openssh/key.c

867 lines
19 KiB
C

/* $OpenBSD: key.c,v 1.67 2006/08/03 03:34:42 deraadt Exp $ */
/*
* read_bignum():
* Copyright (c) 1995 Tatu Ylonen <ylo@cs.hut.fi>, Espoo, Finland
*
* As far as I am concerned, the code I have written for this software
* can be used freely for any purpose. Any derived versions of this
* software must be clearly marked as such, and if the derived work is
* incompatible with the protocol description in the RFC file, it must be
* called by a name other than "ssh" or "Secure Shell".
*
*
* Copyright (c) 2000, 2001 Markus Friedl. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
#include "includes.h"
#include <sys/types.h>
#include <openssl/evp.h>
#include <stdarg.h>
#include <stdio.h>
#include <string.h>
#include "xmalloc.h"
#include "key.h"
#include "rsa.h"
#include "uuencode.h"
#include "buffer.h"
#include "log.h"
Key *
key_new(int type)
{
Key *k;
RSA *rsa;
DSA *dsa;
k = xcalloc(1, sizeof(*k));
k->type = type;
k->dsa = NULL;
k->rsa = NULL;
switch (k->type) {
case KEY_RSA1:
case KEY_RSA:
if ((rsa = RSA_new()) == NULL)
fatal("key_new: RSA_new failed");
if ((rsa->n = BN_new()) == NULL)
fatal("key_new: BN_new failed");
if ((rsa->e = BN_new()) == NULL)
fatal("key_new: BN_new failed");
k->rsa = rsa;
break;
case KEY_DSA:
if ((dsa = DSA_new()) == NULL)
fatal("key_new: DSA_new failed");
if ((dsa->p = BN_new()) == NULL)
fatal("key_new: BN_new failed");
if ((dsa->q = BN_new()) == NULL)
fatal("key_new: BN_new failed");
if ((dsa->g = BN_new()) == NULL)
fatal("key_new: BN_new failed");
if ((dsa->pub_key = BN_new()) == NULL)
fatal("key_new: BN_new failed");
k->dsa = dsa;
break;
case KEY_UNSPEC:
break;
default:
fatal("key_new: bad key type %d", k->type);
break;
}
return k;
}
Key *
key_new_private(int type)
{
Key *k = key_new(type);
switch (k->type) {
case KEY_RSA1:
case KEY_RSA:
if ((k->rsa->d = BN_new()) == NULL)
fatal("key_new_private: BN_new failed");
if ((k->rsa->iqmp = BN_new()) == NULL)
fatal("key_new_private: BN_new failed");
if ((k->rsa->q = BN_new()) == NULL)
fatal("key_new_private: BN_new failed");
if ((k->rsa->p = BN_new()) == NULL)
fatal("key_new_private: BN_new failed");
if ((k->rsa->dmq1 = BN_new()) == NULL)
fatal("key_new_private: BN_new failed");
if ((k->rsa->dmp1 = BN_new()) == NULL)
fatal("key_new_private: BN_new failed");
break;
case KEY_DSA:
if ((k->dsa->priv_key = BN_new()) == NULL)
fatal("key_new_private: BN_new failed");
break;
case KEY_UNSPEC:
break;
default:
break;
}
return k;
}
void
key_free(Key *k)
{
if (k == NULL)
fatal("key_free: key is NULL");
switch (k->type) {
case KEY_RSA1:
case KEY_RSA:
if (k->rsa != NULL)
RSA_free(k->rsa);
k->rsa = NULL;
break;
case KEY_DSA:
if (k->dsa != NULL)
DSA_free(k->dsa);
k->dsa = NULL;
break;
case KEY_UNSPEC:
break;
default:
fatal("key_free: bad key type %d", k->type);
break;
}
xfree(k);
}
int
key_equal(const Key *a, const Key *b)
{
if (a == NULL || b == NULL || a->type != b->type)
return 0;
switch (a->type) {
case KEY_RSA1:
case KEY_RSA:
return a->rsa != NULL && b->rsa != NULL &&
BN_cmp(a->rsa->e, b->rsa->e) == 0 &&
BN_cmp(a->rsa->n, b->rsa->n) == 0;
case KEY_DSA:
return a->dsa != NULL && b->dsa != NULL &&
BN_cmp(a->dsa->p, b->dsa->p) == 0 &&
BN_cmp(a->dsa->q, b->dsa->q) == 0 &&
BN_cmp(a->dsa->g, b->dsa->g) == 0 &&
BN_cmp(a->dsa->pub_key, b->dsa->pub_key) == 0;
default:
fatal("key_equal: bad key type %d", a->type);
break;
}
return 0;
}
u_char*
key_fingerprint_raw(const Key *k, enum fp_type dgst_type,
u_int *dgst_raw_length)
{
const EVP_MD *md = NULL;
EVP_MD_CTX ctx;
u_char *blob = NULL;
u_char *retval = NULL;
u_int len = 0;
int nlen, elen;
*dgst_raw_length = 0;
switch (dgst_type) {
case SSH_FP_MD5:
md = EVP_md5();
break;
case SSH_FP_SHA1:
md = EVP_sha1();
break;
default:
fatal("key_fingerprint_raw: bad digest type %d",
dgst_type);
}
switch (k->type) {
case KEY_RSA1:
nlen = BN_num_bytes(k->rsa->n);
elen = BN_num_bytes(k->rsa->e);
len = nlen + elen;
blob = xmalloc(len);
BN_bn2bin(k->rsa->n, blob);
BN_bn2bin(k->rsa->e, blob + nlen);
break;
case KEY_DSA:
case KEY_RSA:
key_to_blob(k, &blob, &len);
break;
case KEY_UNSPEC:
return retval;
default:
fatal("key_fingerprint_raw: bad key type %d", k->type);
break;
}
if (blob != NULL) {
retval = xmalloc(EVP_MAX_MD_SIZE);
EVP_DigestInit(&ctx, md);
EVP_DigestUpdate(&ctx, blob, len);
EVP_DigestFinal(&ctx, retval, dgst_raw_length);
memset(blob, 0, len);
xfree(blob);
} else {
fatal("key_fingerprint_raw: blob is null");
}
return retval;
}
static char *
key_fingerprint_hex(u_char *dgst_raw, u_int dgst_raw_len)
{
char *retval;
u_int i;
retval = xcalloc(1, dgst_raw_len * 3 + 1);
for (i = 0; i < dgst_raw_len; i++) {
char hex[4];
snprintf(hex, sizeof(hex), "%02x:", dgst_raw[i]);
strlcat(retval, hex, dgst_raw_len * 3 + 1);
}
/* Remove the trailing ':' character */
retval[(dgst_raw_len * 3) - 1] = '\0';
return retval;
}
static char *
key_fingerprint_bubblebabble(u_char *dgst_raw, u_int dgst_raw_len)
{
char vowels[] = { 'a', 'e', 'i', 'o', 'u', 'y' };
char consonants[] = { 'b', 'c', 'd', 'f', 'g', 'h', 'k', 'l', 'm',
'n', 'p', 'r', 's', 't', 'v', 'z', 'x' };
u_int i, j = 0, rounds, seed = 1;
char *retval;
rounds = (dgst_raw_len / 2) + 1;
retval = xcalloc((rounds * 6), sizeof(char));
retval[j++] = 'x';
for (i = 0; i < rounds; i++) {
u_int idx0, idx1, idx2, idx3, idx4;
if ((i + 1 < rounds) || (dgst_raw_len % 2 != 0)) {
idx0 = (((((u_int)(dgst_raw[2 * i])) >> 6) & 3) +
seed) % 6;
idx1 = (((u_int)(dgst_raw[2 * i])) >> 2) & 15;
idx2 = ((((u_int)(dgst_raw[2 * i])) & 3) +
(seed / 6)) % 6;
retval[j++] = vowels[idx0];
retval[j++] = consonants[idx1];
retval[j++] = vowels[idx2];
if ((i + 1) < rounds) {
idx3 = (((u_int)(dgst_raw[(2 * i) + 1])) >> 4) & 15;
idx4 = (((u_int)(dgst_raw[(2 * i) + 1]))) & 15;
retval[j++] = consonants[idx3];
retval[j++] = '-';
retval[j++] = consonants[idx4];
seed = ((seed * 5) +
((((u_int)(dgst_raw[2 * i])) * 7) +
((u_int)(dgst_raw[(2 * i) + 1])))) % 36;
}
} else {
idx0 = seed % 6;
idx1 = 16;
idx2 = seed / 6;
retval[j++] = vowels[idx0];
retval[j++] = consonants[idx1];
retval[j++] = vowels[idx2];
}
}
retval[j++] = 'x';
retval[j++] = '\0';
return retval;
}
char *
key_fingerprint(const Key *k, enum fp_type dgst_type, enum fp_rep dgst_rep)
{
char *retval = NULL;
u_char *dgst_raw;
u_int dgst_raw_len;
dgst_raw = key_fingerprint_raw(k, dgst_type, &dgst_raw_len);
if (!dgst_raw)
fatal("key_fingerprint: null from key_fingerprint_raw()");
switch (dgst_rep) {
case SSH_FP_HEX:
retval = key_fingerprint_hex(dgst_raw, dgst_raw_len);
break;
case SSH_FP_BUBBLEBABBLE:
retval = key_fingerprint_bubblebabble(dgst_raw, dgst_raw_len);
break;
default:
fatal("key_fingerprint_ex: bad digest representation %d",
dgst_rep);
break;
}
memset(dgst_raw, 0, dgst_raw_len);
xfree(dgst_raw);
return retval;
}
/*
* Reads a multiple-precision integer in decimal from the buffer, and advances
* the pointer. The integer must already be initialized. This function is
* permitted to modify the buffer. This leaves *cpp to point just beyond the
* last processed (and maybe modified) character. Note that this may modify
* the buffer containing the number.
*/
static int
read_bignum(char **cpp, BIGNUM * value)
{
char *cp = *cpp;
int old;
/* Skip any leading whitespace. */
for (; *cp == ' ' || *cp == '\t'; cp++)
;
/* Check that it begins with a decimal digit. */
if (*cp < '0' || *cp > '9')
return 0;
/* Save starting position. */
*cpp = cp;
/* Move forward until all decimal digits skipped. */
for (; *cp >= '0' && *cp <= '9'; cp++)
;
/* Save the old terminating character, and replace it by \0. */
old = *cp;
*cp = 0;
/* Parse the number. */
if (BN_dec2bn(&value, *cpp) == 0)
return 0;
/* Restore old terminating character. */
*cp = old;
/* Move beyond the number and return success. */
*cpp = cp;
return 1;
}
static int
write_bignum(FILE *f, BIGNUM *num)
{
char *buf = BN_bn2dec(num);
if (buf == NULL) {
error("write_bignum: BN_bn2dec() failed");
return 0;
}
fprintf(f, " %s", buf);
OPENSSL_free(buf);
return 1;
}
/* returns 1 ok, -1 error */
int
key_read(Key *ret, char **cpp)
{
Key *k;
int success = -1;
char *cp, *space;
int len, n, type;
u_int bits;
u_char *blob;
cp = *cpp;
switch (ret->type) {
case KEY_RSA1:
/* Get number of bits. */
if (*cp < '0' || *cp > '9')
return -1; /* Bad bit count... */
for (bits = 0; *cp >= '0' && *cp <= '9'; cp++)
bits = 10 * bits + *cp - '0';
if (bits == 0)
return -1;
*cpp = cp;
/* Get public exponent, public modulus. */
if (!read_bignum(cpp, ret->rsa->e))
return -1;
if (!read_bignum(cpp, ret->rsa->n))
return -1;
success = 1;
break;
case KEY_UNSPEC:
case KEY_RSA:
case KEY_DSA:
space = strchr(cp, ' ');
if (space == NULL) {
debug3("key_read: missing whitespace");
return -1;
}
*space = '\0';
type = key_type_from_name(cp);
*space = ' ';
if (type == KEY_UNSPEC) {
debug3("key_read: missing keytype");
return -1;
}
cp = space+1;
if (*cp == '\0') {
debug3("key_read: short string");
return -1;
}
if (ret->type == KEY_UNSPEC) {
ret->type = type;
} else if (ret->type != type) {
/* is a key, but different type */
debug3("key_read: type mismatch");
return -1;
}
len = 2*strlen(cp);
blob = xmalloc(len);
n = uudecode(cp, blob, len);
if (n < 0) {
error("key_read: uudecode %s failed", cp);
xfree(blob);
return -1;
}
k = key_from_blob(blob, (u_int)n);
xfree(blob);
if (k == NULL) {
error("key_read: key_from_blob %s failed", cp);
return -1;
}
if (k->type != type) {
error("key_read: type mismatch: encoding error");
key_free(k);
return -1;
}
/*XXXX*/
if (ret->type == KEY_RSA) {
if (ret->rsa != NULL)
RSA_free(ret->rsa);
ret->rsa = k->rsa;
k->rsa = NULL;
success = 1;
#ifdef DEBUG_PK
RSA_print_fp(stderr, ret->rsa, 8);
#endif
} else {
if (ret->dsa != NULL)
DSA_free(ret->dsa);
ret->dsa = k->dsa;
k->dsa = NULL;
success = 1;
#ifdef DEBUG_PK
DSA_print_fp(stderr, ret->dsa, 8);
#endif
}
/*XXXX*/
key_free(k);
if (success != 1)
break;
/* advance cp: skip whitespace and data */
while (*cp == ' ' || *cp == '\t')
cp++;
while (*cp != '\0' && *cp != ' ' && *cp != '\t')
cp++;
*cpp = cp;
break;
default:
fatal("key_read: bad key type: %d", ret->type);
break;
}
return success;
}
int
key_write(const Key *key, FILE *f)
{
int n, success = 0;
u_int len, bits = 0;
u_char *blob;
char *uu;
if (key->type == KEY_RSA1 && key->rsa != NULL) {
/* size of modulus 'n' */
bits = BN_num_bits(key->rsa->n);
fprintf(f, "%u", bits);
if (write_bignum(f, key->rsa->e) &&
write_bignum(f, key->rsa->n)) {
success = 1;
} else {
error("key_write: failed for RSA key");
}
} else if ((key->type == KEY_DSA && key->dsa != NULL) ||
(key->type == KEY_RSA && key->rsa != NULL)) {
key_to_blob(key, &blob, &len);
uu = xmalloc(2*len);
n = uuencode(blob, len, uu, 2*len);
if (n > 0) {
fprintf(f, "%s %s", key_ssh_name(key), uu);
success = 1;
}
xfree(blob);
xfree(uu);
}
return success;
}
const char *
key_type(const Key *k)
{
switch (k->type) {
case KEY_RSA1:
return "RSA1";
case KEY_RSA:
return "RSA";
case KEY_DSA:
return "DSA";
}
return "unknown";
}
const char *
key_ssh_name(const Key *k)
{
switch (k->type) {
case KEY_RSA:
return "ssh-rsa";
case KEY_DSA:
return "ssh-dss";
}
return "ssh-unknown";
}
u_int
key_size(const Key *k)
{
switch (k->type) {
case KEY_RSA1:
case KEY_RSA:
return BN_num_bits(k->rsa->n);
case KEY_DSA:
return BN_num_bits(k->dsa->p);
}
return 0;
}
static RSA *
rsa_generate_private_key(u_int bits)
{
RSA *private;
private = RSA_generate_key(bits, 35, NULL, NULL);
if (private == NULL)
fatal("rsa_generate_private_key: key generation failed.");
return private;
}
static DSA*
dsa_generate_private_key(u_int bits)
{
DSA *private = DSA_generate_parameters(bits, NULL, 0, NULL, NULL, NULL, NULL);
if (private == NULL)
fatal("dsa_generate_private_key: DSA_generate_parameters failed");
if (!DSA_generate_key(private))
fatal("dsa_generate_private_key: DSA_generate_key failed.");
if (private == NULL)
fatal("dsa_generate_private_key: NULL.");
return private;
}
Key *
key_generate(int type, u_int bits)
{
Key *k = key_new(KEY_UNSPEC);
switch (type) {
case KEY_DSA:
k->dsa = dsa_generate_private_key(bits);
break;
case KEY_RSA:
case KEY_RSA1:
k->rsa = rsa_generate_private_key(bits);
break;
default:
fatal("key_generate: unknown type %d", type);
}
k->type = type;
return k;
}
Key *
key_from_private(const Key *k)
{
Key *n = NULL;
switch (k->type) {
case KEY_DSA:
n = key_new(k->type);
BN_copy(n->dsa->p, k->dsa->p);
BN_copy(n->dsa->q, k->dsa->q);
BN_copy(n->dsa->g, k->dsa->g);
BN_copy(n->dsa->pub_key, k->dsa->pub_key);
break;
case KEY_RSA:
case KEY_RSA1:
n = key_new(k->type);
BN_copy(n->rsa->n, k->rsa->n);
BN_copy(n->rsa->e, k->rsa->e);
break;
default:
fatal("key_from_private: unknown type %d", k->type);
break;
}
return n;
}
int
key_type_from_name(char *name)
{
if (strcmp(name, "rsa1") == 0) {
return KEY_RSA1;
} else if (strcmp(name, "rsa") == 0) {
return KEY_RSA;
} else if (strcmp(name, "dsa") == 0) {
return KEY_DSA;
} else if (strcmp(name, "ssh-rsa") == 0) {
return KEY_RSA;
} else if (strcmp(name, "ssh-dss") == 0) {
return KEY_DSA;
}
debug2("key_type_from_name: unknown key type '%s'", name);
return KEY_UNSPEC;
}
int
key_names_valid2(const char *names)
{
char *s, *cp, *p;
if (names == NULL || strcmp(names, "") == 0)
return 0;
s = cp = xstrdup(names);
for ((p = strsep(&cp, ",")); p && *p != '\0';
(p = strsep(&cp, ","))) {
switch (key_type_from_name(p)) {
case KEY_RSA1:
case KEY_UNSPEC:
xfree(s);
return 0;
}
}
debug3("key names ok: [%s]", names);
xfree(s);
return 1;
}
Key *
key_from_blob(const u_char *blob, u_int blen)
{
Buffer b;
int rlen, type;
char *ktype = NULL;
Key *key = NULL;
#ifdef DEBUG_PK
dump_base64(stderr, blob, blen);
#endif
buffer_init(&b);
buffer_append(&b, blob, blen);
if ((ktype = buffer_get_string_ret(&b, NULL)) == NULL) {
error("key_from_blob: can't read key type");
goto out;
}
type = key_type_from_name(ktype);
switch (type) {
case KEY_RSA:
key = key_new(type);
if (buffer_get_bignum2_ret(&b, key->rsa->e) == -1 ||
buffer_get_bignum2_ret(&b, key->rsa->n) == -1) {
error("key_from_blob: can't read rsa key");
key_free(key);
key = NULL;
goto out;
}
#ifdef DEBUG_PK
RSA_print_fp(stderr, key->rsa, 8);
#endif
break;
case KEY_DSA:
key = key_new(type);
if (buffer_get_bignum2_ret(&b, key->dsa->p) == -1 ||
buffer_get_bignum2_ret(&b, key->dsa->q) == -1 ||
buffer_get_bignum2_ret(&b, key->dsa->g) == -1 ||
buffer_get_bignum2_ret(&b, key->dsa->pub_key) == -1) {
error("key_from_blob: can't read dsa key");
key_free(key);
key = NULL;
goto out;
}
#ifdef DEBUG_PK
DSA_print_fp(stderr, key->dsa, 8);
#endif
break;
case KEY_UNSPEC:
key = key_new(type);
break;
default:
error("key_from_blob: cannot handle type %s", ktype);
goto out;
}
rlen = buffer_len(&b);
if (key != NULL && rlen != 0)
error("key_from_blob: remaining bytes in key blob %d", rlen);
out:
if (ktype != NULL)
xfree(ktype);
buffer_free(&b);
return key;
}
int
key_to_blob(const Key *key, u_char **blobp, u_int *lenp)
{
Buffer b;
int len;
if (key == NULL) {
error("key_to_blob: key == NULL");
return 0;
}
buffer_init(&b);
switch (key->type) {
case KEY_DSA:
buffer_put_cstring(&b, key_ssh_name(key));
buffer_put_bignum2(&b, key->dsa->p);
buffer_put_bignum2(&b, key->dsa->q);
buffer_put_bignum2(&b, key->dsa->g);
buffer_put_bignum2(&b, key->dsa->pub_key);
break;
case KEY_RSA:
buffer_put_cstring(&b, key_ssh_name(key));
buffer_put_bignum2(&b, key->rsa->e);
buffer_put_bignum2(&b, key->rsa->n);
break;
default:
error("key_to_blob: unsupported key type %d", key->type);
buffer_free(&b);
return 0;
}
len = buffer_len(&b);
if (lenp != NULL)
*lenp = len;
if (blobp != NULL) {
*blobp = xmalloc(len);
memcpy(*blobp, buffer_ptr(&b), len);
}
memset(buffer_ptr(&b), 0, len);
buffer_free(&b);
return len;
}
int
key_sign(
const Key *key,
u_char **sigp, u_int *lenp,
const u_char *data, u_int datalen)
{
switch (key->type) {
case KEY_DSA:
return ssh_dss_sign(key, sigp, lenp, data, datalen);
case KEY_RSA:
return ssh_rsa_sign(key, sigp, lenp, data, datalen);
default:
error("key_sign: invalid key type %d", key->type);
return -1;
}
}
/*
* key_verify returns 1 for a correct signature, 0 for an incorrect signature
* and -1 on error.
*/
int
key_verify(
const Key *key,
const u_char *signature, u_int signaturelen,
const u_char *data, u_int datalen)
{
if (signaturelen == 0)
return -1;
switch (key->type) {
case KEY_DSA:
return ssh_dss_verify(key, signature, signaturelen, data, datalen);
case KEY_RSA:
return ssh_rsa_verify(key, signature, signaturelen, data, datalen);
default:
error("key_verify: invalid key type %d", key->type);
return -1;
}
}
/* Converts a private to a public key */
Key *
key_demote(const Key *k)
{
Key *pk;
pk = xcalloc(1, sizeof(*pk));
pk->type = k->type;
pk->flags = k->flags;
pk->dsa = NULL;
pk->rsa = NULL;
switch (k->type) {
case KEY_RSA1:
case KEY_RSA:
if ((pk->rsa = RSA_new()) == NULL)
fatal("key_demote: RSA_new failed");
if ((pk->rsa->e = BN_dup(k->rsa->e)) == NULL)
fatal("key_demote: BN_dup failed");
if ((pk->rsa->n = BN_dup(k->rsa->n)) == NULL)
fatal("key_demote: BN_dup failed");
break;
case KEY_DSA:
if ((pk->dsa = DSA_new()) == NULL)
fatal("key_demote: DSA_new failed");
if ((pk->dsa->p = BN_dup(k->dsa->p)) == NULL)
fatal("key_demote: BN_dup failed");
if ((pk->dsa->q = BN_dup(k->dsa->q)) == NULL)
fatal("key_demote: BN_dup failed");
if ((pk->dsa->g = BN_dup(k->dsa->g)) == NULL)
fatal("key_demote: BN_dup failed");
if ((pk->dsa->pub_key = BN_dup(k->dsa->pub_key)) == NULL)
fatal("key_demote: BN_dup failed");
break;
default:
fatal("key_free: bad key type %d", k->type);
break;
}
return (pk);
}