2 * Copyright 2018-2023 The OpenSSL Project Authors. All Rights Reserved.
3 * Copyright (c) 2018-2019, Oracle and/or its affiliates. All rights reserved.
5 * Licensed under the Apache License 2.0 (the "License"). You may not use
6 * this file except in compliance with the License. You can obtain a copy
7 * in the file LICENSE in the source distribution or at
8 * https://www.openssl.org/source/license.html
12 * According to NIST SP800-131A "Transitioning the use of cryptographic
13 * algorithms and key lengths" Generation of 1024 bit RSA keys are no longer
14 * allowed for signatures (Table 2) or key transport (Table 5). In the code
15 * below any attempt to generate 1024 bit RSA keys will result in an error (Note
16 * that digital signature verification can still use deprecated 1024 bit keys).
18 * FIPS 186-4 relies on the use of the auxiliary primes p1, p2, q1 and q2 that
19 * must be generated before the module generates the RSA primes p and q.
20 * Table B.1 in FIPS 186-4 specifies RSA modulus lengths of 2048 and
21 * 3072 bits only, the min/max total length of the auxiliary primes.
22 * FIPS 186-5 Table A.1 includes an additional entry for 4096 which has been
26 #include <openssl/bn.h>
28 #include "crypto/bn.h"
29 #include "internal/nelem.h"
32 # define BN_DEF(lo, hi) (BN_ULONG)hi<<32|lo
34 # define BN_DEF(lo, hi) lo, hi
37 /* 1 / sqrt(2) * 2^256, rounded up */
38 static const BN_ULONG inv_sqrt_2_val[] = {
39 BN_DEF(0x83339916UL, 0xED17AC85UL), BN_DEF(0x893BA84CUL, 0x1D6F60BAUL),
40 BN_DEF(0x754ABE9FUL, 0x597D89B3UL), BN_DEF(0xF9DE6484UL, 0xB504F333UL)
43 const BIGNUM ossl_bn_inv_sqrt_2 = {
44 (BN_ULONG *)inv_sqrt_2_val,
45 OSSL_NELEM(inv_sqrt_2_val),
46 OSSL_NELEM(inv_sqrt_2_val),
52 * FIPS 186-5 Table A.1. "Min length of auxiliary primes p1, p2, q1, q2".
53 * (FIPS 186-5 has an entry for >= 4096 bits).
56 * nbits The key size in bits.
58 * The minimum size of the auxiliary primes or 0 if nbits is invalid.
60 static int bn_rsa_fips186_5_aux_prime_min_size(int nbits)
72 * FIPS 186-5 Table A.1 "Max of len(p1) + len(p2) and
73 * len(q1) + len(q2) for p,q Probable Primes".
74 * (FIPS 186-5 has an entry for >= 4096 bits).
76 * nbits The key size in bits.
78 * The maximum length or 0 if nbits is invalid.
80 static int bn_rsa_fips186_5_aux_prime_max_sum_size_for_prob_primes(int nbits)
92 * Find the first odd integer that is a probable prime.
94 * See section FIPS 186-4 B.3.6 (Steps 4.2/5.2).
97 * Xp1 The passed in starting point to find a probably prime.
98 * p1 The returned probable prime (first odd integer >= Xp1)
99 * ctx A BN_CTX object.
100 * cb An optional BIGNUM callback.
101 * Returns: 1 on success otherwise it returns 0.
103 static int bn_rsa_fips186_4_find_aux_prob_prime(const BIGNUM *Xp1,
104 BIGNUM *p1, BN_CTX *ctx,
111 if (BN_copy(p1, Xp1) == NULL)
113 BN_set_flags(p1, BN_FLG_CONSTTIME);
115 /* Find the first odd number >= Xp1 that is probably prime */
118 BN_GENCB_call(cb, 0, i);
119 /* MR test with trial division */
120 tmp = BN_check_prime(p1, ctx, cb);
125 /* Get next odd number */
126 if (!BN_add_word(p1, 2))
129 BN_GENCB_call(cb, 2, i);
136 * Generate a probable prime (p or q).
138 * See FIPS 186-4 B.3.6 (Steps 4 & 5)
141 * p The returned probable prime.
142 * Xpout An optionally returned random number used during generation of p.
143 * p1, p2 The returned auxiliary primes. If NULL they are not returned.
144 * Xp An optional passed in value (that is random number used during
146 * Xp1, Xp2 Optional passed in values that are normally generated
147 * internally. Used to find p1, p2.
148 * nlen The bit length of the modulus (the key size).
149 * e The public exponent.
150 * ctx A BN_CTX object.
151 * cb An optional BIGNUM callback.
152 * Returns: 1 on success otherwise it returns 0.
154 int ossl_bn_rsa_fips186_4_gen_prob_primes(BIGNUM *p, BIGNUM *Xpout,
155 BIGNUM *p1, BIGNUM *p2,
156 const BIGNUM *Xp, const BIGNUM *Xp1,
157 const BIGNUM *Xp2, int nlen,
158 const BIGNUM *e, BN_CTX *ctx,
162 BIGNUM *p1i = NULL, *p2i = NULL, *Xp1i = NULL, *Xp2i = NULL;
165 if (p == NULL || Xpout == NULL)
170 p1i = (p1 != NULL) ? p1 : BN_CTX_get(ctx);
171 p2i = (p2 != NULL) ? p2 : BN_CTX_get(ctx);
172 Xp1i = (Xp1 != NULL) ? (BIGNUM *)Xp1 : BN_CTX_get(ctx);
173 Xp2i = (Xp2 != NULL) ? (BIGNUM *)Xp2 : BN_CTX_get(ctx);
174 if (p1i == NULL || p2i == NULL || Xp1i == NULL || Xp2i == NULL)
177 bitlen = bn_rsa_fips186_5_aux_prime_min_size(nlen);
181 /* (Steps 4.1/5.1): Randomly generate Xp1 if it is not passed in */
183 /* Set the top and bottom bits to make it odd and the correct size */
184 if (!BN_priv_rand_ex(Xp1i, bitlen, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD,
188 /* (Steps 4.1/5.1): Randomly generate Xp2 if it is not passed in */
190 /* Set the top and bottom bits to make it odd and the correct size */
191 if (!BN_priv_rand_ex(Xp2i, bitlen, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD,
196 /* (Steps 4.2/5.2) - find first auxiliary probable primes */
197 if (!bn_rsa_fips186_4_find_aux_prob_prime(Xp1i, p1i, ctx, cb)
198 || !bn_rsa_fips186_4_find_aux_prob_prime(Xp2i, p2i, ctx, cb))
200 /* (Table B.1) auxiliary prime Max length check */
201 if ((BN_num_bits(p1i) + BN_num_bits(p2i)) >=
202 bn_rsa_fips186_5_aux_prime_max_sum_size_for_prob_primes(nlen))
204 /* (Steps 4.3/5.3) - generate prime */
205 if (!ossl_bn_rsa_fips186_4_derive_prime(p, Xpout, Xp, p1i, p2i, nlen, e,
210 /* Zeroize any internally generated values that are not returned */
224 * Constructs a probable prime (a candidate for p or q) using 2 auxiliary
225 * prime numbers and the Chinese Remainder Theorem.
227 * See FIPS 186-4 C.9 "Compute a Probable Prime Factor Based on Auxiliary
228 * Primes". Used by FIPS 186-4 B.3.6 Section (4.3) for p and Section (5.3) for q.
231 * Y The returned prime factor (private_prime_factor) of the modulus n.
232 * X The returned random number used during generation of the prime factor.
233 * Xin An optional passed in value for X used for testing purposes.
234 * r1 An auxiliary prime.
235 * r2 An auxiliary prime.
236 * nlen The desired length of n (the RSA modulus).
237 * e The public exponent.
238 * ctx A BN_CTX object.
239 * cb An optional BIGNUM callback object.
240 * Returns: 1 on success otherwise it returns 0.
242 * Y, X, r1, r2, e are not NULL.
244 int ossl_bn_rsa_fips186_4_derive_prime(BIGNUM *Y, BIGNUM *X, const BIGNUM *Xin,
245 const BIGNUM *r1, const BIGNUM *r2,
246 int nlen, const BIGNUM *e, BN_CTX *ctx,
251 int bits = nlen >> 1;
252 BIGNUM *tmp, *R, *r1r2x2, *y1, *r1x2;
253 BIGNUM *base, *range;
257 base = BN_CTX_get(ctx);
258 range = BN_CTX_get(ctx);
260 tmp = BN_CTX_get(ctx);
261 r1r2x2 = BN_CTX_get(ctx);
262 y1 = BN_CTX_get(ctx);
263 r1x2 = BN_CTX_get(ctx);
267 if (Xin != NULL && BN_copy(X, Xin) == NULL)
271 * We need to generate a random number X in the range
272 * 1/sqrt(2) * 2^(nlen/2) <= X < 2^(nlen/2).
273 * We can rewrite that as:
274 * base = 1/sqrt(2) * 2^(nlen/2)
275 * range = ((2^(nlen/2))) - (1/sqrt(2) * 2^(nlen/2))
276 * X = base + random(range)
277 * We only have the first 256 bit of 1/sqrt(2)
280 if (bits < BN_num_bits(&ossl_bn_inv_sqrt_2))
282 if (!BN_lshift(base, &ossl_bn_inv_sqrt_2,
283 bits - BN_num_bits(&ossl_bn_inv_sqrt_2))
284 || !BN_lshift(range, BN_value_one(), bits)
285 || !BN_sub(range, range, base))
289 if (!(BN_lshift1(r1x2, r1)
290 /* (Step 1) GCD(2r1, r2) = 1 */
291 && BN_gcd(tmp, r1x2, r2, ctx)
293 /* (Step 2) R = ((r2^-1 mod 2r1) * r2) - ((2r1^-1 mod r2)*2r1) */
294 && BN_mod_inverse(R, r2, r1x2, ctx)
295 && BN_mul(R, R, r2, ctx) /* R = (r2^-1 mod 2r1) * r2 */
296 && BN_mod_inverse(tmp, r1x2, r2, ctx)
297 && BN_mul(tmp, tmp, r1x2, ctx) /* tmp = (2r1^-1 mod r2)*2r1 */
299 /* Calculate 2r1r2 */
300 && BN_mul(r1r2x2, r1x2, r2, ctx)))
302 /* Make positive by adding the modulus */
303 if (BN_is_negative(R) && !BN_add(R, R, r1r2x2))
307 * In FIPS 186-4 imax was set to 5 * nlen/2.
308 * Analysis by Allen Roginsky (See https://csrc.nist.gov/CSRC/media/Publications/fips/186/4/final/documents/comments-received-fips186-4-december-2015.pdf
309 * page 68) indicates this has a 1 in 2 million chance of failure.
310 * The number has been updated to 20 * nlen/2 as used in
311 * FIPS186-5 Appendix B.9 Step 9.
313 imax = 20 * bits; /* max = 20/2 * nbits */
317 * (Step 3) Choose Random X such that
318 * sqrt(2) * 2^(nlen/2-1) <= Random X <= (2^(nlen/2)) - 1.
320 if (!BN_priv_rand_range_ex(X, range, 0, ctx) || !BN_add(X, X, base))
323 /* (Step 4) Y = X + ((R - X) mod 2r1r2) */
324 if (!BN_mod_sub(Y, R, X, r1r2x2, ctx) || !BN_add(Y, Y, X))
330 if (BN_num_bits(Y) > bits) {
332 break; /* Randomly Generated X so Go back to Step 3 */
334 goto err; /* X is not random so it will always fail */
336 BN_GENCB_call(cb, 0, 2);
338 /* (Step 7) If GCD(Y-1) == 1 & Y is probably prime then return Y */
339 if (BN_copy(y1, Y) == NULL
340 || !BN_sub_word(y1, 1)
341 || !BN_gcd(tmp, y1, e, ctx))
343 if (BN_is_one(tmp)) {
344 int rv = BN_check_prime(Y, ctx, cb);
353 ERR_raise(ERR_LIB_BN, BN_R_NO_PRIME_CANDIDATE);
356 if (!BN_add(Y, Y, r1r2x2))
362 BN_GENCB_call(cb, 3, 0);