schnorr.c - security/openssh-library - Rivoreo Source Code Repositories

 /* $OpenBSD: schnorr.c,v 1.1 2008/11/04 08:22:13 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 "jpake.h"

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

 /* XXX */
 /* Parametise signature hash? (sha256, sha1, etc.) */
 /* Signature format - include type name, hash type, group params? */

 #ifndef SCHNORR_DEBUG
 # define SCHNORR_DEBUG_BN(a)
 # define SCHNORR_DEBUG_BUF(a)
 #else
 # define SCHNORR_DEBUG_BN(a)	jpake_debug3_bn a
 # define SCHNORR_DEBUG_BUF(a)	jpake_debug3_buf a
 #endif /* SCHNORR_DEBUG */

 /*
  * Calculate hash component of Schnorr signature H(g || g^v || g^x || id)
  * using SHA1. Returns signature as bignum or NULL on error.
  */
 static BIGNUM *
 schnorr_hash(const BIGNUM *p, const BIGNUM *q, const BIGNUM *g,
     const BIGNUM *g_v, const BIGNUM *g_x,
     const u_char *id, u_int idlen)
 {
 	u_char *digest;
 	u_int digest_len;
 	BIGNUM *h;
 	EVP_MD_CTX evp_md_ctx;
 	Buffer b;
 	int success = -1;

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

 	buffer_init(&b);
 	EVP_MD_CTX_init(&evp_md_ctx);

 	/* h = H(g || g^v || g^x || id) */
 	buffer_put_bignum2(&b, g);
 	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_sha256(),
 	    &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);
 	EVP_MD_CTX_cleanup(&evp_md_ctx);
 	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'
  * '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(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)
 {
 	int success = -1;
 	Buffer b;
 	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, 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((r, "%s: r = ", __func__));

 	/* Signature is (g_v, r) */
 	buffer_init(&b);
 	/* XXX sigtype-hash as string? */
 	buffer_put_bignum2(&b, g_v);
 	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);
 	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(r);
 	BN_clear_free(g_v);
 	BN_clear_free(tmp);

 	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'.
  * 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 BIGNUM *g_x, const u_char *id, u_int idlen,
     const u_char *sig, u_int siglen)
 {
 	int success = -1;
 	Buffer b;
 	BIGNUM *g_v, *h, *r, *g_xh, *g_r, *expected;
 	BN_CTX *bn_ctx;
 	u_int rlen;

 	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;
 	}

 	g_v = h = r = g_xh = g_r = expected = 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 ||
 	    (g_xh = BN_new()) == NULL ||
 	    (g_r = BN_new()) == NULL ||
 	    (expected = 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, g_v);
 	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;
 	}
 	buffer_free(&b);
 	SCHNORR_DEBUG_BN((g_v, "%s: g_v = ", __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, g_v, 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 g_v == expected */
 	success = BN_cmp(expected, g_v) == 0;
  out:
 	BN_CTX_free(bn_ctx);
 	if (h != NULL)
 		BN_clear_free(h);
 	BN_clear_free(g_v);
 	BN_clear_free(r);
 	BN_clear_free(g_xh);
 	BN_clear_free(g_r);
 	BN_clear_free(expected);
 	return success;
 }

 #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(grp_p, grp_q, grp_g, x, g_x, "junk", 4, &sig, &siglen))
 		fatal("%s: schnorr_sign", __func__);
 	if (schnorr_verify(grp_p, grp_q, grp_g, g_x, "junk", 4,
 	    sig, siglen) != 1)
 		fatal("%s: verify fail", __func__);
 	if (schnorr_verify(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(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 jpake_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
	/* $OpenBSD: schnorr.c,v 1.1 2008/11/04 08:22:13 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 "jpake.h"

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

	/* XXX */
	/* Parametise signature hash? (sha256, sha1, etc.) */
	/* Signature format - include type name, hash type, group params? */

	#ifndef SCHNORR_DEBUG
	# define SCHNORR_DEBUG_BN(a)
	# define SCHNORR_DEBUG_BUF(a)
	#else
	# define SCHNORR_DEBUG_BN(a) jpake_debug3_bn a
	# define SCHNORR_DEBUG_BUF(a) jpake_debug3_buf a
	#endif /* SCHNORR_DEBUG */

	/*
	* Calculate hash component of Schnorr signature H(g \|\| g^v \|\| g^x \|\| id)
	* using SHA1. Returns signature as bignum or NULL on error.
	*/
	static BIGNUM *
	schnorr_hash(const BIGNUM p, const BIGNUM q, const BIGNUM *g,
	const BIGNUM g_v, const BIGNUM g_x,
	const u_char *id, u_int idlen)
	{
	u_char *digest;
	u_int digest_len;
	BIGNUM *h;
	EVP_MD_CTX evp_md_ctx;
	Buffer b;
	int success = -1;

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

	buffer_init(&b);
	EVP_MD_CTX_init(&evp_md_ctx);

	/* h = H(g \|\| g^v \|\| g^x \|\| id) */
	buffer_put_bignum2(&b, g);
	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_sha256(),
	&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);
	EVP_MD_CTX_cleanup(&evp_md_ctx);
	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'
	* '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(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)
	{
	int success = -1;
	Buffer b;
	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, 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((r, "%s: r = ", __func__));

	/* Signature is (g_v, r) */
	buffer_init(&b);
	/* XXX sigtype-hash as string? */
	buffer_put_bignum2(&b, g_v);
	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);
	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(r);
	BN_clear_free(g_v);
	BN_clear_free(tmp);

	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'.
	* 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 BIGNUM g_x, const u_char id, u_int idlen,
	const u_char *sig, u_int siglen)
	{
	int success = -1;
	Buffer b;
	BIGNUM g_v, h, r, g_xh, g_r, expected;
	BN_CTX *bn_ctx;
	u_int rlen;

	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;
	}

	g_v = h = r = g_xh = g_r = expected = 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 \|\|
	(g_xh = BN_new()) == NULL \|\|
	(g_r = BN_new()) == NULL \|\|
	(expected = 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, g_v);
	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;
	}
	buffer_free(&b);
	SCHNORR_DEBUG_BN((g_v, "%s: g_v = ", __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, g_v, 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 g_v == expected */
	success = BN_cmp(expected, g_v) == 0;
	out:
	BN_CTX_free(bn_ctx);
	if (h != NULL)
	BN_clear_free(h);
	BN_clear_free(g_v);
	BN_clear_free(r);
	BN_clear_free(g_xh);
	BN_clear_free(g_r);
	BN_clear_free(expected);
	return success;
	}

	#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(grp_p, grp_q, grp_g, x, g_x, "junk", 4, &sig, &siglen))
	fatal("%s: schnorr_sign", __func__);
	if (schnorr_verify(grp_p, grp_q, grp_g, g_x, "junk", 4,
	sig, siglen) != 1)
	fatal("%s: verify fail", __func__);
	if (schnorr_verify(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(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 jpake_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