Fixed comment.

[BearSSL] / src / rsa / rsa_i15_keygen.c

/*
 * Copyright (c) 2018 Thomas Pornin <pornin@bolet.org>
 *
 * 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, uint16_t *x, uint32_t esize)
{
        size_t u, len;
        unsigned m;

        len = (esize + 15) >> 4;
        (*rng)->generate(rng, x + 1, len * sizeof(uint16_t));
        for (u = 1; u < len; u ++) {
                x[u] &= 0x7FFF;
        }
        m = esize & 15;
        if (m == 0) {
                x[len] &= 0x7FFF;
        } else {
                x[len] &= 0x7FFF >> (15 - 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 1024 (which is a count in 16-bit words) when BR_MAX_RSA_SIZE
 * is greater than 4350 (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 TEMPS       MAX(1024, 7 * ((((BR_MAX_RSA_SIZE + 1) >> 1) + 29) / 15))

/*
 * Perform trial division on a candidate prime. This computes
 * y = SMALL_PRIMES mod x, then tries to compute y/y mod x. The
 * br_i15_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 uint16_t *x, uint16_t *t)
{
        uint16_t *y;
        uint16_t x0i;

        y = t;
        t += 1 + ((x[0] + 15) >> 4);
        x0i = br_i15_ninv15(x[1]);
        br_i15_decode_reduce(y, SMALL_PRIMES, sizeof SMALL_PRIMES, x);
        return br_i15_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 uint16_t *x, int n,
        uint16_t *t, size_t tlen)
{
        /*
         * 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_u16, u;
        uint32_t asize;
        unsigned cc;
        uint16_t x0i;

        /*
         * Compute (x-1)/2 (encoded).
         */
        xm1d2 = (unsigned char *)t;
        xm1d2_len = ((x[0] - (x[0] >> 4)) + 7) >> 3;
        br_i15_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_u16 = (xm1d2_len + 1) >> 1;
        t += xm1d2_len_u16;
        tlen -= xm1d2_len_u16;

        xlen = (x[0] + 15) >> 4;
        asize = x[0] - 1 - EQ0(x[0] & 15);
        x0i = br_i15_ninv15(x[1]);
        while (n -- > 0) {
                uint16_t *a;
                uint32_t eq1, eqm1;

                /*
                 * 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).
                 */
                br_i15_modpow_opt(a, xm1d2, xm1d2_len,
                        x, x0i, t + 1 + xlen, tlen - 1 - xlen);

                /*
                 * 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, uint16_t *x, uint32_t esize,
        uint32_t pubexp, uint16_t *t, size_t tlen)
{
        size_t len;

        x[0] = esize;
        len = (esize + 15) >> 4;
        for (;;) {
                size_t u;
                uint32_t m3, m5, m7, m11;
                int rounds;

                /*
                 * Generate random bits. We force the two top bits and the
                 * two bottom bits to 1.
                 */
                mkrand(rng, x, esize);
                if ((esize & 15) == 0) {
                        x[len] |= 0x6000;
                } else if ((esize & 15) == 1) {
                        x[len] |= 0x0001;
                        x[len - 1] |= 0x4000;
                } else {
                        x[len] |= 0x0003 << ((esize & 15) - 2);
                }
                x[1] |= 0x0003;

                /*
                 * 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;
                for (u = 0; u < len; u ++) {
                        uint32_t w;

                        w = x[1 + u];
                        m3 += w << (u & 1);
                        m3 = (m3 & 0xFF) + (m3 >> 8);
                        m5 += w << ((4 - u) & 3);
                        m5 = (m5 & 0xFF) + (m5 >> 8);
                        m7 += w;
                        m7 = (m7 & 0x1FF) + (m7 >> 9);
                        m11 += w << (5 & -(u & 1));
                        m11 = (m11 & 0x3FF) + (m11 >> 10);
                }

                /*
                 * Maximum values of m* at this point:
                 *  m3:   511
                 *  m5:   2310
                 *  m7:   510
                 *  m11:  2047
                 * We use the same properties to make further reductions.
                 */

                m3 = (m3 & 0x0F) + (m3 >> 4);      /* max: 46 */
                m3 = (m3 & 0x0F) + (m3 >> 4);      /* max: 16 */
                m3 = ((m3 * 43) >> 5) & 3;

                m5 = (m5 & 0xFF) + (m5 >> 8);      /* max: 263 */
                m5 = (m5 & 0x0F) + (m5 >> 4);      /* max: 30 */
                m5 = (m5 & 0x0F) + (m5 >> 4);      /* max: 15 */
                m5 -= 10 & -GT(m5, 9);
                m5 -= 5 & -GT(m5, 4);

                m7 = (m7 & 0x3F) + (m7 >> 6);      /* max: 69 */
                m7 = (m7 & 7) + (m7 >> 3);         /* max: 14 */
                m7 = ((m7 * 147) >> 7) & 7;

                /*
                 * 2^5 = 32 = -1 mod 11.
                 */
                m11 = (m11 & 0x1F) + 66 - (m11 >> 5);   /* max: 97 */
                m11 -= 88 & -GT(m11, 87);
                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 == 1)
                        || (pubexp == 7 && m7 == 1)
                        || (pubexp == 11 && m11 == 1))
                {
                        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 < 320) {
                        rounds = 12;
                } else if (esize < 480) {
                        rounds = 9;
                } else if (esize < 693) {
                        rounds = 6;
                } else if (esize < 906) {
                        rounds = 4;
                } else if (esize < 1386) {
                        rounds = 3;
                } else {
                        rounds = 2;
                }

                if (miller_rabin(rng, x, rounds, t, tlen)) {
                        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(uint16_t *d, const uint16_t *m, uint32_t e, uint16_t *t)
{
        uint16_t *f;
        uint32_t r;

        f = t;
        t += 1 + ((m[0] + 15) >> 4);

        /*
         * Compute d = 1/e mod m. Since p = 3 mod 4, m is odd.
         */
        br_i15_zero(d, m[0]);
        d[1] = 1;
        br_i15_zero(f, m[0]);
        f[1] = e & 0x7FFF;
        f[2] = (e >> 15) & 0x7FFF;
        f[3] = e >> 30;
        r = br_i15_moddiv(d, f, m, br_i15_ninv15(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_i15_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 bearssl_rsa.h */
uint32_t
br_rsa_i15_keygen(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)
{
        uint32_t esize_p, esize_q;
        size_t plen, qlen, tlen;
        uint16_t *p, *q, *t;
        uint16_t tmp[TEMPS];
        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 * 17477) / (2^18)) is equal to floor(x/15) for all
         * integers x from 0 to 23833.
         */
        esize_p += MUL15(esize_p, 17477) >> 18;
        esize_q += MUL15(esize_q, 17477) >> 18;
        plen = (esize_p + 15) >> 4;
        qlen = (esize_q + 15) >> 4;
        p = tmp;
        q = p + 1 + plen;
        t = q + 1 + qlen;
        tlen = ((sizeof tmp) / sizeof(uint16_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);
                br_i15_rshift(p, 1);
                if (invert_pubexp(t, p, pubexp, t + 1 + plen)) {
                        br_i15_add(p, p, 1);
                        p[1] |= 1;
                        br_i15_encode(sk->p, sk->plen, p);
                        br_i15_encode(sk->dp, sk->dplen, t);
                        break;
                }
        }

        for (;;) {
                mkprime(rng, q, esize_q, pubexp, t, tlen);
                br_i15_rshift(q, 1);
                if (invert_pubexp(t, q, pubexp, t + 1 + qlen)) {
                        br_i15_add(q, q, 1);
                        q[1] |= 1;
                        br_i15_encode(sk->q, sk->qlen, q);
                        br_i15_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_i15_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_i15_zero(t, p[0]);
        t[1] = 1;
        r = br_i15_moddiv(t, q, p, br_i15_ninv15(p[1]), t + 1 + plen);
        br_i15_encode(sk->iq, sk->iqlen, t);

        /*
         * Compute the public modulus too, if required.
         */
        if (pk != NULL) {
                br_i15_zero(t, p[0]);
                br_i15_mulacc(t, p, q);
                br_i15_encode(pk->n, pk->nlen, t);
        }

        return r;
}