN_new()) == NULL) {
347		error("%s: BN_new", __func__);
348		goto out;
349	}
350
351	/* Extract g^v and r from signature blob */
352	buffer_init(&b);
353	buffer_append(&b, sig, siglen);
354	SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
355	    "%s: sigblob", __func__));
356	buffer_get_bignum2(&b, e);
357	buffer_get_bignum2(&b, r);
358	rlen = buffer_len(&b);
359	buffer_free(&b);
360	if (rlen != 0) {
361		error("%s: remaining bytes in signature %d", __func__, rlen);
362		goto out;
363	}
364
365	ret = schnorr_verify(grp_p, grp_q, grp_g, EVP_sha256(),
366	    g_x, id, idlen, r, e);
367 out:
368	BN_clear_free(e);
369	BN_clear_free(r);
370
371	return ret;
372}
373
374/* Helper functions */
375
376/*
377 * Generate uniformly distributed random number in range (1, high).
378 * Return number on success, NULL on failure.
379 */
380BIGNUM *
381bn_rand_range_gt_one(const BIGNUM *high)
382{
383	BIGNUM *r, *tmp;
384	int success = -1;
385
386	if ((tmp = BN_new()) == NULL) {
387		error("%s: BN_new", __func__);
388		return NULL;
389	}
390	if ((r = BN_new()) == NULL) {
391		error("%s: BN_new failed", __func__);
392		goto out;
393	}
394	if (BN_set_word(tmp, 2) != 1) {
395		error("%s: BN_set_word(tmp, 2)", __func__);
396		goto out;
397	}
398	if (BN_sub(tmp, high, tmp) == -1) {
399		error("%s: BN_sub failed (tmp = high - 2)", __func__);
400		goto out;
401	}
402	if (BN_rand_range(r, tmp) == -1) {
403		error("%s: BN_rand_range failed", __func__);
404		goto out;
405	}
406	if (BN_set_word(tmp, 2) != 1) {
407		error("%s: BN_set_word(tmp, 2)", __func__);
408		goto out;
409	}
410	if (BN_add(r, r, tmp) == -1) {
411		error("%s: BN_add failed (r = r + 2)", __func__);
412		goto out;
413	}
414	success = 0;
415 out:
416	BN_clear_free(tmp);
417	if (success == 0)
418		return r;
419	BN_clear_free(r);
420	return NULL;
421}
422
423/*
424 * Hash contents of buffer 'b' with hash 'md'. Returns 0 on success,
425 * with digest via 'digestp' (caller to free) and length via 'lenp'.
426 * Returns -1 on failure.
427 */
428int
429hash_buffer(const u_char *buf, u_int len, const EVP_MD *md,
430    u_char **digestp, u_int *lenp)
431{
432	u_char digest[EVP_MAX_MD_SIZE];
433	u_int digest_len;
434	EVP_MD_CTX evp_md_ctx;
435	int success = -1;
436
437	EVP_MD_CTX_init(&evp_md_ctx);
438
439	if (EVP_DigestInit_ex(&evp_md_ctx, md, NULL) != 1) {
440		error("%s: EVP_DigestInit_ex", __func__);
441		goto out;
442	}
443	if (EVP_DigestUpdate(&evp_md_ctx, buf, len) != 1) {
444		error("%s: EVP_DigestUpdate", __func__);
445		goto out;
446	}
447	if (EVP_DigestFinal_ex(&evp_md_ctx, digest, &digest_len) != 1) {
448		error("%s: EVP_DigestFinal_ex", __func__);
449		goto out;
450	}
451	*digestp = xmalloc(digest_len);
452	*lenp = digest_len;
453	memcpy(*digestp, digest, *lenp);
454	success = 0;
455 out:
456	EVP_MD_CTX_cleanup(&evp_md_ctx);
457	bzero(digest, sizeof(digest));
458	digest_len = 0;
459	return success;
460}
461
462/* print formatted string followed by bignum */
463void
464debug3_bn(const BIGNUM *n, const char *fmt, ...)
465{
466	char *out, *h;
467	va_list args;
468
469	out = NULL;
470	va_start(args, fmt);
471	vasprintf(&out, fmt, args);
472	va_end(args);
473	if (out == NULL)
474		fatal("%s: vasprintf failed", __func__);
475
476	if (n == NULL)
477		debug3("%s(null)", out);
478	else {
479		h = BN_bn2hex(n);
480		debug3("%s0x%s", out, h);
481		free(h);
482	}
483	free(out);
484}
485
486/* print formatted string followed by buffer contents in hex */
487void
488debug3_buf(const u_char *buf, u_int len, const char *fmt, ...)
489{
490	char *out, h[65];
491	u_int i, j;
492	va_list args;
493
494	out = NULL;
495	va_start(args, fmt);
496	vasprintf(&out, fmt, args);
497	va_end(args);
498	if (out == NULL)
499		fatal("%s: vasprintf failed", __func__);
500
501	debug3("%s length %u%s", out, len, buf == NULL ? " (null)" : "");
502	free(out);
503	if (buf == NULL)
504		return;
505
506	*h = '\0';
507	for (i = j = 0; i < len; i++) {
508		snprintf(h + j, sizeof(h) - j, "%02x", buf[i]);
509		j += 2;
510		if (j >= sizeof(h) - 1 || i == len - 1) {
511			debug3("    %s", h);
512			*h = '\0';
513			j = 0;
514		}
515	}
516}
517
518/*
519 * Construct a MODP group from hex strings p (which must be a safe
520 * prime) and g, automatically calculating subgroup q as (p / 2)
521 */
522struct modp_group *
523modp_group_from_g_and_safe_p(const char *grp_g, const char *grp_p)
524{
525	struct modp_group *ret;
526
527	ret = xmalloc(sizeof(*ret));
528	ret->p = ret->q = ret->g = NULL;
529	if (BN_hex2bn(&ret->p, grp_p) == 0 ||
530	    BN_hex2bn(&ret->g, grp_g) == 0)
531		fatal("%s: BN_hex2bn", __func__);
532	/* Subgroup order is p/2 (p is a safe prime) */
533	if ((ret->q = BN_new()) == NULL)
534		fatal("%s: BN_new", __func__);
535	if (BN_rshift1(ret->q, ret->p) != 1)
536		fatal("%s: BN_rshift1", __func__);
537
538	return ret;
539}
540
541void
542modp_group_free(struct modp_group *grp)
543{
544	if (grp->g != NULL)
545		BN_clear_free(grp->g);
546	if (grp->p != NULL)
547		BN_clear_free(grp->p);
548	if (grp->q != NULL)
549		BN_clear_free(grp->q);
550	bzero(grp, sizeof(*grp));
551	xfree(grp);
552}
553
554/* main() function for self-test */
555
556#ifdef SCHNORR_MAIN
557static void
558schnorr_selftest_one(const BIGNUM *grp_p, const BIGNUM *grp_q,
559    const BIGNUM *grp_g, const BIGNUM *x)
560{
561	BIGNUM *g_x;
562	u_char *sig;
563	u_int siglen;
564	BN_CTX *bn_ctx;
565
566	if ((bn_ctx = BN_CTX_new()) == NULL)
567		fatal("%s: BN_CTX_new", __func__);
568	if ((g_x = BN_new()) == NULL)
569		fatal("%s: BN_new", __func__);
570
571	if (BN_mod_exp(g_x, grp_g, x, grp_p, bn_ctx) == -1)
572		fatal("%s: g_x", __func__);
573	if (schnorr_sign_buf(grp_p, grp_q, grp_g, x, g_x, "junk", 4,
574	    &sig, &siglen))
575		fatal("%s: schnorr_sign", __func__);
576	if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
577	    sig, siglen) != 1)
578		fatal("%s: verify fail", __func__);
579	if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "JUNK", 4,
580	    sig, siglen) != 0)
581		fatal("%s: verify should have failed (bad ID)", __func__);
582	sig[4] ^= 1;
583	if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
584	    sig, siglen) != 0)
585		fatal("%s: verify should have failed (bit error)", __func__);
586	xfree(sig);
587	BN_free(g_x);
588	BN_CTX_free(bn_ctx);
589}
590
591static void
592schnorr_selftest(void)
593{
594	BIGNUM *x;
595	struct modp_group *grp;
596	u_int i;
597	char *hh;
598
599	grp = jpake_default_group();
600	if ((x = BN_new()) == NULL)
601		fatal("%s: BN_new", __func__);
602	SCHNORR_DEBUG_BN((grp->p, "%s: grp->p = ", __func__));
603	SCHNORR_DEBUG_BN((grp->q, "%s: grp->q = ", __func__));
604	SCHNORR_DEBUG_BN((grp->g, "%s: grp->g = ", __func__));
605
606	/* [1, 20) */
607	for (i = 1; i < 20; i++) {
608		printf("x = %u\n", i);
609		fflush(stdout);
610		if (BN_set_word(x, i) != 1)
611			fatal("%s: set x word", __func__);
612		schnorr_selftest_one(grp->p, grp->q, grp->g, x);
613	}
614
615	/* 100 x random [0, p) */
616	for (i = 0; i < 100; i++) {
617		if (BN_rand_range(x, grp->p) != 1)
618			fatal("%s: BN_rand_range", __func__);
619		hh = BN_bn2hex(x);
620		printf("x = (random) 0x%s\n", hh);
621		free(hh);
622		fflush(stdout);
623		schnorr_selftest_one(grp->p, grp->q, grp->g, x);
624	}
625
626	/* [q-20, q) */
627	if (BN_set_word(x, 20) != 1)
628		fatal("%s: BN_set_word (x = 20)", __func__);
629	if (BN_sub(x, grp->q, x) != 1)
630		fatal("%s: BN_sub (q - x)", __func__);
631	for (i = 0; i < 19; i++) {
632		hh = BN_bn2hex(x);
633		printf("x = (q - %d) 0x%s\n", 20 - i, hh);
634		free(hh);
635		fflush(stdout);
636		schnorr_selftest_one(grp->p, grp->q, grp->g, x);
637		if (BN_add(x, x, BN_value_one()) != 1)
638			fatal("%s: BN_add (x + 1)", __func__);
639	}
640	BN_free(x);
641}
642
643int
644main(int argc, char **argv)
645{
646	log_init(argv[0], SYSLOG_LEVEL_DEBUG3, SYSLOG_FACILITY_USER, 1);
647
648	schnorr_selftest();
649	return 0;
650}
651#endif
652
653#endif
654