[openssh.git] / schnorr.c

/* $OpenBSD: schnorr.c,v 1.3 2009/03/05 07:18:19 djm Exp $ */
/*
 * Copyright (c) 2008 Damien Miller.  All rights reserved.
 *
 * Permission to use, copy, modify, and distribute this software for any
 * purpose with or without fee is hereby granted, provided that the above
 * copyright notice and this permission notice appear in all copies.
 *
 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
 */

/*
 * Implementation of Schnorr signatures / zero-knowledge proofs, based on
 * description in:
 * 	
 * F. Hao, P. Ryan, "Password Authenticated Key Exchange by Juggling",
 * 16th Workshop on Security Protocols, Cambridge, April 2008
 *
 * http://grouper.ieee.org/groups/1363/Research/contributions/hao-ryan-2008.pdf
 */

#include "includes.h"

#include <sys/types.h>

#include <string.h>
#include <stdarg.h>
#include <stdio.h>

#include <openssl/evp.h>
#include <openssl/bn.h>

#include "xmalloc.h"
#include "buffer.h"
#include "log.h"

#include "schnorr.h"

/* #define SCHNORR_DEBUG */		/* Privacy-violating debugging */
/* #define SCHNORR_MAIN */		/* Include main() selftest */

#ifndef SCHNORR_DEBUG
# define SCHNORR_DEBUG_BN(a)
# define SCHNORR_DEBUG_BUF(a)
#else
# define SCHNORR_DEBUG_BN(a)	debug3_bn a
# define SCHNORR_DEBUG_BUF(a)	debug3_buf a
#endif /* SCHNORR_DEBUG */

/*
 * Calculate hash component of Schnorr signature H(g || g^v || g^x || id)
 * using the hash function defined by "evp_md". Returns signature as
 * bignum or NULL on error.
 */
static BIGNUM *
schnorr_hash(const BIGNUM *p, const BIGNUM *q, const BIGNUM *g,
    const EVP_MD *evp_md, const BIGNUM *g_v, const BIGNUM *g_x,
    const u_char *id, u_int idlen)
{
	u_char *digest;
	u_int digest_len;
	BIGNUM *h;
	Buffer b;
	int success = -1;

	if ((h = BN_new()) == NULL) {
		error("%s: BN_new", __func__);
		return NULL;
	}

	buffer_init(&b);

	/* h = H(g || p || q || g^v || g^x || id) */
	buffer_put_bignum2(&b, g);
	buffer_put_bignum2(&b, p);
	buffer_put_bignum2(&b, q);
	buffer_put_bignum2(&b, g_v);
	buffer_put_bignum2(&b, g_x);
	buffer_put_string(&b, id, idlen);

	SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
	    "%s: hashblob", __func__));
	if (hash_buffer(buffer_ptr(&b), buffer_len(&b), evp_md,
	    &digest, &digest_len) != 0) {
		error("%s: hash_buffer", __func__);
		goto out;
	}
	if (BN_bin2bn(digest, (int)digest_len, h) == NULL) {
		error("%s: BN_bin2bn", __func__);
		goto out;
	}
	success = 0;
	SCHNORR_DEBUG_BN((h, "%s: h = ", __func__));
 out:
	buffer_free(&b);
	bzero(digest, digest_len);
	xfree(digest);
	digest_len = 0;
	if (success == 0)
		return h;
	BN_clear_free(h);
	return NULL;
}

/*
 * Generate Schnorr signature to prove knowledge of private value 'x' used
 * in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
 * using the hash function "evp_md".
 * 'idlen' bytes from 'id' will be included in the signature hash as an anti-
 * replay salt.
 * 
 * On success, 0 is returned. The signature values are returned as *e_p
 * (g^v mod p) and *r_p (v - xh mod q). The caller must free these values.
 * On failure, -1 is returned.
 */
int
schnorr_sign(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
    const EVP_MD *evp_md, const BIGNUM *x, const BIGNUM *g_x,
    const u_char *id, u_int idlen, BIGNUM **r_p, BIGNUM **e_p)
{
	int success = -1;
	BIGNUM *h, *tmp, *v, *g_v, *r;
	BN_CTX *bn_ctx;

	SCHNORR_DEBUG_BN((x, "%s: x = ", __func__));
	SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));

	/* Avoid degenerate cases: g^0 yields a spoofable signature */
	if (BN_cmp(g_x, BN_value_one()) <= 0) {
		error("%s: g_x < 1", __func__);
		return -1;
	}

	h = g_v = r = tmp = v = NULL;
	if ((bn_ctx = BN_CTX_new()) == NULL) {
		error("%s: BN_CTX_new", __func__);
		goto out;
	}
	if ((g_v = BN_new()) == NULL ||
	    (r = BN_new()) == NULL ||
	    (tmp = BN_new()) == NULL) {
		error("%s: BN_new", __func__);
		goto out;
	}

	/*
	 * v must be a random element of Zq, so 1 <= v < q
	 * we also exclude v = 1, since g^1 looks dangerous
	 */
	if ((v = bn_rand_range_gt_one(grp_p)) == NULL) {
		error("%s: bn_rand_range2", __func__);
		goto out;
	}
	SCHNORR_DEBUG_BN((v, "%s: v = ", __func__));

	/* g_v = g^v mod p */
	if (BN_mod_exp(g_v, grp_g, v, grp_p, bn_ctx) == -1) {
		error("%s: BN_mod_exp (g^v mod p)", __func__);
		goto out;
	}
	SCHNORR_DEBUG_BN((g_v, "%s: g_v = ", __func__));

	/* h = H(g || g^v || g^x || id) */
	if ((h = schnorr_hash(grp_p, grp_q, grp_g, evp_md, g_v, g_x,
	    id, idlen)) == NULL) {
		error("%s: schnorr_hash failed", __func__);
		goto out;
	}

	/* r = v - xh mod q */
	if (BN_mod_mul(tmp, x, h, grp_q, bn_ctx) == -1) {
		error("%s: BN_mod_mul (tmp = xv mod q)", __func__);
		goto out;
	}
	if (BN_mod_sub(r, v, tmp, grp_q, bn_ctx) == -1) {
		error("%s: BN_mod_mul (r = v - tmp)", __func__);
		goto out;
	}
	SCHNORR_DEBUG_BN((g_v, "%s: e = ", __func__));
	SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));

	*e_p = g_v;
	*r_p = r;

	success = 0;
 out:
	BN_CTX_free(bn_ctx);
	if (h != NULL)
		BN_clear_free(h);
	if (v != NULL)
		BN_clear_free(v);
	BN_clear_free(tmp);

	return success;
}

/*
 * Generate Schnorr signature to prove knowledge of private value 'x' used
 * in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
 * using a SHA256 hash.
 * 'idlen' bytes from 'id' will be included in the signature hash as an anti-
 * replay salt.
 * On success, 0 is returned and *siglen bytes of signature are returned in
 * *sig (caller to free). Returns -1 on failure.
 */
int
schnorr_sign_buf(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
    const BIGNUM *x, const BIGNUM *g_x, const u_char *id, u_int idlen,
    u_char **sig, u_int *siglen)
{
	Buffer b;
	BIGNUM *r, *e;

	if (schnorr_sign(grp_p, grp_q, grp_g, EVP_sha256(),
	    x, g_x, id, idlen, &r, &e) != 0)
		return -1;

	/* Signature is (e, r) */
	buffer_init(&b);
	/* XXX sigtype-hash as string? */
	buffer_put_bignum2(&b, e);
	buffer_put_bignum2(&b, r);
	*siglen = buffer_len(&b);
	*sig = xmalloc(*siglen);
	memcpy(*sig, buffer_ptr(&b), *siglen);
	SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
	    "%s: sigblob", __func__));
	buffer_free(&b);

	BN_clear_free(r);
	BN_clear_free(e);

	return 0;
}

/*
 * Verify Schnorr signature { r (v - xh mod q), e (g^v mod p) } against
 * public exponent g_x (g^x) under group defined by 'grp_p', 'grp_q' and
 * 'grp_g' using hash "evp_md".
 * Signature hash will be salted with 'idlen' bytes from 'id'.
 * Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
 */
int
schnorr_verify(const BIGNUM *grp_p, const BIGNUM *grp_q, const BIGNUM *grp_g,
    const EVP_MD *evp_md, const BIGNUM *g_x, const u_char *id, u_int idlen,
    const BIGNUM *r, const BIGNUM *e)
{
	int success = -1;
	BIGNUM *h, *g_xh, *g_r, *expected;
	BN_CTX *bn_ctx;

	SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));

	/* Avoid degenerate cases: g^0 yields a spoofable signature */
	if (BN_cmp(g_x, BN_value_one()) <= 0) {
		error("%s: g_x < 1", __func__);
		return -1;
	}

	h = g_xh = g_r = expected = NULL;
	if ((bn_ctx = BN_CTX_new()) == NULL) {
		error("%s: BN_CTX_new", __func__);
		goto out;
	}
	if ((g_xh = BN_new()) == NULL ||
	    (g_r = BN_new()) == NULL ||
	    (expected = BN_new()) == NULL) {
		error("%s: BN_new", __func__);
		goto out;
	}

	SCHNORR_DEBUG_BN((e, "%s: e = ", __func__));
	SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));

	/* h = H(g || g^v || g^x || id) */
	if ((h = schnorr_hash(grp_p, grp_q, grp_g, evp_md, e, g_x,
	    id, idlen)) == NULL) {
		error("%s: schnorr_hash failed", __func__);
		goto out;
	}

	/* g_xh = (g^x)^h */
	if (BN_mod_exp(g_xh, g_x, h, grp_p, bn_ctx) == -1) {
		error("%s: BN_mod_exp (g_x^h mod p)", __func__);
		goto out;
	}
	SCHNORR_DEBUG_BN((g_xh, "%s: g_xh = ", __func__));

	/* g_r = g^r */
	if (BN_mod_exp(g_r, grp_g, r, grp_p, bn_ctx) == -1) {
		error("%s: BN_mod_exp (g_x^h mod p)", __func__);
		goto out;
	}
	SCHNORR_DEBUG_BN((g_r, "%s: g_r = ", __func__));

	/* expected = g^r * g_xh */
	if (BN_mod_mul(expected, g_r, g_xh, grp_p, bn_ctx) == -1) {
		error("%s: BN_mod_mul (expected = g_r mod p)", __func__);
		goto out;
	}
	SCHNORR_DEBUG_BN((expected, "%s: expected = ", __func__));

	/* Check e == expected */
	success = BN_cmp(expected, e) == 0;
 out:
	BN_CTX_free(bn_ctx);
	if (h != NULL)
		BN_clear_free(h);
	BN_clear_free(g_xh);
	BN_clear_free(g_r);
	BN_clear_free(expected);
	return success;
}

/*
 * Verify Schnorr signature 'sig' of length 'siglen' against public exponent
 * g_x (g^x) under group defined by 'grp_p', 'grp_q' and 'grp_g' using a
 * SHA256 hash.
 * Signature hash will be salted with 'idlen' bytes from 'id'.
 * Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
 */
int
schnorr_verify_buf(const BIGNUM *grp_p, const BIGNUM *grp_q,
    const BIGNUM *grp_g,
    const BIGNUM *g_x, const u_char *id, u_int idlen,
    const u_char *sig, u_int siglen)
{
	Buffer b;
	int ret = -1;
	u_int rlen;
	BIGNUM *r, *e;

	e = r = NULL;
	if ((e = BN_new()) == NULL ||
	    (r = BN_new()) == NULL) {
		error("%s: BN_new", __func__);
		goto out;
	}

	/* Extract g^v and r from signature blob */
	buffer_init(&b);
	buffer_append(&b, sig, siglen);
	SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
	    "%s: sigblob", __func__));
	buffer_get_bignum2(&b, e);
	buffer_get_bignum2(&b, r);
	rlen = buffer_len(&b);
	buffer_free(&b);
	if (rlen != 0) {
		error("%s: remaining bytes in signature %d", __func__, rlen);
		goto out;
	}

	ret = schnorr_verify(grp_p, grp_q, grp_g, EVP_sha256(),
	    g_x, id, idlen, r, e);
 out:
	BN_clear_free(e);
	BN_clear_free(r);

	return ret;
}

/* Helper functions */

/*
 * Generate uniformly distributed random number in range (1, high).
 * Return number on success, NULL on failure.
 */
BIGNUM *
bn_rand_range_gt_one(const BIGNUM *high)
{
	BIGNUM *r, *tmp;
	int success = -1;

	if ((tmp = BN_new()) == NULL) {
		error("%s: BN_new", __func__);
		return NULL;
	}
	if ((r = BN_new()) == NULL) {
		error("%s: BN_new failed", __func__);
		goto out;
	}
	if (BN_set_word(tmp, 2) != 1) {
		error("%s: BN_set_word(tmp, 2)", __func__);
		goto out;
	}
	if (BN_sub(tmp, high, tmp) == -1) {
		error("%s: BN_sub failed (tmp = high - 2)", __func__);
		goto out;
	}
	if (BN_rand_range(r, tmp) == -1) {
		error("%s: BN_rand_range failed", __func__);
		goto out;
	}
	if (BN_set_word(tmp, 2) != 1) {
		error("%s: BN_set_word(tmp, 2)", __func__);
		goto out;
	}
	if (BN_add(r, r, tmp) == -1) {
		error("%s: BN_add failed (r = r + 2)", __func__);
		goto out;
	}
	success = 0;
 out:
	BN_clear_free(tmp);
	if (success == 0)
		return r;
	BN_clear_free(r);
	return NULL;
}

/*
 * Hash contents of buffer 'b' with hash 'md'. Returns 0 on success,
 * with digest via 'digestp' (caller to free) and length via 'lenp'.
 * Returns -1 on failure.
 */
int
hash_buffer(const u_char *buf, u_int len, const EVP_MD *md,
    u_char **digestp, u_int *lenp)
{
	u_char digest[EVP_MAX_MD_SIZE];
	u_int digest_len;
	EVP_MD_CTX evp_md_ctx;
	int success = -1;

	EVP_MD_CTX_init(&evp_md_ctx);

	if (EVP_DigestInit_ex(&evp_md_ctx, md, NULL) != 1) {
		error("%s: EVP_DigestInit_ex", __func__);
		goto out;
	}
	if (EVP_DigestUpdate(&evp_md_ctx, buf, len) != 1) {
		error("%s: EVP_DigestUpdate", __func__);
		goto out;
	}
	if (EVP_DigestFinal_ex(&evp_md_ctx, digest, &digest_len) != 1) {
		error("%s: EVP_DigestFinal_ex", __func__);
		goto out;
	}
	*digestp = xmalloc(digest_len);
	*lenp = digest_len;
	memcpy(*digestp, digest, *lenp);
	success = 0;
 out:
	EVP_MD_CTX_cleanup(&evp_md_ctx);
	bzero(digest, sizeof(digest));
	digest_len = 0;
	return success;
}

/* print formatted string followed by bignum */
void
debug3_bn(const BIGNUM *n, const char *fmt, ...)
{
	char *out, *h;
	va_list args;

	out = NULL;
	va_start(args, fmt);
	vasprintf(&out, fmt, args);
	va_end(args);
	if (out == NULL)
		fatal("%s: vasprintf failed", __func__);

	if (n == NULL)
		debug3("%s(null)", out);
	else {
		h = BN_bn2hex(n);
		debug3("%s0x%s", out, h);
		free(h);
	}
	free(out);
}

/* print formatted string followed by buffer contents in hex */
void
debug3_buf(const u_char *buf, u_int len, const char *fmt, ...)
{
	char *out, h[65];
	u_int i, j;
	va_list args;

	out = NULL;
	va_start(args, fmt);
	vasprintf(&out, fmt, args);
	va_end(args);
	if (out == NULL)
		fatal("%s: vasprintf failed", __func__);

	debug3("%s length %u%s", out, len, buf == NULL ? " (null)" : "");
	free(out);
	if (buf == NULL)
		return;

	*h = '\0';
	for (i = j = 0; i < len; i++) {
		snprintf(h + j, sizeof(h) - j, "%02x", buf[i]);
		j += 2;
		if (j >= sizeof(h) - 1 || i == len - 1) {
			debug3("    %s", h);
			*h = '\0';
			j = 0;
		}
	}
}

/*
 * Construct a MODP group from hex strings p (which must be a safe
 * prime) and g, automatically calculating subgroup q as (p / 2)
 */
struct modp_group *
modp_group_from_g_and_safe_p(const char *grp_g, const char *grp_p)
{
	struct modp_group *ret;

	ret = xmalloc(sizeof(*ret));
	ret->p = ret->q = ret->g = NULL;
	if (BN_hex2bn(&ret->p, grp_p) == 0 ||
	    BN_hex2bn(&ret->g, grp_g) == 0)
		fatal("%s: BN_hex2bn", __func__);
	/* Subgroup order is p/2 (p is a safe prime) */
	if ((ret->q = BN_new()) == NULL)
		fatal("%s: BN_new", __func__);
	if (BN_rshift1(ret->q, ret->p) != 1)
		fatal("%s: BN_rshift1", __func__);

	return ret;
}

void
modp_group_free(struct modp_group *grp)
{
	if (grp->g != NULL)
		BN_clear_free(grp->g);
	if (grp->p != NULL)
		BN_clear_free(grp->p);
	if (grp->q != NULL)
		BN_clear_free(grp->q);
	bzero(grp, sizeof(*grp));
	xfree(grp);
}

/* main() function for self-test */

#ifdef SCHNORR_MAIN
static void
schnorr_selftest_one(const BIGNUM *grp_p, const BIGNUM *grp_q,
    const BIGNUM *grp_g, const BIGNUM *x)
{
	BIGNUM *g_x;
	u_char *sig;
	u_int siglen;
	BN_CTX *bn_ctx;

	if ((bn_ctx = BN_CTX_new()) == NULL)
		fatal("%s: BN_CTX_new", __func__);
	if ((g_x = BN_new()) == NULL)
		fatal("%s: BN_new", __func__);

	if (BN_mod_exp(g_x, grp_g, x, grp_p, bn_ctx) == -1)
		fatal("%s: g_x", __func__);
	if (schnorr_sign_buf(grp_p, grp_q, grp_g, x, g_x, "junk", 4,
	    &sig, &siglen))
		fatal("%s: schnorr_sign", __func__);
	if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
	    sig, siglen) != 1)
		fatal("%s: verify fail", __func__);
	if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "JUNK", 4,
	    sig, siglen) != 0)
		fatal("%s: verify should have failed (bad ID)", __func__);
	sig[4] ^= 1;
	if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
	    sig, siglen) != 0)
		fatal("%s: verify should have failed (bit error)", __func__);
	xfree(sig);
	BN_free(g_x);
	BN_CTX_free(bn_ctx);
}

static void
schnorr_selftest(void)
{
	BIGNUM *x;
	struct modp_group *grp;
	u_int i;
	char *hh;

	grp = jpake_default_group();
	if ((x = BN_new()) == NULL)
		fatal("%s: BN_new", __func__);
	SCHNORR_DEBUG_BN((grp->p, "%s: grp->p = ", __func__));
	SCHNORR_DEBUG_BN((grp->q, "%s: grp->q = ", __func__));
	SCHNORR_DEBUG_BN((grp->g, "%s: grp->g = ", __func__));

	/* [1, 20) */
	for (i = 1; i < 20; i++) {
		printf("x = %u\n", i);
		fflush(stdout);
		if (BN_set_word(x, i) != 1)
			fatal("%s: set x word", __func__);
		schnorr_selftest_one(grp->p, grp->q, grp->g, x);
	}

	/* 100 x random [0, p) */
	for (i = 0; i < 100; i++) {
		if (BN_rand_range(x, grp->p) != 1)
			fatal("%s: BN_rand_range", __func__);
		hh = BN_bn2hex(x);
		printf("x = (random) 0x%s\n", hh);
		free(hh);
		fflush(stdout);
		schnorr_selftest_one(grp->p, grp->q, grp->g, x);
	}

	/* [q-20, q) */
	if (BN_set_word(x, 20) != 1)
		fatal("%s: BN_set_word (x = 20)", __func__);
	if (BN_sub(x, grp->q, x) != 1)
		fatal("%s: BN_sub (q - x)", __func__);
	for (i = 0; i < 19; i++) {
		hh = BN_bn2hex(x);
		printf("x = (q - %d) 0x%s\n", 20 - i, hh);
		free(hh);
		fflush(stdout);
		schnorr_selftest_one(grp->p, grp->q, grp->g, x);
		if (BN_add(x, x, BN_value_one()) != 1)
			fatal("%s: BN_add (x + 1)", __func__);
	}
	BN_free(x);
}

int
main(int argc, char **argv)
{
	log_init(argv[0], SYSLOG_LEVEL_DEBUG3, SYSLOG_FACILITY_USER, 1);

	schnorr_selftest();
	return 0;
}
#endif
Commit	Line	Data
5b01421b	1	/* $OpenBSD: schnorr.c,v 1.3 2009/03/05 07:18:19 djm Exp $ */
5adf6b9a	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
5b01421b	43	#include "schnorr.h"
5adf6b9a	44
	45	/* #define SCHNORR_DEBUG / / Privacy-violating debugging */
	46	/* #define SCHNORR_MAIN / / Include main() selftest */
	47
5adf6b9a	48	#ifndef SCHNORR_DEBUG
	49	# define SCHNORR_DEBUG_BN(a)
	50	# define SCHNORR_DEBUG_BUF(a)
	51	#else
5b01421b	52	# define SCHNORR_DEBUG_BN(a) debug3_bn a
5b01421b	53	# define SCHNORR_DEBUG_BUF(a) debug3_buf a
5adf6b9a	54	#endif /* SCHNORR_DEBUG */
	55
	56	/*
	57	* Calculate hash component of Schnorr signature H(g \|\| g^v \|\| g^x \|\| id)
5b01421b	58	* using the hash function defined by "evp_md". Returns signature as
5b01421b	59	* bignum or NULL on error.
5adf6b9a	60	*/
	61	static BIGNUM *
	62	schnorr_hash(const BIGNUM p, const BIGNUM q, const BIGNUM *g,
5b01421b	63	const EVP_MD evp_md, const BIGNUM g_v, const BIGNUM *g_x,
5adf6b9a	64	const u_char *id, u_int idlen)
	65	{
	66	u_char *digest;
	67	u_int digest_len;
	68	BIGNUM *h;
5adf6b9a	69	Buffer b;
	70	int success = -1;
	71
	72	if ((h = BN_new()) == NULL) {
	73	error("%s: BN_new", __func__);
	74	return NULL;
	75	}
	76
	77	buffer_init(&b);
5adf6b9a	78
dca75d4b	79	/* h = H(g \|\| p \|\| q \|\| g^v \|\| g^x \|\| id) */
5adf6b9a	80	buffer_put_bignum2(&b, g);
dca75d4b	81	buffer_put_bignum2(&b, p);
dca75d4b	82	buffer_put_bignum2(&b, q);
5adf6b9a	83	buffer_put_bignum2(&b, g_v);
	84	buffer_put_bignum2(&b, g_x);
	85	buffer_put_string(&b, id, idlen);
	86
	87	SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
	88	"%s: hashblob", __func__));
5b01421b	89	if (hash_buffer(buffer_ptr(&b), buffer_len(&b), evp_md,
5adf6b9a	90	&digest, &digest_len) != 0) {
	91	error("%s: hash_buffer", __func__);
	92	goto out;
	93	}
	94	if (BN_bin2bn(digest, (int)digest_len, h) == NULL) {
	95	error("%s: BN_bin2bn", __func__);
	96	goto out;
	97	}
	98	success = 0;
	99	SCHNORR_DEBUG_BN((h, "%s: h = ", __func__));
	100	out:
	101	buffer_free(&b);
5adf6b9a	102	bzero(digest, digest_len);
	103	xfree(digest);
	104	digest_len = 0;
	105	if (success == 0)
	106	return h;
	107	BN_clear_free(h);
	108	return NULL;
	109	}
	110
	111	/*
	112	* Generate Schnorr signature to prove knowledge of private value 'x' used
	113	* in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
5b01421b	114	* using the hash function "evp_md".
5adf6b9a	115	* 'idlen' bytes from 'id' will be included in the signature hash as an anti-
5adf6b9a	116	* replay salt.
5b01421b	117	*
	118	* On success, 0 is returned. The signature values are returned as *e_p
	119	* (g^v mod p) and *r_p (v - xh mod q). The caller must free these values.
	120	* On failure, -1 is returned.
5adf6b9a	121	*/
	122	int
	123	schnorr_sign(const BIGNUM grp_p, const BIGNUM grp_q, const BIGNUM *grp_g,
5b01421b	124	const EVP_MD evp_md, const BIGNUM x, const BIGNUM *g_x,
5b01421b	125	const u_char id, u_int idlen, BIGNUM r_p, BIGNUM *e_p)
5adf6b9a	126	{
5adf6b9a	127	int success = -1;
5adf6b9a	128	BIGNUM h, tmp, v, g_v, *r;
	129	BN_CTX *bn_ctx;
	130
	131	SCHNORR_DEBUG_BN((x, "%s: x = ", __func__));
	132	SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));
	133
	134	/* Avoid degenerate cases: g^0 yields a spoofable signature */
	135	if (BN_cmp(g_x, BN_value_one()) <= 0) {
	136	error("%s: g_x < 1", __func__);
	137	return -1;
	138	}
	139
	140	h = g_v = r = tmp = v = NULL;
	141	if ((bn_ctx = BN_CTX_new()) == NULL) {
	142	error("%s: BN_CTX_new", __func__);
	143	goto out;
	144	}
	145	if ((g_v = BN_new()) == NULL \|\|
	146	(r = BN_new()) == NULL \|\|
	147	(tmp = BN_new()) == NULL) {
	148	error("%s: BN_new", __func__);
	149	goto out;
	150	}
	151
	152	/*
	153	* v must be a random element of Zq, so 1 <= v < q
	154	* we also exclude v = 1, since g^1 looks dangerous
	155	*/
	156	if ((v = bn_rand_range_gt_one(grp_p)) == NULL) {
	157	error("%s: bn_rand_range2", __func__);
	158	goto out;
	159	}
	160	SCHNORR_DEBUG_BN((v, "%s: v = ", __func__));
	161
	162	/* g_v = g^v mod p */
	163	if (BN_mod_exp(g_v, grp_g, v, grp_p, bn_ctx) == -1) {
	164	error("%s: BN_mod_exp (g^v mod p)", __func__);
	165	goto out;
	166	}
	167	SCHNORR_DEBUG_BN((g_v, "%s: g_v = ", __func__));
	168
	169	/* h = H(g \|\| g^v \|\| g^x \|\| id) */
5b01421b	170	if ((h = schnorr_hash(grp_p, grp_q, grp_g, evp_md, g_v, g_x,
5adf6b9a	171	id, idlen)) == NULL) {
	172	error("%s: schnorr_hash failed", __func__);
	173	goto out;
	174	}
	175
	176	/* r = v - xh mod q */
	177	if (BN_mod_mul(tmp, x, h, grp_q, bn_ctx) == -1) {
	178	error("%s: BN_mod_mul (tmp = xv mod q)", __func__);
	179	goto out;
	180	}
	181	if (BN_mod_sub(r, v, tmp, grp_q, bn_ctx) == -1) {
	182	error("%s: BN_mod_mul (r = v - tmp)", __func__);
	183	goto out;
	184	}
5b01421b	185	SCHNORR_DEBUG_BN((g_v, "%s: e = ", __func__));
5adf6b9a	186	SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));
5adf6b9a	187
5b01421b	188	*e_p = g_v;
	189	*r_p = r;
	190
	191	success = 0;
	192	out:
	193	BN_CTX_free(bn_ctx);
	194	if (h != NULL)
	195	BN_clear_free(h);
	196	if (v != NULL)
	197	BN_clear_free(v);
	198	BN_clear_free(tmp);
	199
	200	return success;
	201	}
	202
	203	/*
	204	* Generate Schnorr signature to prove knowledge of private value 'x' used
	205	* in public exponent g^x, under group defined by 'grp_p', 'grp_q' and 'grp_g'
	206	* using a SHA256 hash.
	207	* 'idlen' bytes from 'id' will be included in the signature hash as an anti-
	208	* replay salt.
	209	* On success, 0 is returned and *siglen bytes of signature are returned in
	210	* *sig (caller to free). Returns -1 on failure.
	211	*/
	212	int
	213	schnorr_sign_buf(const BIGNUM grp_p, const BIGNUM grp_q, const BIGNUM *grp_g,
	214	const BIGNUM x, const BIGNUM g_x, const u_char *id, u_int idlen,
	215	u_char *sig, u_int siglen)
	216	{
	217	Buffer b;
	218	BIGNUM r, e;
	219
	220	if (schnorr_sign(grp_p, grp_q, grp_g, EVP_sha256(),
	221	x, g_x, id, idlen, &r, &e) != 0)
	222	return -1;
	223
	224	/* Signature is (e, r) */
5adf6b9a	225	buffer_init(&b);
5adf6b9a	226	/* XXX sigtype-hash as string? */
5b01421b	227	buffer_put_bignum2(&b, e);
5adf6b9a	228	buffer_put_bignum2(&b, r);
	229	*siglen = buffer_len(&b);
	230	sig = xmalloc(siglen);
	231	memcpy(sig, buffer_ptr(&b), siglen);
	232	SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
	233	"%s: sigblob", __func__));
	234	buffer_free(&b);
5b01421b	235
5adf6b9a	236	BN_clear_free(r);
5b01421b	237	BN_clear_free(e);
5adf6b9a	238
5b01421b	239	return 0;
5adf6b9a	240	}
	241
	242	/*
5b01421b	243	* Verify Schnorr signature { r (v - xh mod q), e (g^v mod p) } against
	244	* public exponent g_x (g^x) under group defined by 'grp_p', 'grp_q' and
	245	* 'grp_g' using hash "evp_md".
5adf6b9a	246	* Signature hash will be salted with 'idlen' bytes from 'id'.
	247	* Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
	248	*/
	249	int
	250	schnorr_verify(const BIGNUM grp_p, const BIGNUM grp_q, const BIGNUM *grp_g,
5b01421b	251	const EVP_MD evp_md, const BIGNUM g_x, const u_char *id, u_int idlen,
5b01421b	252	const BIGNUM r, const BIGNUM e)
5adf6b9a	253	{
5adf6b9a	254	int success = -1;
5b01421b	255	BIGNUM h, g_xh, g_r, expected;
5adf6b9a	256	BN_CTX *bn_ctx;
5adf6b9a	257
	258	SCHNORR_DEBUG_BN((g_x, "%s: g_x = ", __func__));
	259
	260	/* Avoid degenerate cases: g^0 yields a spoofable signature */
	261	if (BN_cmp(g_x, BN_value_one()) <= 0) {
	262	error("%s: g_x < 1", __func__);
	263	return -1;
	264	}
	265
5b01421b	266	h = g_xh = g_r = expected = NULL;
5adf6b9a	267	if ((bn_ctx = BN_CTX_new()) == NULL) {
	268	error("%s: BN_CTX_new", __func__);
	269	goto out;
	270	}
5b01421b	271	if ((g_xh = BN_new()) == NULL \|\|
5adf6b9a	272	(g_r = BN_new()) == NULL \|\|
	273	(expected = BN_new()) == NULL) {
	274	error("%s: BN_new", __func__);
	275	goto out;
	276	}
	277
5b01421b	278	SCHNORR_DEBUG_BN((e, "%s: e = ", __func__));
5adf6b9a	279	SCHNORR_DEBUG_BN((r, "%s: r = ", __func__));
	280
	281	/* h = H(g \|\| g^v \|\| g^x \|\| id) */
5b01421b	282	if ((h = schnorr_hash(grp_p, grp_q, grp_g, evp_md, e, g_x,
5adf6b9a	283	id, idlen)) == NULL) {
	284	error("%s: schnorr_hash failed", __func__);
	285	goto out;
	286	}
	287
	288	/* g_xh = (g^x)^h */
	289	if (BN_mod_exp(g_xh, g_x, h, grp_p, bn_ctx) == -1) {
	290	error("%s: BN_mod_exp (g_x^h mod p)", __func__);
	291	goto out;
	292	}
	293	SCHNORR_DEBUG_BN((g_xh, "%s: g_xh = ", __func__));
	294
	295	/* g_r = g^r */
	296	if (BN_mod_exp(g_r, grp_g, r, grp_p, bn_ctx) == -1) {
	297	error("%s: BN_mod_exp (g_x^h mod p)", __func__);
	298	goto out;
	299	}
	300	SCHNORR_DEBUG_BN((g_r, "%s: g_r = ", __func__));
	301
	302	/* expected = g^r * g_xh */
	303	if (BN_mod_mul(expected, g_r, g_xh, grp_p, bn_ctx) == -1) {
	304	error("%s: BN_mod_mul (expected = g_r mod p)", __func__);
	305	goto out;
	306	}
	307	SCHNORR_DEBUG_BN((expected, "%s: expected = ", __func__));
	308
5b01421b	309	/* Check e == expected */
5b01421b	310	success = BN_cmp(expected, e) == 0;
5adf6b9a	311	out:
	312	BN_CTX_free(bn_ctx);
	313	if (h != NULL)
	314	BN_clear_free(h);
5adf6b9a	315	BN_clear_free(g_xh);
	316	BN_clear_free(g_r);
	317	BN_clear_free(expected);
	318	return success;
	319	}
	320
5b01421b	321	/*
	322	* Verify Schnorr signature 'sig' of length 'siglen' against public exponent
	323	* g_x (g^x) under group defined by 'grp_p', 'grp_q' and 'grp_g' using a
	324	* SHA256 hash.
	325	* Signature hash will be salted with 'idlen' bytes from 'id'.
	326	* Returns -1 on failure, 0 on incorrect signature or 1 on matching signature.
	327	*/
	328	int
	329	schnorr_verify_buf(const BIGNUM grp_p, const BIGNUM grp_q,
	330	const BIGNUM *grp_g,
	331	const BIGNUM g_x, const u_char id, u_int idlen,
	332	const u_char *sig, u_int siglen)
	333	{
	334	Buffer b;
	335	int ret = -1;
	336	u_int rlen;
	337	BIGNUM r, e;
	338
	339	e = r = NULL;
	340	if ((e = BN_new()) == NULL \|\|
	341	(r = BN_new()) == NULL) {
	342	error("%s: BN_new", __func__);
	343	goto out;
	344	}
	345
	346	/* Extract g^v and r from signature blob */
	347	buffer_init(&b);
	348	buffer_append(&b, sig, siglen);
	349	SCHNORR_DEBUG_BUF((buffer_ptr(&b), buffer_len(&b),
	350	"%s: sigblob", __func__));
	351	buffer_get_bignum2(&b, e);
	352	buffer_get_bignum2(&b, r);
	353	rlen = buffer_len(&b);
	354	buffer_free(&b);
	355	if (rlen != 0) {
	356	error("%s: remaining bytes in signature %d", __func__, rlen);
	357	goto out;
	358	}
	359
	360	ret = schnorr_verify(grp_p, grp_q, grp_g, EVP_sha256(),
	361	g_x, id, idlen, r, e);
	362	out:
	363	BN_clear_free(e);
	364	BN_clear_free(r);
	365
	366	return ret;
	367	}
	368
	369	/* Helper functions */
	370
	371	/*
	372	* Generate uniformly distributed random number in range (1, high).
	373	* Return number on success, NULL on failure.
	374	*/
	375	BIGNUM *
	376	bn_rand_range_gt_one(const BIGNUM *high)
	377	{
	378	BIGNUM r, tmp;
	379	int success = -1;
	380
	381	if ((tmp = BN_new()) == NULL) {
	382	error("%s: BN_new", __func__);
	383	return NULL;
	384	}
385	if ((r = BN_new()) == NULL) {
386	error("%s: BN_new failed", __func__);
387	goto out;
388	}
389	if (BN_set_word(tmp, 2) != 1) {
390	error("%s: BN_set_word(tmp, 2)", __func__);
391	goto out;
392	}
393	if (BN_sub(tmp, high, tmp) == -1) {
394	error("%s: BN_sub failed (tmp = high - 2)", __func__);
395	goto out;
396	}
397	if (BN_rand_range(r, tmp) == -1) {
398	error("%s: BN_rand_range failed", __func__);
399	goto out;
400	}
401	if (BN_set_word(tmp, 2) != 1) {
402	error("%s: BN_set_word(tmp, 2)", __func__);
403	goto out;
404	}
405	if (BN_add(r, r, tmp) == -1) {
406	error("%s: BN_add failed (r = r + 2)", __func__);
407	goto out;
408	}
409	success = 0;
410	out:
411	BN_clear_free(tmp);
412	if (success == 0)
413	return r;
414	BN_clear_free(r);
415	return NULL;
416	}
417
418	/*
419	* Hash contents of buffer 'b' with hash 'md'. Returns 0 on success,
420	* with digest via 'digestp' (caller to free) and length via 'lenp'.
421	* Returns -1 on failure.
422	*/
423	int
424	hash_buffer(const u_char buf, u_int len, const EVP_MD md,
425	u_char *digestp, u_int lenp)
426	{
427	u_char digest[EVP_MAX_MD_SIZE];
428	u_int digest_len;
429	EVP_MD_CTX evp_md_ctx;
430	int success = -1;
431
432	EVP_MD_CTX_init(&evp_md_ctx);
433
434	if (EVP_DigestInit_ex(&evp_md_ctx, md, NULL) != 1) {
435	error("%s: EVP_DigestInit_ex", __func__);
436	goto out;
437	}
438	if (EVP_DigestUpdate(&evp_md_ctx, buf, len) != 1) {
439	error("%s: EVP_DigestUpdate", __func__);
440	goto out;
441	}
442	if (EVP_DigestFinal_ex(&evp_md_ctx, digest, &digest_len) != 1) {
443	error("%s: EVP_DigestFinal_ex", __func__);
444	goto out;
445	}
446	*digestp = xmalloc(digest_len);
447	*lenp = digest_len;
448	memcpy(digestp, digest, lenp);
449	success = 0;
450	out:
451	EVP_MD_CTX_cleanup(&evp_md_ctx);
452	bzero(digest, sizeof(digest));
453	digest_len = 0;
454	return success;
455	}
456
457	/* print formatted string followed by bignum */
458	void
459	debug3_bn(const BIGNUM n, const char fmt, ...)
460	{
461	char out, h;
462	va_list args;
463
464	out = NULL;
465	va_start(args, fmt);
466	vasprintf(&out, fmt, args);
467	va_end(args);
468	if (out == NULL)
469	fatal("%s: vasprintf failed", __func__);
470
471	if (n == NULL)
472	debug3("%s(null)", out);
473	else {
474	h = BN_bn2hex(n);
475	debug3("%s0x%s", out, h);
476	free(h);
477	}
478	free(out);
479	}
480
481	/* print formatted string followed by buffer contents in hex */
482	void
483	debug3_buf(const u_char buf, u_int len, const char fmt, ...)
484	{
485	char *out, h[65];
486	u_int i, j;
487	va_list args;
488
489	out = NULL;
490	va_start(args, fmt);
491	vasprintf(&out, fmt, args);
492	va_end(args);
493	if (out == NULL)
494	fatal("%s: vasprintf failed", __func__);
495
496	debug3("%s length %u%s", out, len, buf == NULL ? " (null)" : "");
497	free(out);
498	if (buf == NULL)
499	return;
500
501	*h = '\0';
502	for (i = j = 0; i < len; i++) {
503	snprintf(h + j, sizeof(h) - j, "%02x", buf[i]);
504	j += 2;
505	if (j >= sizeof(h) - 1 \|\| i == len - 1) {
506	debug3(" %s", h);
507	*h = '\0';
508	j = 0;
509	}
510	}
511	}
512
513	/*
514	* Construct a MODP group from hex strings p (which must be a safe
515	* prime) and g, automatically calculating subgroup q as (p / 2)
516	*/
517	struct modp_group *
518	modp_group_from_g_and_safe_p(const char grp_g, const char grp_p)
519	{
520	struct modp_group *ret;
521
522	ret = xmalloc(sizeof(*ret));
523	ret->p = ret->q = ret->g = NULL;
524	if (BN_hex2bn(&ret->p, grp_p) == 0 \|\|
525	BN_hex2bn(&ret->g, grp_g) == 0)
526	fatal("%s: BN_hex2bn", __func__);
527	/* Subgroup order is p/2 (p is a safe prime) */
528	if ((ret->q = BN_new()) == NULL)
529	fatal("%s: BN_new", __func__);
530	if (BN_rshift1(ret->q, ret->p) != 1)
531	fatal("%s: BN_rshift1", __func__);
532
533	return ret;
534	}
535
536	void
537	modp_group_free(struct modp_group *grp)
538	{
539	if (grp->g != NULL)
540	BN_clear_free(grp->g);
541	if (grp->p != NULL)
542	BN_clear_free(grp->p);
543	if (grp->q != NULL)
544	BN_clear_free(grp->q);
545	bzero(grp, sizeof(*grp));
546	xfree(grp);
547	}
548
549	/* main() function for self-test */
550
5adf6b9a	551	#ifdef SCHNORR_MAIN
	552	static void
	553	schnorr_selftest_one(const BIGNUM grp_p, const BIGNUM grp_q,
	554	const BIGNUM grp_g, const BIGNUM x)
	555	{
	556	BIGNUM *g_x;
	557	u_char *sig;
	558	u_int siglen;
	559	BN_CTX *bn_ctx;
	560
	561	if ((bn_ctx = BN_CTX_new()) == NULL)
	562	fatal("%s: BN_CTX_new", __func__);
	563	if ((g_x = BN_new()) == NULL)
	564	fatal("%s: BN_new", __func__);
	565
	566	if (BN_mod_exp(g_x, grp_g, x, grp_p, bn_ctx) == -1)
	567	fatal("%s: g_x", __func__);
5b01421b	568	if (schnorr_sign_buf(grp_p, grp_q, grp_g, x, g_x, "junk", 4,
5b01421b	569	&sig, &siglen))
5adf6b9a	570	fatal("%s: schnorr_sign", __func__);
5b01421b	571	if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
5adf6b9a	572	sig, siglen) != 1)
5adf6b9a	573	fatal("%s: verify fail", __func__);
5b01421b	574	if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "JUNK", 4,
5adf6b9a	575	sig, siglen) != 0)
	576	fatal("%s: verify should have failed (bad ID)", __func__);
	577	sig[4] ^= 1;
5b01421b	578	if (schnorr_verify_buf(grp_p, grp_q, grp_g, g_x, "junk", 4,
5adf6b9a	579	sig, siglen) != 0)
	580	fatal("%s: verify should have failed (bit error)", __func__);
	581	xfree(sig);
	582	BN_free(g_x);
	583	BN_CTX_free(bn_ctx);
	584	}
	585
	586	static void
	587	schnorr_selftest(void)
	588	{
	589	BIGNUM *x;
5b01421b	590	struct modp_group *grp;
5adf6b9a	591	u_int i;
	592	char *hh;
	593
	594	grp = jpake_default_group();
	595	if ((x = BN_new()) == NULL)
	596	fatal("%s: BN_new", __func__);
	597	SCHNORR_DEBUG_BN((grp->p, "%s: grp->p = ", __func__));
	598	SCHNORR_DEBUG_BN((grp->q, "%s: grp->q = ", __func__));
	599	SCHNORR_DEBUG_BN((grp->g, "%s: grp->g = ", __func__));
	600
	601	/* [1, 20) */
	602	for (i = 1; i < 20; i++) {
	603	printf("x = %u\n", i);
	604	fflush(stdout);
	605	if (BN_set_word(x, i) != 1)
	606	fatal("%s: set x word", __func__);
	607	schnorr_selftest_one(grp->p, grp->q, grp->g, x);
	608	}
	609
	610	/* 100 x random [0, p) */
	611	for (i = 0; i < 100; i++) {
	612	if (BN_rand_range(x, grp->p) != 1)
	613	fatal("%s: BN_rand_range", __func__);
	614	hh = BN_bn2hex(x);
	615	printf("x = (random) 0x%s\n", hh);
	616	free(hh);
	617	fflush(stdout);
	618	schnorr_selftest_one(grp->p, grp->q, grp->g, x);
	619	}
	620
	621	/* [q-20, q) */
	622	if (BN_set_word(x, 20) != 1)
	623	fatal("%s: BN_set_word (x = 20)", __func__);
	624	if (BN_sub(x, grp->q, x) != 1)
	625	fatal("%s: BN_sub (q - x)", __func__);
	626	for (i = 0; i < 19; i++) {
	627	hh = BN_bn2hex(x);
	628	printf("x = (q - %d) 0x%s\n", 20 - i, hh);
	629	free(hh);
	630	fflush(stdout);
	631	schnorr_selftest_one(grp->p, grp->q, grp->g, x);
	632	if (BN_add(x, x, BN_value_one()) != 1)
	633	fatal("%s: BN_add (x + 1)", __func__);
	634	}
	635	BN_free(x);
	636	}
	637
	638	int
	639	main(int argc, char **argv)
	640	{
	641	log_init(argv[0], SYSLOG_LEVEL_DEBUG3, SYSLOG_FACILITY_USER, 1);
	642
	643	schnorr_selftest();
	644	return 0;
	645	}
	646	#endif
	647