Make number of Miller-Rabin tests for a prime tests depend on the security level of the prime

The old numbers where all generated for an 80 bit security level. But
the number should depend on security level you want to reach. For bigger
primes we want a higher security level and so need to do more tests.

Reviewed-by: Richard Levitte <levitte@openssl.org>
Reviewed-by: Matthias St. Pierre <Matthias.St.Pierre@ncp-e.com>
Reviewed-by: Paul Dale <paul.dale@oracle.com>
GH: #6075
Fixes: #6012

1 files changed, 69 insertions, 18 deletions

diff --git a/include/openssl/bn.h b/include/openssl/bn.h
index 4678bb0b21..8af05d00e5 100644
--- a/include/openssl/bn.h
+++ b/include/openssl/bn.h
@@ -107,25 +107,76 @@ void *BN_GENCB_get_arg(BN_GENCB *cb);
                                  * on the size of the number */
 
 /*
- * 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)
+ * BN_prime_checks_for_size() returns the number of Miller-Rabin iterations
+ * that will be done for checking that a random number is probably prime. The
+ * error rate for accepting a composite number as prime depends on the size of
+ * the prime |b|. The error rates used are for calculating an RSA key with 2 primes,
+ * and so the level is what you would expect for a key of double the size of the
+ * prime.
+ *
+ * This table is generated using the algorithm of FIPS PUB 186-4
+ * Digital Signature Standard (DSS), section F.1, page 117.
+ * (https://dx.doi.org/10.6028/NIST.FIPS.186-4)
+ *
+ * The following magma script was used to generate the output:
+ * securitybits:=125;
+ * k:=1024;
+ * for t:=1 to 65 do
+ *   for M:=3 to Floor(2*Sqrt(k-1)-1) do
+ *     S:=0;
+ *     // Sum over m
+ *     for m:=3 to M do
+ *       s:=0;
+ *       // Sum over j
+ *       for j:=2 to m do
+ *         s+:=(RealField(32)!2)^-(j+(k-1)/j);
+ *       end for;
+ *       S+:=2^(m-(m-1)*t)*s;
+ *     end for;
+ *     A:=2^(k-2-M*t);
+ *     B:=8*(Pi(RealField(32))^2-6)/3*2^(k-2)*S;
+ *     pkt:=2.00743*Log(2)*k*2^-k*(A+B);
+ *     seclevel:=Floor(-Log(2,pkt));
+ *     if seclevel ge securitybits then
+ *       printf "k: %5o, security: %o bits  (t: %o, M: %o)\n",k,seclevel,t,M;
+ *       break;
+ *     end if;
+ *   end for;
+ *   if seclevel ge securitybits then break; end if;
+ * end for;
+ *
+ * It can be run online at:
+ * http://magma.maths.usyd.edu.au/calc
+ *
+ * And will output:
+ * k:  1024, security: 129 bits  (t: 6, M: 23)
+ *
+ * k is the number of bits of the prime, securitybits is the level we want to
+ * reach.
+ *
+ * prime length | RSA key size | # MR tests | security level
+ * -------------+--------------|------------+---------------
+ *  (b) >= 6394 |     >= 12788 |          3 |        256 bit
+ *  (b) >= 3747 |     >=  7494 |          3 |        192 bit
+ *  (b) >= 1345 |     >=  2690 |          4 |        128 bit
+ *  (b) >= 1080 |     >=  2160 |          5 |        128 bit
+ *  (b) >=  852 |     >=  1704 |          5 |        112 bit
+ *  (b) >=  476 |     >=   952 |          5 |         80 bit
+ *  (b) >=  400 |     >=   800 |          6 |         80 bit
+ *  (b) >=  347 |     >=   694 |          7 |         80 bit
+ *  (b) >=  308 |     >=   616 |          8 |         80 bit
+ *  (b) >=   55 |     >=   110 |         27 |         64 bit
+ *  (b) >=    6 |     >=    12 |         34 |         64 bit
  */
-# 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)
+
+# define BN_prime_checks_for_size(b) ((b) >= 3747 ?  3 : \
+                                (b) >=  1345 ?  4 : \
+                                (b) >=  476 ?  5 : \
+                                (b) >=  400 ?  6 : \
+                                (b) >=  347 ?  7 : \
+                                (b) >=  308 ?  8 : \
+                                (b) >=  55  ? 27 : \
+                                /* b >= 6 */ 34)
 
 # define BN_num_bytes(a) ((BN_num_bits(a)+7)/8)