/* * Copyright (c) 2018 Thomas Pornin * * Permission is hereby granted, free of charge, to any person obtaining * a copy of this software and associated documentation files (the * "Software"), to deal in the Software without restriction, including * without limitation the rights to use, copy, modify, merge, publish, * distribute, sublicense, and/or sell copies of the Software, and to * permit persons to whom the Software is furnished to do so, subject to * the following conditions: * * The above copyright notice and this permission notice shall be * included in all copies or substantial portions of the Software. * * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS * BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN * ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE * SOFTWARE. */ #include "inner.h" /* * Make a random integer of the provided size. The size is encoded. * The header word is untouched. */ static void mkrand(const br_prng_class **rng, uint32_t *x, uint32_t esize) { size_t u, len; unsigned m; len = (esize + 31) >> 5; (*rng)->generate(rng, x + 1, len * sizeof(uint32_t)); for (u = 1; u < len; u ++) { x[u] &= 0x7FFFFFFF; } m = esize & 31; if (m == 0) { x[len] &= 0x7FFFFFFF; } else { x[len] &= 0x7FFFFFFF >> (31 - m); } } /* * This is the big-endian unsigned representation of the product of * all small primes from 13 to 1481. */ static const unsigned char SMALL_PRIMES[] = { 0x2E, 0xAB, 0x92, 0xD1, 0x8B, 0x12, 0x47, 0x31, 0x54, 0x0A, 0x99, 0x5D, 0x25, 0x5E, 0xE2, 0x14, 0x96, 0x29, 0x1E, 0xB7, 0x78, 0x70, 0xCC, 0x1F, 0xA5, 0xAB, 0x8D, 0x72, 0x11, 0x37, 0xFB, 0xD8, 0x1E, 0x3F, 0x5B, 0x34, 0x30, 0x17, 0x8B, 0xE5, 0x26, 0x28, 0x23, 0xA1, 0x8A, 0xA4, 0x29, 0xEA, 0xFD, 0x9E, 0x39, 0x60, 0x8A, 0xF3, 0xB5, 0xA6, 0xEB, 0x3F, 0x02, 0xB6, 0x16, 0xC3, 0x96, 0x9D, 0x38, 0xB0, 0x7D, 0x82, 0x87, 0x0C, 0xF7, 0xBE, 0x24, 0xE5, 0x5F, 0x41, 0x04, 0x79, 0x76, 0x40, 0xE7, 0x00, 0x22, 0x7E, 0xB5, 0x85, 0x7F, 0x8D, 0x01, 0x50, 0xE9, 0xD3, 0x29, 0x42, 0x08, 0xB3, 0x51, 0x40, 0x7B, 0xD7, 0x8D, 0xCC, 0x10, 0x01, 0x64, 0x59, 0x28, 0xB6, 0x53, 0xF3, 0x50, 0x4E, 0xB1, 0xF2, 0x58, 0xCD, 0x6E, 0xF5, 0x56, 0x3E, 0x66, 0x2F, 0xD7, 0x07, 0x7F, 0x52, 0x4C, 0x13, 0x24, 0xDC, 0x8E, 0x8D, 0xCC, 0xED, 0x77, 0xC4, 0x21, 0xD2, 0xFD, 0x08, 0xEA, 0xD7, 0xC0, 0x5C, 0x13, 0x82, 0x81, 0x31, 0x2F, 0x2B, 0x08, 0xE4, 0x80, 0x04, 0x7A, 0x0C, 0x8A, 0x3C, 0xDC, 0x22, 0xE4, 0x5A, 0x7A, 0xB0, 0x12, 0x5E, 0x4A, 0x76, 0x94, 0x77, 0xC2, 0x0E, 0x92, 0xBA, 0x8A, 0xA0, 0x1F, 0x14, 0x51, 0x1E, 0x66, 0x6C, 0x38, 0x03, 0x6C, 0xC7, 0x4A, 0x4B, 0x70, 0x80, 0xAF, 0xCA, 0x84, 0x51, 0xD8, 0xD2, 0x26, 0x49, 0xF5, 0xA8, 0x5E, 0x35, 0x4B, 0xAC, 0xCE, 0x29, 0x92, 0x33, 0xB7, 0xA2, 0x69, 0x7D, 0x0C, 0xE0, 0x9C, 0xDB, 0x04, 0xD6, 0xB4, 0xBC, 0x39, 0xD7, 0x7F, 0x9E, 0x9D, 0x78, 0x38, 0x7F, 0x51, 0x54, 0x50, 0x8B, 0x9E, 0x9C, 0x03, 0x6C, 0xF5, 0x9D, 0x2C, 0x74, 0x57, 0xF0, 0x27, 0x2A, 0xC3, 0x47, 0xCA, 0xB9, 0xD7, 0x5C, 0xFF, 0xC2, 0xAC, 0x65, 0x4E, 0xBD }; /* * We need temporary values for at least 7 integers of the same size * as a factor (including header word); more space helps with performance * (in modular exponentiations), but we much prefer to remain under * 2 kilobytes in total, to save stack space. The macro TEMPS below * exceeds 512 (which is a count in 32-bit words) when BR_MAX_RSA_SIZE * is greater than 4464 (default value is 4096, so the 2-kB limit is * maintained unless BR_MAX_RSA_SIZE was modified). */ #define MAX(x, y) ((x) > (y) ? (x) : (y)) #define ROUND2(x) ((((x) + 1) >> 1) << 1) #define TEMPS MAX(512, ROUND2(7 * ((((BR_MAX_RSA_SIZE + 1) >> 1) + 61) / 31))) /* * Perform trial division on a candidate prime. This computes * y = SMALL_PRIMES mod x, then tries to compute y/y mod x. The * br_i31_moddiv() function will report an error if y is not invertible * modulo x. Returned value is 1 on success (none of the small primes * divides x), 0 on error (a non-trivial GCD is obtained). * * This function assumes that x is odd. */ static uint32_t trial_divisions(const uint32_t *x, uint32_t *t) { uint32_t *y; uint32_t x0i; y = t; t += 1 + ((x[0] + 31) >> 5); x0i = br_i31_ninv31(x[1]); br_i31_decode_reduce(y, SMALL_PRIMES, sizeof SMALL_PRIMES, x); return br_i31_moddiv(y, y, x, x0i, t); } /* * Perform n rounds of Miller-Rabin on the candidate prime x. This * function assumes that x = 3 mod 4. * * Returned value is 1 on success (all rounds completed successfully), * 0 otherwise. */ static uint32_t miller_rabin(const br_prng_class **rng, const uint32_t *x, int n, uint32_t *t, size_t tlen, br_i31_modpow_opt_type mp31) { /* * Since x = 3 mod 4, the Miller-Rabin test is simple: * - get a random base a (such that 1 < a < x-1) * - compute z = a^((x-1)/2) mod x * - if z != 1 and z != x-1, the number x is composite * * We generate bases 'a' randomly with a size which is * one bit less than x, which ensures that a < x-1. It * is not useful to verify that a > 1 because the probability * that we get a value a equal to 0 or 1 is much smaller * than the probability of our Miller-Rabin tests not to * detect a composite, which is already quite smaller than the * probability of the hardware misbehaving and return a * composite integer because of some glitch (e.g. bad RAM * or ill-timed cosmic ray). */ unsigned char *xm1d2; size_t xlen, xm1d2_len, xm1d2_len_u32, u; uint32_t asize; unsigned cc; uint32_t x0i; /* * Compute (x-1)/2 (encoded). */ xm1d2 = (unsigned char *)t; xm1d2_len = ((x[0] - (x[0] >> 5)) + 7) >> 3; br_i31_encode(xm1d2, xm1d2_len, x); cc = 0; for (u = 0; u < xm1d2_len; u ++) { unsigned w; w = xm1d2[u]; xm1d2[u] = (unsigned char)((w >> 1) | cc); cc = w << 7; } /* * We used some words of the provided buffer for (x-1)/2. */ xm1d2_len_u32 = (xm1d2_len + 3) >> 2; t += xm1d2_len_u32; tlen -= xm1d2_len_u32; xlen = (x[0] + 31) >> 5; asize = x[0] - 1 - EQ0(x[0] & 31); x0i = br_i31_ninv31(x[1]); while (n -- > 0) { uint32_t *a, *t2; uint32_t eq1, eqm1; size_t t2len; /* * Generate a random base. We don't need the base to be * really uniform modulo x, so we just get a random * number which is one bit shorter than x. */ a = t; a[0] = x[0]; a[xlen] = 0; mkrand(rng, a, asize); /* * Compute a^((x-1)/2) mod x. We assume here that the * function will not fail (the temporary array is large * enough). */ t2 = t + 1 + xlen; t2len = tlen - 1 - xlen; if ((t2len & 1) != 0) { /* * Since the source array is 64-bit aligned and * has an even number of elements (TEMPS), we * can use the parity of the remaining length to * detect and adjust alignment. */ t2 ++; t2len --; } mp31(a, xm1d2, xm1d2_len, x, x0i, t2, t2len); /* * We must obtain either 1 or x-1. Note that x is odd, * hence x-1 differs from x only in its low word (no * carry). */ eq1 = a[1] ^ 1; eqm1 = a[1] ^ (x[1] - 1); for (u = 2; u <= xlen; u ++) { eq1 |= a[u]; eqm1 |= a[u] ^ x[u]; } if ((EQ0(eq1) | EQ0(eqm1)) == 0) { return 0; } } return 1; } /* * Create a random prime of the provided size. 'size' is the _encoded_ * bit length. The two top bits and the two bottom bits are set to 1. */ static void mkprime(const br_prng_class **rng, uint32_t *x, uint32_t esize, uint32_t pubexp, uint32_t *t, size_t tlen, br_i31_modpow_opt_type mp31) { size_t len; x[0] = esize; len = (esize + 31) >> 5; for (;;) { size_t u; uint32_t m3, m5, m7, m11; int rounds, s7, s11; /* * Generate random bits. We force the two top bits and the * two bottom bits to 1. */ mkrand(rng, x, esize); if ((esize & 31) == 0) { x[len] |= 0x60000000; } else if ((esize & 31) == 1) { x[len] |= 0x00000001; x[len - 1] |= 0x40000000; } else { x[len] |= 0x00000003 << ((esize & 31) - 2); } x[1] |= 0x00000003; /* * Trial division with low primes (3, 5, 7 and 11). We * use the following properties: * * 2^2 = 1 mod 3 * 2^4 = 1 mod 5 * 2^3 = 1 mod 7 * 2^10 = 1 mod 11 */ m3 = 0; m5 = 0; m7 = 0; m11 = 0; s7 = 0; s11 = 0; for (u = 0; u < len; u ++) { uint32_t w, w3, w5, w7, w11; w = x[1 + u]; w3 = (w & 0xFFFF) + (w >> 16); /* max: 98302 */ w5 = (w & 0xFFFF) + (w >> 16); /* max: 98302 */ w7 = (w & 0x7FFF) + (w >> 15); /* max: 98302 */ w11 = (w & 0xFFFFF) + (w >> 20); /* max: 1050622 */ m3 += w3 << (u & 1); m3 = (m3 & 0xFF) + (m3 >> 8); /* max: 1025 */ m5 += w5 << ((4 - u) & 3); m5 = (m5 & 0xFFF) + (m5 >> 12); /* max: 4479 */ m7 += w7 << s7; m7 = (m7 & 0x1FF) + (m7 >> 9); /* max: 1280 */ if (++ s7 == 3) { s7 = 0; } m11 += w11 << s11; if (++ s11 == 10) { s11 = 0; } m11 = (m11 & 0x3FF) + (m11 >> 10); /* max: 526847 */ } m3 = (m3 & 0x3F) + (m3 >> 6); /* max: 78 */ m3 = (m3 & 0x0F) + (m3 >> 4); /* max: 18 */ m3 = ((m3 * 43) >> 5) & 3; m5 = (m5 & 0xFF) + (m5 >> 8); /* max: 271 */ m5 = (m5 & 0x0F) + (m5 >> 4); /* max: 31 */ m5 -= 20 & -GT(m5, 19); m5 -= 10 & -GT(m5, 9); m5 -= 5 & -GT(m5, 4); m7 = (m7 & 0x3F) + (m7 >> 6); /* max: 82 */ m7 = (m7 & 0x07) + (m7 >> 3); /* max: 16 */ m7 = ((m7 * 147) >> 7) & 7; /* * 2^5 = 32 = -1 mod 11. */ m11 = (m11 & 0x3FF) + (m11 >> 10); /* max: 1536 */ m11 = (m11 & 0x3FF) + (m11 >> 10); /* max: 1023 */ m11 = (m11 & 0x1F) + 33 - (m11 >> 5); /* max: 64 */ m11 -= 44 & -GT(m11, 43); m11 -= 22 & -GT(m11, 21); m11 -= 11 & -GT(m11, 10); /* * If any of these modulo is 0, then the candidate is * not prime. Also, if pubexp is 3, 5, 7 or 11, and the * corresponding modulus is 1, then the candidate must * be rejected, because we need e to be invertible * modulo p-1. We can use simple comparisons here * because they won't leak information on a candidate * that we keep, only on one that we reject (and is thus * not secret). */ if (m3 == 0 || m5 == 0 || m7 == 0 || m11 == 0) { continue; } if ((pubexp == 3 && m3 == 1) || (pubexp == 5 && m5 == 5) || (pubexp == 7 && m5 == 7) || (pubexp == 11 && m5 == 11)) { continue; } /* * More trial divisions. */ if (!trial_divisions(x, t)) { continue; } /* * Miller-Rabin algorithm. Since we selected a random * integer, not a maliciously crafted integer, we can use * relatively few rounds to lower the risk of a false * positive (i.e. declaring prime a non-prime) under * 2^(-80). It is not useful to lower the probability much * below that, since that would be substantially below * the probability of the hardware misbehaving. Sufficient * numbers of rounds are extracted from the Handbook of * Applied Cryptography, note 4.49 (page 149). * * Since we work on the encoded size (esize), we need to * compare with encoded thresholds. */ if (esize < 309) { rounds = 12; } else if (esize < 464) { rounds = 9; } else if (esize < 670) { rounds = 6; } else if (esize < 877) { rounds = 4; } else if (esize < 1341) { rounds = 3; } else { rounds = 2; } if (miller_rabin(rng, x, rounds, t, tlen, mp31)) { return; } } } /* * Let p be a prime (p > 2^33, p = 3 mod 4). Let m = (p-1)/2, provided * as parameter (with announced bit length equal to that of p). This * function computes d = 1/e mod p-1 (for an odd integer e). Returned * value is 1 on success, 0 on error (an error is reported if e is not * invertible modulo p-1). * * The temporary buffer (t) must have room for at least 4 integers of * the size of p. */ static uint32_t invert_pubexp(uint32_t *d, const uint32_t *m, uint32_t e, uint32_t *t) { uint32_t *f; uint32_t r; f = t; t += 1 + ((m[0] + 31) >> 5); /* * Compute d = 1/e mod m. Since p = 3 mod 4, m is odd. */ br_i31_zero(d, m[0]); d[1] = 1; br_i31_zero(f, m[0]); f[1] = e & 0x7FFFFFFF; f[2] = e >> 31; r = br_i31_moddiv(d, f, m, br_i31_ninv31(m[1]), t); /* * We really want d = 1/e mod p-1, with p = 2m. By the CRT, * the result is either the d we got, or d + m. * * Let's write e*d = 1 + k*m, for some integer k. Integers e * and m are odd. If d is odd, then e*d is odd, which implies * that k must be even; in that case, e*d = 1 + (k/2)*2m, and * thus d is already fine. Conversely, if d is even, then k * is odd, and we must add m to d in order to get the correct * result. */ br_i31_add(d, m, (uint32_t)(1 - (d[1] & 1))); return r; } /* * Swap two buffers in RAM. They must be disjoint. */ static void bufswap(void *b1, void *b2, size_t len) { size_t u; unsigned char *buf1, *buf2; buf1 = b1; buf2 = b2; for (u = 0; u < len; u ++) { unsigned w; w = buf1[u]; buf1[u] = buf2[u]; buf2[u] = w; } } /* see inner.h */ uint32_t br_rsa_i31_keygen_inner(const br_prng_class **rng, br_rsa_private_key *sk, void *kbuf_priv, br_rsa_public_key *pk, void *kbuf_pub, unsigned size, uint32_t pubexp, br_i31_modpow_opt_type mp31) { uint32_t esize_p, esize_q; size_t plen, qlen, tlen; uint32_t *p, *q, *t; union { uint32_t t32[TEMPS]; uint64_t t64[TEMPS >> 1]; /* for 64-bit alignment */ } tmp; uint32_t r; if (size < BR_MIN_RSA_SIZE || size > BR_MAX_RSA_SIZE) { return 0; } if (pubexp == 0) { pubexp = 3; } else if (pubexp == 1 || (pubexp & 1) == 0) { return 0; } esize_p = (size + 1) >> 1; esize_q = size - esize_p; sk->n_bitlen = size; sk->p = kbuf_priv; sk->plen = (esize_p + 7) >> 3; sk->q = sk->p + sk->plen; sk->qlen = (esize_q + 7) >> 3; sk->dp = sk->q + sk->qlen; sk->dplen = sk->plen; sk->dq = sk->dp + sk->dplen; sk->dqlen = sk->qlen; sk->iq = sk->dq + sk->dqlen; sk->iqlen = sk->plen; if (pk != NULL) { pk->n = kbuf_pub; pk->nlen = (size + 7) >> 3; pk->e = pk->n + pk->nlen; pk->elen = 4; br_enc32be(pk->e, pubexp); while (*pk->e == 0) { pk->e ++; pk->elen --; } } /* * We now switch to encoded sizes. * * floor((x * 16913) / (2^19)) is equal to floor(x/31) for all * integers x from 0 to 34966; the intermediate product fits on * 30 bits, thus we can use MUL31(). */ esize_p += MUL31(esize_p, 16913) >> 19; esize_q += MUL31(esize_q, 16913) >> 19; plen = (esize_p + 31) >> 5; qlen = (esize_q + 31) >> 5; p = tmp.t32; q = p + 1 + plen; t = q + 1 + qlen; tlen = ((sizeof tmp.t32) / sizeof(uint32_t)) - (2 + plen + qlen); /* * When looking for primes p and q, we temporarily divide * candidates by 2, in order to compute the inverse of the * public exponent. */ for (;;) { mkprime(rng, p, esize_p, pubexp, t, tlen, mp31); br_i31_rshift(p, 1); if (invert_pubexp(t, p, pubexp, t + 1 + plen)) { br_i31_add(p, p, 1); p[1] |= 1; br_i31_encode(sk->p, sk->plen, p); br_i31_encode(sk->dp, sk->dplen, t); break; } } for (;;) { mkprime(rng, q, esize_q, pubexp, t, tlen, mp31); br_i31_rshift(q, 1); if (invert_pubexp(t, q, pubexp, t + 1 + qlen)) { br_i31_add(q, q, 1); q[1] |= 1; br_i31_encode(sk->q, sk->qlen, q); br_i31_encode(sk->dq, sk->dqlen, t); break; } } /* * If p and q have the same size, then it is possible that q > p * (when the target modulus size is odd, we generate p with a * greater bit length than q). If q > p, we want to swap p and q * (and also dp and dq) for two reasons: * - The final step below (inversion of q modulo p) is easier if * p > q. * - While BearSSL's RSA code is perfectly happy with RSA keys such * that p < q, some other implementations have restrictions and * require p > q. * * Note that we can do a simple non-constant-time swap here, * because the only information we leak here is that we insist on * returning p and q such that p > q, which is not a secret. */ if (esize_p == esize_q && br_i31_sub(p, q, 0) == 1) { bufswap(p, q, (1 + plen) * sizeof *p); bufswap(sk->p, sk->q, sk->plen); bufswap(sk->dp, sk->dq, sk->dplen); } /* * We have produced p, q, dp and dq. We can now compute iq = 1/d mod p. * * We ensured that p >= q, so this is just a matter of updating the * header word for q (and possibly adding an extra word). * * Theoretically, the call below may fail, in case we were * extraordinarily unlucky, and p = q. Another failure case is if * Miller-Rabin failed us _twice_, and p and q are non-prime and * have a factor is common. We report the error mostly because it * is cheap and we can, but in practice this never happens (or, at * least, it happens way less often than hardware glitches). */ q[0] = p[0]; if (plen > qlen) { q[plen] = 0; t ++; tlen --; } br_i31_zero(t, p[0]); t[1] = 1; r = br_i31_moddiv(t, q, p, br_i31_ninv31(p[1]), t + 1 + plen); br_i31_encode(sk->iq, sk->iqlen, t); /* * Compute the public modulus too, if required. */ if (pk != NULL) { br_i31_zero(t, p[0]); br_i31_mulacc(t, p, q); br_i31_encode(pk->n, pk->nlen, t); } return r; }