schnorr.c

   1 /* $OpenBSD: schnorr.c,v 1.3 2009/03/05 07:18:19 djm Exp $ */
   2 /*
   3  * Copyright (c) 2008 Damien Miller.  All rights reserved.
   4  *
   5  * Permission to use, copy, modify, and distribute this software for any
   6  * purpose with or without fee is hereby granted, provided that the above
   7  * copyright notice and this permission notice appear in all copies.
   8  *
   9  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  10  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  11  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  12  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  13  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  14  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  15  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  16  */
  17
  18 /*
  19  * Implementation of Schnorr signatures / zero-knowledge proofs, based on
  20  * description in:
  21  *
  22  * F. Hao, P. Ryan, "Password Authenticated Key Exchange by Juggling",
  23  * 16th Workshop on Security Protocols, Cambridge, April 2008
  24  *
  25  * http://grouper.ieee.org/groups/1363/Research/contributions/hao-ryan-2008.pdf
  26  */
  27
  28 #include "includes.h"
  29
  30 #include <sys/types.h>
  31
  32 #include <string.h>
  33 #include <stdarg.h>
  34 #include <stdio.h>
  35
  36 #include <openssl/evp.h>
  37 #include <openssl/bn.h>
  38
  39 #include "xmalloc.h"
  40 #include "buffer.h"
  41 #include "log.h"
  42
  43 #include "schnorr.h"
  44
  45 #include "openbsd-compat/openssl-compat.h"
  46
  47 /* #define SCHNORR_DEBUG */             /* Privacy-violating debugging */
  48 /* #define SCHNORR_MAIN */              /* Include main() selftest */
  49
  50 #ifndef SCHNORR_DEBUG
  51 # define SCHNORR_DEBUG_BN(a)
  52 # define SCHNORR_DEBUG_BUF(a)
  53 #else
  54 # define SCHNORR_DEBUG_BN(a)    debug3_bn a
  55 # define SCHNORR_DEBUG_BUF(a)   debug3_buf a
  56 #endif /* SCHNORR_DEBUG */
  57
  58 /*
  59  * Calculate hash component of Schnorr signature H(g || g^v || g^x || id)
  60  * using the hash function defined by "evp_md". Returns signature as
  61  * bignum or NULL on error.
  62  */
  63 static BIGNUM *
  64 schnorr_hash(const BIGNUM *p, const BIGNUM *q, const BIGNUM *g,
  65     const EVP_MD *evp_md, const BIGNUM *g_v, const BIGNUM *g_x,
  66     const u_char *id, u_int idlen)
  67 {
  68         u_char *digest;
  69         u_int digest_len;
  70         BIGNUM *h;
  71         Buffer b;
  72         int success = -1;
  73
  74         if ((h = BN_new()) == NULL) {
  75                 error("%s: BN_new", __func__);
  76                 return NULL;
  77         }
  78
  79         buffer_init(&b);
  80
  81         /* h = H(g || p || q || g^v || g^x || id) */
  82         buffer_put_bignum2(&b, g);
  83         buffer_put_bignum2(&b, p);
  84         buffer_put_bignum2(&b, q);
  85         buffer_put_bignum2(&b, g_v);
  86         buffer_put_bignum2(&b, g_x);
  87         buffer_put_string(&b, id, idlen);
  88
  89         SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
  90             "%s: hashblob", __func__));
  91         if (hash_buffer(buffer_ptr(&b), buffer_len(&b), evp_md,
  92             &digest, &digest_len) != 0) {
  93                 error("%s: hash_buffer", __func__);
  94                 goto out;
  95         }
  96         if (BN_bin2bn(digest, (int)digest_len, h) == NULL) {
  97                 error("%s: BN_bin2bn", __func__);
  98                 goto out;
  99         }
 100         success = 0;
 101         SCHNORR_DEBUG_BN((h, "%s: h = ", __func__));
 102  out:
 103         buffer_free(&b);
 104         bzero(digest, digest_len);
 105         xfree(digest);
 106         digest_len = 0;
 107         if (success == 0)
 108                 return h;
 109         BN_clear_free(h);
 110         return NULL;
 111 }
 112
 113 /*
 114  * Generate Schnorr signature to prove knowledge of private value 'x' used
 115  * in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
 116  * using the hash function "evp_md".
 117  * 'idlen' bytes from 'id' will be included in the signature hash as an anti-
 118  * replay salt.
 119  *
 120  * On success, 0 is returned. The signature values are returned as *e_p
 121  * (g^v mod p) and *r_p (v - xh mod q). The caller must free these values.
 122  * On failure, -1 is returned.
 123  */
 124 int
 125 schnorr_sign(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
 126     const EVP_MD *evp_md, const BIGNUM *x, const BIGNUM *g_x,
 127     const u_char *id, u_int idlen, BIGNUM **r_p, BIGNUM **e_p)
 128 {
 129         int success = -1;
 130         BIGNUM *h, *tmp, *v, *g_v, *r;
 131         BN_CTX *bn_ctx;
 132
 133         SCHNORR_DEBUG_BN((x, "%s: x = ", __func__));
 134         SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));
 135
 136         /* Avoid degenerate cases: g^0 yields a spoofable signature */
 137         if (BN_cmp(g_x, BN_value_one()) <= 0) {
 138                 error("%s: g_x < 1", __func__);
 139                 return -1;
 140         }
 141
 142         h = g_v = r = tmp = v = NULL;
 143         if ((bn_ctx = BN_CTX_new()) == NULL) {
 144                 error("%s: BN_CTX_new", __func__);
 145                 goto out;
 146         }
 147         if ((g_v = BN_new()) == NULL ||
 148             (r = BN_new()) == NULL ||
 149             (tmp = BN_new()) == NULL) {
 150                 error("%s: BN_new", __func__);
 151                 goto out;
 152         }
 153
 154         /*
 155          * v must be a random element of Zq, so 1 <= v < q
 156          * we also exclude v = 1, since g^1 looks dangerous
 157          */
 158         if ((v = bn_rand_range_gt_one(grp_p)) == NULL) {
 159                 error("%s: bn_rand_range2", __func__);
 160                 goto out;
 161         }
 162         SCHNORR_DEBUG_BN((v, "%s: v = ", __func__));
 163
 164         /* g_v = g^v mod p */
 165         if (BN_mod_exp(g_v, grp_g, v, grp_p, bn_ctx) == -1) {
 166                 error("%s: BN_mod_exp (g^v mod p)", __func__);
 167                 goto out;
 168         }
 169         SCHNORR_DEBUG_BN((g_v, "%s: g_v = ", __func__));
 170
 171         /* h = H(g || g^v || g^x || id) */
 172         if ((h = schnorr_hash(grp_p, grp_q, grp_g, evp_md, g_v, g_x,
 173             id, idlen)) == NULL) {
 174                 error("%s: schnorr_hash failed", __func__);
 175                 goto out;
 176         }
 177
 178         /* r = v - xh mod q */
 179         if (BN_mod_mul(tmp, x, h, grp_q, bn_ctx) == -1) {
 180                 error("%s: BN_mod_mul (tmp = xv mod q)", __func__);
 181                 goto out;
 182         }
 183         if (BN_mod_sub(r, v, tmp, grp_q, bn_ctx) == -1) {
 184                 error("%s: BN_mod_mul (r = v - tmp)", __func__);
 185                 goto out;
 186         }
 187         SCHNORR_DEBUG_BN((g_v, "%s: e = ", __func__));
 188         SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));
 189
 190         *e_p = g_v;
 191         *r_p = r;
 192
 193         success = 0;
 194  out:
 195         BN_CTX_free(bn_ctx);
 196         if (h != NULL)
 197                 BN_clear_free(h);
 198         if (v != NULL)
 199                 BN_clear_free(v);
 200         BN_clear_free(tmp);
 201
 202         return success;
 203 }
 204
 205 /*
 206  * Generate Schnorr signature to prove knowledge of private value 'x' used
 207  * in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
 208  * using a SHA256 hash.
 209  * 'idlen' bytes from 'id' will be included in the signature hash as an anti-
 210  * replay salt.
 211  * On success, 0 is returned and *siglen bytes of signature are returned in
 212  * *sig (caller to free). Returns -1 on failure.
 213  */
 214 int
 215 schnorr_sign_buf(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
 216     const BIGNUM *x, const BIGNUM *g_x, const u_char *id, u_int idlen,
 217     u_char **sig, u_int *siglen)
 218 {
 219         Buffer b;
 220         BIGNUM *r, *e;
 221
 222         if (schnorr_sign(grp_p, grp_q, grp_g, EVP_sha256(),
 223             x, g_x, id, idlen, &r, &e) != 0)
 224                 return -1;
 225
 226         /* Signature is (e, r) */
 227         buffer_init(&b);
 228         /* XXX sigtype-hash as string? */
 229         buffer_put_bignum2(&b, e);
 230         buffer_put_bignum2(&b, r);
 231         *siglen = buffer_len(&b);
 232         *sig = xmalloc(*siglen);
 233         memcpy(*sig, buffer_ptr(&b), *siglen);
 234         SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
 235             "%s: sigblob", __func__));
 236         buffer_free(&b);
 237
 238         BN_clear_free(r);
 239         BN_clear_free(e);
 240
 241         return 0;
 242 }
 243
 244 /*
 245  * Verify Schnorr signature { r (v - xh mod q), e (g^v mod p) } against
 246  * public exponent g_x (g^x) under group defined by 'grp_p', 'grp_q' and
 247  * 'grp_g' using hash "evp_md".
 248  * Signature hash will be salted with 'idlen' bytes from 'id'.
 249  * Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
 250  */
 251 int
 252 schnorr_verify(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
 253     const EVP_MD *evp_md, const BIGNUM *g_x, const u_char *id, u_int idlen,
 254     const BIGNUM *r, const BIGNUM *e)
 255 {
 256         int success = -1;
 257         BIGNUM *h, *g_xh, *g_r, *expected;
 258         BN_CTX *bn_ctx;
 259
 260         SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));
 261
 262         /* Avoid degenerate cases: g^0 yields a spoofable signature */
 263         if (BN_cmp(g_x, BN_value_one()) <= 0) {
 264                 error("%s: g_x < 1", __func__);
 265                 return -1;
 266         }
 267
 268         h = g_xh = g_r = expected = NULL;
 269         if ((bn_ctx = BN_CTX_new()) == NULL) {
 270                 error("%s: BN_CTX_new", __func__);
 271                 goto out;
 272         }
 273         if ((g_xh = BN_new()) == NULL ||
 274             (g_r = BN_new()) == NULL ||
 275             (expected = BN_new()) == NULL) {
 276                 error("%s: BN_new", __func__);
 277                 goto out;
 278         }
 279
 280         SCHNORR_DEBUG_BN((e, "%s: e = ", __func__));
 281         SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));
 282
 283         /* h = H(g || g^v || g^x || id) */
 284         if ((h = schnorr_hash(grp_p, grp_q, grp_g, evp_md, e, g_x,
 285             id, idlen)) == NULL) {
 286                 error("%s: schnorr_hash failed", __func__);
 287                 goto out;
 288         }
 289
 290         /* g_xh = (g^x)^h */
 291         if (BN_mod_exp(g_xh, g_x, h, grp_p, bn_ctx) == -1) {
 292                 error("%s: BN_mod_exp (g_x^h mod p)", __func__);
 293                 goto out;
 294         }
 295         SCHNORR_DEBUG_BN((g_xh, "%s: g_xh = ", __func__));
 296
 297         /* g_r = g^r */
 298         if (BN_mod_exp(g_r, grp_g, r, grp_p, bn_ctx) == -1) {
 299                 error("%s: BN_mod_exp (g_x^h mod p)", __func__);
 300                 goto out;
 301         }
 302         SCHNORR_DEBUG_BN((g_r, "%s: g_r = ", __func__));
 303
 304         /* expected = g^r * g_xh */
 305         if (BN_mod_mul(expected, g_r, g_xh, grp_p, bn_ctx) == -1) {
 306                 error("%s: BN_mod_mul (expected = g_r mod p)", __func__);
 307                 goto out;
 308         }
 309         SCHNORR_DEBUG_BN((expected, "%s: expected = ", __func__));
 310
 311         /* Check e == expected */
 312         success = BN_cmp(expected, e) == 0;
 313  out:
 314         BN_CTX_free(bn_ctx);
 315         if (h != NULL)
 316                 BN_clear_free(h);
 317         BN_clear_free(g_xh);
 318         BN_clear_free(g_r);
 319         BN_clear_free(expected);
 320         return success;
 321 }
 322
 323 /*
 324  * Verify Schnorr signature 'sig' of length 'siglen' against public exponent
 325  * g_x (g^x) under group defined by 'grp_p', 'grp_q' and 'grp_g' using a
 326  * SHA256 hash.
 327  * Signature hash will be salted with 'idlen' bytes from 'id'.
 328  * Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
 329  */
 330 int
 331 schnorr_verify_buf(const BIGNUM *grp_p, const BIGNUM *grp_q,
 332     const BIGNUM *grp_g,
 333     const BIGNUM *g_x, const u_char *id, u_int idlen,
 334     const u_char *sig, u_int siglen)
 335 {
 336         Buffer b;
 337         int ret = -1;
 338         u_int rlen;
 339         BIGNUM *r, *e;
 340
 341         e = r = NULL;
 342         if ((e = BN_new()) == NULL ||
 343             (r = BN_new()) == NULL) {
 344                 error("%s: BN_new", __func__);
 345                 goto out;
 346         }
 347
 348         /* Extract g^v and r from signature blob */
 349         buffer_init(&b);
 350         buffer_append(&b, sig, siglen);
 351         SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
 352             "%s: sigblob", __func__));
 353         buffer_get_bignum2(&b, e);
 354         buffer_get_bignum2(&b, r);
 355         rlen = buffer_len(&b);
 356         buffer_free(&b);
 357         if (rlen != 0) {
 358                 error("%s: remaining bytes in signature %d", __func__, rlen);
 359                 goto out;
 360         }
 361
 362         ret = schnorr_verify(grp_p, grp_q, grp_g, EVP_sha256(),
 363             g_x, id, idlen, r, e);
 364  out:
 365         BN_clear_free(e);
 366         BN_clear_free(r);
 367
 368         return ret;
 369 }
 370
 371 /* Helper functions */
 372
 373 /*
 374  * Generate uniformly distributed random number in range (1, high).
 375  * Return number on success, NULL on failure.
 376  */
 377 BIGNUM *
 378 bn_rand_range_gt_one(const BIGNUM *high)
 379 {
 380         BIGNUM *r, *tmp;
 381         int success = -1;
 382
 383         if ((tmp = BN_new()) == NULL) {
 384                 error("%s: BN_new", __func__);
 385                 return NULL;
 386         }
 387         if ((r = BN_new()) == NULL) {
 388                 error("%s: BN_new failed", __func__);
 389                 goto out;
 390         }
 391         if (BN_set_word(tmp, 2) != 1) {
 392                 error("%s: BN_set_word(tmp, 2)", __func__);
 393                 goto out;
 394         }
 395         if (BN_sub(tmp, high, tmp) == -1) {
 396                 error("%s: BN_sub failed (tmp = high - 2)", __func__);
 397                 goto out;
 398         }
 399         if (BN_rand_range(r, tmp) == -1) {
 400                 error("%s: BN_rand_range failed", __func__);
 401                 goto out;
 402         }
 403         if (BN_set_word(tmp, 2) != 1) {
 404                 error("%s: BN_set_word(tmp, 2)", __func__);
 405                 goto out;
 406         }
 407         if (BN_add(r, r, tmp) == -1) {
 408                 error("%s: BN_add failed (r = r + 2)", __func__);
 409                 goto out;
 410         }
 411         success = 0;
 412  out:
 413         BN_clear_free(tmp);
 414         if (success == 0)
 415                 return r;
 416         BN_clear_free(r);
 417         return NULL;
 418 }
 419
 420 /*
 421  * Hash contents of buffer 'b' with hash 'md'. Returns 0 on success,
 422  * with digest via 'digestp' (caller to free) and length via 'lenp'.
 423  * Returns -1 on failure.
 424  */
 425 int
 426 hash_buffer(const u_char *buf, u_int len, const EVP_MD *md,
 427     u_char **digestp, u_int *lenp)
 428 {
 429         u_char digest[EVP_MAX_MD_SIZE];
 430         u_int digest_len;
 431         EVP_MD_CTX evp_md_ctx;
 432         int success = -1;
 433
 434         EVP_MD_CTX_init(&evp_md_ctx);
 435
 436         if (EVP_DigestInit_ex(&evp_md_ctx, md, NULL) != 1) {
 437                 error("%s: EVP_DigestInit_ex", __func__);
 438                 goto out;
 439         }
 440         if (EVP_DigestUpdate(&evp_md_ctx, buf, len) != 1) {
 441                 error("%s: EVP_DigestUpdate", __func__);
 442                 goto out;
 443         }
 444         if (EVP_DigestFinal_ex(&evp_md_ctx, digest, &digest_len) != 1) {
 445                 error("%s: EVP_DigestFinal_ex", __func__);
 446                 goto out;
 447         }
 448         *digestp = xmalloc(digest_len);
 449         *lenp = digest_len;
 450         memcpy(*digestp, digest, *lenp);
 451         success = 0;
 452  out:
 453         EVP_MD_CTX_cleanup(&evp_md_ctx);
 454         bzero(digest, sizeof(digest));
 455         digest_len = 0;
 456         return success;
 457 }
 458
 459 /* print formatted string followed by bignum */
 460 void
 461 debug3_bn(const BIGNUM *n, const char *fmt, ...)
 462 {
 463         char *out, *h;
 464         va_list args;
 465
 466         out = NULL;
 467         va_start(args, fmt);
 468         vasprintf(&out, fmt, args);
 469         va_end(args);
 470         if (out == NULL)
 471                 fatal("%s: vasprintf failed", __func__);
 472
 473         if (n == NULL)
 474                 debug3("%s(null)", out);
 475         else {
 476                 h = BN_bn2hex(n);
 477                 debug3("%s0x%s", out, h);
 478                 free(h);
 479         }
 480         free(out);
 481 }
 482
 483 /* print formatted string followed by buffer contents in hex */
 484 void
 485 debug3_buf(const u_char *buf, u_int len, const char *fmt, ...)
 486 {
 487         char *out, h[65];
 488         u_int i, j;
 489         va_list args;
 490
 491         out = NULL;
 492         va_start(args, fmt);
 493         vasprintf(&out, fmt, args);
 494         va_end(args);
 495         if (out == NULL)
 496                 fatal("%s: vasprintf failed", __func__);
 497
 498         debug3("%s length %u%s", out, len, buf == NULL ? " (null)" : "");
 499         free(out);
 500         if (buf == NULL)
 501                 return;
 502
 503         *h = '\0';
 504         for (i = j = 0; i < len; i++) {
 505                 snprintf(h + j, sizeof(h) - j, "%02x", buf[i]);
 506                 j += 2;
 507                 if (j >= sizeof(h) - 1 || i == len - 1) {
 508                         debug3("    %s", h);
 509                         *h = '\0';
 510                         j = 0;
 511                 }
 512         }
 513 }
 514
 515 /*
 516  * Construct a MODP group from hex strings p (which must be a safe
 517  * prime) and g, automatically calculating subgroup q as (p / 2)
 518  */
 519 struct modp_group *
 520 modp_group_from_g_and_safe_p(const char *grp_g, const char *grp_p)
 521 {
 522         struct modp_group *ret;
 523
 524         ret = xmalloc(sizeof(*ret));
 525         ret->p = ret->q = ret->g = NULL;
 526         if (BN_hex2bn(&ret->p, grp_p) == 0 ||
 527             BN_hex2bn(&ret->g, grp_g) == 0)
 528                 fatal("%s: BN_hex2bn", __func__);
 529         /* Subgroup order is p/2 (p is a safe prime) */
 530         if ((ret->q = BN_new()) == NULL)
 531                 fatal("%s: BN_new", __func__);
 532         if (BN_rshift1(ret->q, ret->p) != 1)
 533                 fatal("%s: BN_rshift1", __func__);
 534
 535         return ret;
 536 }
 537
 538 void
 539 modp_group_free(struct modp_group *grp)
 540 {
 541         if (grp->g != NULL)
 542                 BN_clear_free(grp->g);
 543         if (grp->p != NULL)
 544                 BN_clear_free(grp->p);
 545         if (grp->q != NULL)
 546                 BN_clear_free(grp->q);
 547         bzero(grp, sizeof(*grp));
 548         xfree(grp);
 549 }
 550
 551 /* main() function for self-test */
 552
 553 #ifdef SCHNORR_MAIN
 554 static void
 555 schnorr_selftest_one(const BIGNUM *grp_p, const BIGNUM *grp_q,
 556     const BIGNUM *grp_g, const BIGNUM *x)
 557 {
 558         BIGNUM *g_x;
 559         u_char *sig;
 560         u_int siglen;
 561         BN_CTX *bn_ctx;
 562
 563         if ((bn_ctx = BN_CTX_new()) == NULL)
 564                 fatal("%s: BN_CTX_new", __func__);
 565         if ((g_x = BN_new()) == NULL)
 566                 fatal("%s: BN_new", __func__);
 567
 568         if (BN_mod_exp(g_x, grp_g, x, grp_p, bn_ctx) == -1)
 569                 fatal("%s: g_x", __func__);
 570         if (schnorr_sign_buf(grp_p, grp_q, grp_g, x, g_x, "junk", 4,
 571             &sig, &siglen))
 572                 fatal("%s: schnorr_sign", __func__);
 573         if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
 574             sig, siglen) != 1)
 575                 fatal("%s: verify fail", __func__);
 576         if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "JUNK", 4,
 577             sig, siglen) != 0)
 578                 fatal("%s: verify should have failed (bad ID)", __func__);
 579         sig[4] ^= 1;
 580         if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
 581             sig, siglen) != 0)
 582                 fatal("%s: verify should have failed (bit error)", __func__);
 583         xfree(sig);
 584         BN_free(g_x);
 585         BN_CTX_free(bn_ctx);
 586 }
 587
 588 static void
 589 schnorr_selftest(void)
 590 {
 591         BIGNUM *x;
 592         struct modp_group *grp;
 593         u_int i;
 594         char *hh;
 595
 596         grp = jpake_default_group();
 597         if ((x = BN_new()) == NULL)
 598                 fatal("%s: BN_new", __func__);
 599         SCHNORR_DEBUG_BN((grp->p, "%s: grp->p = ", __func__));
 600         SCHNORR_DEBUG_BN((grp->q, "%s: grp->q = ", __func__));
 601         SCHNORR_DEBUG_BN((grp->g, "%s: grp->g = ", __func__));
 602
 603         /* [1, 20) */
 604         for (i = 1; i < 20; i++) {
 605                 printf("x = %u\n", i);
 606                 fflush(stdout);
 607                 if (BN_set_word(x, i) != 1)
 608                         fatal("%s: set x word", __func__);
 609                 schnorr_selftest_one(grp->p, grp->q, grp->g, x);
 610         }
 611
 612         /* 100 x random [0, p) */
 613         for (i = 0; i < 100; i++) {
 614                 if (BN_rand_range(x, grp->p) != 1)
 615                         fatal("%s: BN_rand_range", __func__);
 616                 hh = BN_bn2hex(x);
 617                 printf("x = (random) 0x%s\n", hh);
 618                 free(hh);
 619                 fflush(stdout);
 620                 schnorr_selftest_one(grp->p, grp->q, grp->g, x);
 621         }
 622
 623         /* [q-20, q) */
 624         if (BN_set_word(x, 20) != 1)
 625                 fatal("%s: BN_set_word (x = 20)", __func__);
 626         if (BN_sub(x, grp->q, x) != 1)
 627                 fatal("%s: BN_sub (q - x)", __func__);
 628         for (i = 0; i < 19; i++) {
 629                 hh = BN_bn2hex(x);
 630                 printf("x = (q - %d) 0x%s\n", 20 - i, hh);
 631                 free(hh);
 632                 fflush(stdout);
 633                 schnorr_selftest_one(grp->p, grp->q, grp->g, x);
 634                 if (BN_add(x, x, BN_value_one()) != 1)
 635                         fatal("%s: BN_add (x + 1)", __func__);
 636         }
 637         BN_free(x);
 638 }
 639
 640 int
 641 main(int argc, char **argv)
 642 {
 643         log_init(argv[0], SYSLOG_LEVEL_DEBUG3, SYSLOG_FACILITY_USER, 1);
 644
 645         schnorr_selftest();
 646         return 0;
 647 }
 648 #endif
 649