xref: /external/chromium_org/third_party/boringssl/src/crypto/bn/prime.c

/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
 * All rights reserved.
 *
 * This package is an SSL implementation written
 * by Eric Young (eay@cryptsoft.com).
 * The implementation was written so as to conform with Netscapes SSL.
 *
 * This library is free for commercial and non-commercial use as long as
 * the following conditions are aheared to.  The following conditions
 * apply to all code found in this distribution, be it the RC4, RSA,
 * lhash, DES, etc., code; not just the SSL code.  The SSL documentation
 * included with this distribution is covered by the same copyright terms
 * except that the holder is Tim Hudson (tjh@cryptsoft.com).
 *
 * Copyright remains Eric Young's, and as such any Copyright notices in
 * the code are not to be removed.
 * If this package is used in a product, Eric Young should be given attribution
 * as the author of the parts of the library used.
 * This can be in the form of a textual message at program startup or
 * in documentation (online or textual) provided with the package.
 *
 * 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 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.
 * 3. All advertising materials mentioning features or use of this software
 *    must display the following acknowledgement:
 *    "This product includes cryptographic software written by
 *     Eric Young (eay@cryptsoft.com)"
 *    The word 'cryptographic' can be left out if the rouines from the library
 *    being used are not cryptographic related :-).
 * 4. If you include any Windows specific code (or a derivative thereof) from
 *    the apps directory (application code) you must include an acknowledgement:
 *    "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
 *
 * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``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 OR CONTRIBUTORS 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.
 *
 * The licence and distribution terms for any publically available version or
 * derivative of this code cannot be changed.  i.e. this code cannot simply be
 * copied and put under another distribution licence
 * [including the GNU Public Licence.]
 */
/* ====================================================================
 * Copyright (c) 1998-2001 The OpenSSL Project.  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.
 *
 * 3. All advertising materials mentioning features or use of this
 *    software must display the following acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
 *
 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
 *    endorse or promote products derived from this software without
 *    prior written permission. For written permission, please contact
 *    openssl-core@openssl.org.
 *
 * 5. Products derived from this software may not be called "OpenSSL"
 *    nor may "OpenSSL" appear in their names without prior written
 *    permission of the OpenSSL Project.
 *
 * 6. Redistributions of any form whatsoever must retain the following
 *    acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
 *
 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
 * EXPRESSED 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 OpenSSL PROJECT OR
 * ITS CONTRIBUTORS 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.
 * ====================================================================
 *
 * This product includes cryptographic software written by Eric Young
 * (eay@cryptsoft.com).  This product includes software written by Tim
 * Hudson (tjh@cryptsoft.com). */

#include <openssl/bn.h>

#include <openssl/err.h>
#include <openssl/mem.h>

#include "internal.h"

/* number of Miller-Rabin iterations for an error rate  of less than 2^-80
 * for random 'b'-bit input, b >= 100 (taken from table 4.4 in the Handbook
 * of Applied Cryptography [Menezes, van Oorschot, Vanstone; CRC Press 1996];
 * original paper: Damgaard, Landrock, Pomerance: Average case error estimates
 * for the strong probable prime test. -- Math. Comp. 61 (1993) 177-194) */
#define BN_prime_checks_for_size(b) ((b) >= 1300 ?  2 : \
                                (b) >=  850 ?  3 : \
                                (b) >=  650 ?  4 : \
                                (b) >=  550 ?  5 : \
                                (b) >=  450 ?  6 : \
                                (b) >=  400 ?  7 : \
                                (b) >=  350 ?  8 : \
                                (b) >=  300 ?  9 : \
                                (b) >=  250 ? 12 : \
                                (b) >=  200 ? 15 : \
                                (b) >=  150 ? 18 : \
                                /* b >= 100 */ 27)

/* The quick sieve algorithm approach to weeding out primes is Philip
 * Zimmermann's, as implemented in PGP.  I have had a read of his comments and
 * implemented my own version. */

/* NUMPRIMES is the number of primes that fit into a uint16_t. */
#define NUMPRIMES 2048

/* primes is defined at the bottom of the file and contains all the primes that
 * fit into a uint16_t. */
static const uint16_t primes[NUMPRIMES];

static int witness(BIGNUM *w, const BIGNUM *a, const BIGNUM *a1,
                   const BIGNUM *a1_odd, int k, BN_CTX *ctx, BN_MONT_CTX *mont);
static int probable_prime(BIGNUM *rnd, int bits);
static int probable_prime_dh(BIGNUM *rnd, int bits, const BIGNUM *add,
                             const BIGNUM *rem, BN_CTX *ctx);
static int probable_prime_dh_safe(BIGNUM *rnd, int bits, const BIGNUM *add,
                                  const BIGNUM *rem, BN_CTX *ctx);

void BN_GENCB_set(BN_GENCB *callback,
                  int (*f)(int event, int n, struct bn_gencb_st *),
                  void *arg) {
  callback->callback = f;
  callback->arg = arg;
}

int BN_GENCB_call(BN_GENCB *callback, int event, int n) {
  if (!callback) {
    return 1;
  }

  return callback->callback(event, n, callback);
}

int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe, const BIGNUM *add,
                         const BIGNUM *rem, BN_GENCB *cb) {
  BIGNUM *t;
  int found = 0;
  int i, j, c1 = 0;
  BN_CTX *ctx;
  int checks = BN_prime_checks_for_size(bits);

  if (bits < 2) {
    /* There are no prime numbers this small. */
    OPENSSL_PUT_ERROR(BN, BN_generate_prime_ex, BN_R_BITS_TOO_SMALL);
    return 0;
  } else if (bits == 2 && safe) {
    /* The smallest safe prime (7) is three bits. */
    OPENSSL_PUT_ERROR(BN, BN_generate_prime_ex, BN_R_BITS_TOO_SMALL);
    return 0;
  }

  ctx = BN_CTX_new();
  if (ctx == NULL) {
    goto err;
  }
  BN_CTX_start(ctx);
  t = BN_CTX_get(ctx);
  if (!t) {
    goto err;
  }

loop:
  /* make a random number and set the top and bottom bits */
  if (add == NULL) {
    if (!probable_prime(ret, bits)) {
      goto err;
    }
  } else {
    if (safe) {
      if (!probable_prime_dh_safe(ret, bits, add, rem, ctx)) {
        goto err;
      }
    } else {
      if (!probable_prime_dh(ret, bits, add, rem, ctx)) {
        goto err;
      }
    }
  }

  if (!BN_GENCB_call(cb, BN_GENCB_GENERATED, c1++)) {
    /* aborted */
    goto err;
  }

  if (!safe) {
    i = BN_is_prime_fasttest_ex(ret, checks, ctx, 0, cb);
    if (i == -1) {
      goto err;
    } else if (i == 0) {
      goto loop;
    }
  } else {
    /* for "safe prime" generation, check that (p-1)/2 is prime. Since a prime
     * is odd, We just need to divide by 2 */
    if (!BN_rshift1(t, ret)) {
      goto err;
    }

    for (i = 0; i < checks; i++) {
      j = BN_is_prime_fasttest_ex(ret, 1, ctx, 0, NULL);
      if (j == -1) {
        goto err;
      } else if (j == 0) {
        goto loop;
      }

      j = BN_is_prime_fasttest_ex(t, 1, ctx, 0, NULL);
      if (j == -1) {
        goto err;
      } else if (j == 0) {
        goto loop;
      }

      if (!BN_GENCB_call(cb, i, c1 - 1)) {
        goto err;
      }
      /* We have a safe prime test pass */
    }
  }

  /* we have a prime :-) */
  found = 1;

err:
  if (ctx != NULL) {
    BN_CTX_end(ctx);
    BN_CTX_free(ctx);
  }

  return found;
}

int BN_primality_test(int *is_probably_prime, const BIGNUM *candidate,
                      int checks, BN_CTX *ctx, int do_trial_division,
                      BN_GENCB *cb) {
  switch (BN_is_prime_fasttest_ex(candidate, checks, ctx, do_trial_division, cb)) {
    case 1:
      *is_probably_prime = 1;
      return 1;
    case 0:
      *is_probably_prime = 0;
      return 1;
    default:
      *is_probably_prime = 0;
      return 0;
  }
}

int BN_is_prime_ex(const BIGNUM *candidate, int checks, BN_CTX *ctx, BN_GENCB *cb) {
  return BN_is_prime_fasttest_ex(candidate, checks, ctx, 0, cb);
}

int BN_is_prime_fasttest_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
                            int do_trial_division, BN_GENCB *cb) {
  int i, j, ret = -1;
  int k;
  BN_CTX *ctx = NULL;
  BIGNUM *A1, *A1_odd, *check; /* taken from ctx */
  BN_MONT_CTX *mont = NULL;
  const BIGNUM *A = NULL;

  if (BN_cmp(a, BN_value_one()) <= 0) {
    return 0;
  }

  if (checks == BN_prime_checks) {
    checks = BN_prime_checks_for_size(BN_num_bits(a));
  }

  /* first look for small factors */
  if (!BN_is_odd(a)) {
    /* a is even => a is prime if and only if a == 2 */
    return BN_is_word(a, 2);
  }

  if (do_trial_division) {
    for (i = 1; i < NUMPRIMES; i++) {
      if (BN_mod_word(a, primes[i]) == 0) {
        return 0;
      }
    }

    if (!BN_GENCB_call(cb, 1, -1)) {
      goto err;
    }
  }

  if (ctx_passed != NULL) {
    ctx = ctx_passed;
  } else if ((ctx = BN_CTX_new()) == NULL) {
    goto err;
  }
  BN_CTX_start(ctx);

  /* A := abs(a) */
  if (a->neg) {
    BIGNUM *t;
    if ((t = BN_CTX_get(ctx)) == NULL) {
      goto err;
    }
    BN_copy(t, a);
    t->neg = 0;
    A = t;
  } else {
    A = a;
  }

  A1 = BN_CTX_get(ctx);
  A1_odd = BN_CTX_get(ctx);
  check = BN_CTX_get(ctx);
  if (check == NULL) {
    goto err;
  }

  /* compute A1 := A - 1 */
  if (!BN_copy(A1, A)) {
    goto err;
  }
  if (!BN_sub_word(A1, 1)) {
    goto err;
  }
  if (BN_is_zero(A1)) {
    ret = 0;
    goto err;
  }

  /* write  A1  as  A1_odd * 2^k */
  k = 1;
  while (!BN_is_bit_set(A1, k)) {
    k++;
  }
  if (!BN_rshift(A1_odd, A1, k)) {
    goto err;
  }

  /* Montgomery setup for computations mod A */
  mont = BN_MONT_CTX_new();
  if (mont == NULL) {
    goto err;
  }
  if (!BN_MONT_CTX_set(mont, A, ctx)) {
    goto err;
  }

  for (i = 0; i < checks; i++) {
    if (!BN_pseudo_rand_range(check, A1)) {
      goto err;
    }
    if (!BN_add_word(check, 1)) {
      goto err;
    }
    /* now 1 <= check < A */

    j = witness(check, A, A1, A1_odd, k, ctx, mont);
    if (j == -1) {
      goto err;
    }
    if (j) {
      ret = 0;
      goto err;
    }
    if (!BN_GENCB_call(cb, 1, i)) {
      goto err;
    }
  }
  ret = 1;

err:
  if (ctx != NULL) {
    BN_CTX_end(ctx);
    if (ctx_passed == NULL) {
      BN_CTX_free(ctx);
    }
  }
  if (mont != NULL) {
    BN_MONT_CTX_free(mont);
  }

  return ret;
}

static int witness(BIGNUM *w, const BIGNUM *a, const BIGNUM *a1,
                   const BIGNUM *a1_odd, int k, BN_CTX *ctx,
                   BN_MONT_CTX *mont) {
  if (!BN_mod_exp_mont(w, w, a1_odd, a, ctx, mont)) { /* w := w^a1_odd mod a */
    return -1;
  }
  if (BN_is_one(w)) {
    return 0; /* probably prime */
  }
  if (BN_cmp(w, a1) == 0) {
    return 0; /* w == -1 (mod a),  'a' is probably prime */
  }

  while (--k) {
    if (!BN_mod_mul(w, w, w, a, ctx)) { /* w := w^2 mod a */
      return -1;
    }

    if (BN_is_one(w)) {
      return 1; /* 'a' is composite, otherwise a previous 'w' would
                 * have been == -1 (mod 'a') */
    }

    if (BN_cmp(w, a1) == 0) {
      return 0; /* w == -1 (mod a), 'a' is probably prime */
    }
  }

  /* If we get here, 'w' is the (a-1)/2-th power of the original 'w',
   * and it is neither -1 nor +1 -- so 'a' cannot be prime */
  return 1;
}

static BN_ULONG get_word(const BIGNUM *bn) {
  if (bn->top == 1) {
    return bn->d[0];
  }
  return 0;
}

static int probable_prime(BIGNUM *rnd, int bits) {
  int i;
  uint16_t mods[NUMPRIMES];
  BN_ULONG delta;
  BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 1];
  char is_single_word = bits <= BN_BITS2;

again:
  if (!BN_rand(rnd, bits, 1, 1)) {
    return 0;
  }

  /* we now have a random number 'rnd' to test. */
  for (i = 1; i < NUMPRIMES; i++) {
    mods[i] = (uint16_t)BN_mod_word(rnd, (BN_ULONG)primes[i]);
  }
  /* If bits is so small that it fits into a single word then we
   * additionally don't want to exceed that many bits. */
  if (is_single_word) {
    BN_ULONG size_limit = (((BN_ULONG)1) << bits) - get_word(rnd) - 1;
    if (size_limit < maxdelta) {
      maxdelta = size_limit;
    }
  }
  delta = 0;

loop:
  if (is_single_word) {
    BN_ULONG rnd_word = get_word(rnd);

    /* In the case that the candidate prime is a single word then
     * we check that:
     *   1) It's greater than primes[i] because we shouldn't reject
     *      3 as being a prime number because it's a multiple of
     *      three.
     *   2) That it's not a multiple of a known prime. We don't
     *      check that rnd-1 is also coprime to all the known
     *      primes because there aren't many small primes where
     *      that's true. */
    for (i = 1; i < NUMPRIMES && primes[i] < rnd_word; i++) {
      if ((mods[i] + delta) % primes[i] == 0) {
        delta += 2;
        if (delta > maxdelta)
          goto again;
        goto loop;
      }
    }
  } else {
    for (i = 1; i < NUMPRIMES; i++) {
      /* check that rnd is not a prime and also
       * that gcd(rnd-1,primes) == 1 (except for 2) */
      if (((mods[i] + delta) % primes[i]) <= 1) {
        delta += 2;
        if (delta > maxdelta)
          goto again;
        goto loop;
      }
    }
  }

  if (!BN_add_word(rnd, delta)) {
    return 0;
  }
  if (BN_num_bits(rnd) != bits) {
    goto again;
  }

  return 1;
}

static int probable_prime_dh(BIGNUM *rnd, int bits, const BIGNUM *add,
                             const BIGNUM *rem, BN_CTX *ctx) {
  int i, ret = 0;
  BIGNUM *t1;

  BN_CTX_start(ctx);
  if ((t1 = BN_CTX_get(ctx)) == NULL) {
    goto err;
  }

  if (!BN_rand(rnd, bits, 0, 1)) {
    goto err;
  }

  /* we need ((rnd-rem) % add) == 0 */

  if (!BN_mod(t1, rnd, add, ctx)) {
    goto err;
  }
  if (!BN_sub(rnd, rnd, t1)) {
    goto err;
  }
  if (rem == NULL) {
    if (!BN_add_word(rnd, 1)) {
      goto err;
    }
  } else {
    if (!BN_add(rnd, rnd, rem)) {
      goto err;
    }
  }
  /* we now have a random number 'rand' to test. */

loop:
  for (i = 1; i < NUMPRIMES; i++) {
    /* check that rnd is a prime */
    if (BN_mod_word(rnd, (BN_ULONG)primes[i]) <= 1) {
      if (!BN_add(rnd, rnd, add)) {
        goto err;
      }
      goto loop;
    }
  }

  ret = 1;

err:
  BN_CTX_end(ctx);
  return ret;
}

static int probable_prime_dh_safe(BIGNUM *p, int bits, const BIGNUM *padd,
                                  const BIGNUM *rem, BN_CTX *ctx) {
  int i, ret = 0;
  BIGNUM *t1, *qadd, *q;

  bits--;
  BN_CTX_start(ctx);
  t1 = BN_CTX_get(ctx);
  q = BN_CTX_get(ctx);
  qadd = BN_CTX_get(ctx);
  if (qadd == NULL) {
    goto err;
  }

  if (!BN_rshift1(qadd, padd)) {
    goto err;
  }

  if (!BN_rand(q, bits, 0, 1)) {
    goto err;
  }

  /* we need ((rnd-rem) % add) == 0 */
  if (!BN_mod(t1, q, qadd, ctx)) {
    goto err;
  }

  if (!BN_sub(q, q, t1)) {
    goto err;
  }

  if (rem == NULL) {
    if (!BN_add_word(q, 1)) {
      goto err;
    }
  } else {
    if (!BN_rshift1(t1, rem)) {
      goto err;
    }
    if (!BN_add(q, q, t1)) {
      goto err;
    }
  }

  /* we now have a random number 'rand' to test. */
  if (!BN_lshift1(p, q)) {
    goto err;
  }
  if (!BN_add_word(p, 1)) {
    goto err;
  }

loop:
  for (i = 1; i < NUMPRIMES; i++) {
    /* check that p and q are prime */
    /* check that for p and q
     * gcd(p-1,primes) == 1 (except for 2) */
    if ((BN_mod_word(p, (BN_ULONG)primes[i]) == 0) ||
        (BN_mod_word(q, (BN_ULONG)primes[i]) == 0)) {
      if (!BN_add(p, p, padd)) {
        goto err;
      }
      if (!BN_add(q, q, qadd)) {
        goto err;
      }
      goto loop;
    }
  }

  ret = 1;

err:
  BN_CTX_end(ctx);
  return ret;
}

static const uint16_t primes[NUMPRIMES] = {
    2,     3,     5,     7,     11,    13,    17,    19,    23,    29,    31,
    37,    41,    43,    47,    53,    59,    61,    67,    71,    73,    79,
    83,    89,    97,    101,   103,   107,   109,   113,   127,   131,   137,
    139,   149,   151,   157,   163,   167,   173,   179,   181,   191,   193,
    197,   199,   211,   223,   227,   229,   233,   239,   241,   251,   257,
    263,   269,   271,   277,   281,   283,   293,   307,   311,   313,   317,
    331,   337,   347,   349,   353,   359,   367,   373,   379,   383,   389,
    397,   401,   409,   419,   421,   431,   433,   439,   443,   449,   457,
    461,   463,   467,   479,   487,   491,   499,   503,   509,   521,   523,
    541,   547,   557,   563,   569,   571,   577,   587,   593,   599,   601,
    607,   613,   617,   619,   631,   641,   643,   647,   653,   659,   661,
    673,   677,   683,   691,   701,   709,   719,   727,   733,   739,   743,
    751,   757,   761,   769,   773,   787,   797,   809,   811,   821,   823,
    827,   829,   839,   853,   857,   859,   863,   877,   881,   883,   887,
    907,   911,   919,   929,   937,   941,   947,   953,   967,   971,   977,
    983,   991,   997,   1009,  1013,  1019,  1021,  1031,  1033,  1039,  1049,
    1051,  1061,  1063,  1069,  1087,  1091,  1093,  1097,  1103,  1109,  1117,
    1123,  1129,  1151,  1153,  1163,  1171,  1181,  1187,  1193,  1201,  1213,
    1217,  1223,  1229,  1231,  1237,  1249,  1259,  1277,  1279,  1283,  1289,
    1291,  1297,  1301,  1303,  1307,  1319,  1321,  1327,  1361,  1367,  1373,
    1381,  1399,  1409,  1423,  1427,  1429,  1433,  1439,  1447,  1451,  1453,
    1459,  1471,  1481,  1483,  1487,  1489,  1493,  1499,  1511,  1523,  1531,
    1543,  1549,  1553,  1559,  1567,  1571,  1579,  1583,  1597,  1601,  1607,
    1609,  1613,  1619,  1621,  1627,  1637,  1657,  1663,  1667,  1669,  1693,
    1697,  1699,  1709,  1721,  1723,  1733,  1741,  1747,  1753,  1759,  1777,
    1783,  1787,  1789,  1801,  1811,  1823,  1831,  1847,  1861,  1867,  1871,
    1873,  1877,  1879,  1889,  1901,  1907,  1913,  1931,  1933,  1949,  1951,
    1973,  1979,  1987,  1993,  1997,  1999,  2003,  2011,  2017,  2027,  2029,
    2039,  2053,  2063,  2069,  2081,  2083,  2087,  2089,  2099,  2111,  2113,
    2129,  2131,  2137,  2141,  2143,  2153,  2161,  2179,  2203,  2207,  2213,
    2221,  2237,  2239,  2243,  2251,  2267,  2269,  2273,  2281,  2287,  2293,
    2297,  2309,  2311,  2333,  2339,  2341,  2347,  2351,  2357,  2371,  2377,
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