7 #define RSA_EXPONENT 37 /* we like this prime */
9 #if 0 /* bignum diagnostic function */
10 static void diagbn(char *prefix, Bignum md)
12 int i, nibbles, morenibbles;
13 static const char hex[] = "0123456789ABCDEF";
15 printf("%s0x", prefix ? prefix : "");
17 nibbles = (3 + bignum_bitcount(md)) / 4;
20 morenibbles = 4 * md[0] - nibbles;
21 for (i = 0; i < morenibbles; i++)
23 for (i = nibbles; i--;)
24 putchar(hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF]);
31 int rsa_generate(struct RSAKey *key, int bits, progfn_t pfn,
34 Bignum pm1, qm1, phi_n;
37 * Set up the phase limits for the progress report. We do this
38 * by passing minus the phase number.
40 * For prime generation: our initial filter finds things
41 * coprime to everything below 2^16. Computing the product of
42 * (p-1)/p for all prime p below 2^16 gives about 20.33; so
43 * among B-bit integers, one in every 20.33 will get through
44 * the initial filter to be a candidate prime.
46 * Meanwhile, we are searching for primes in the region of 2^B;
47 * since pi(x) ~ x/log(x), when x is in the region of 2^B, the
48 * prime density will be d/dx pi(x) ~ 1/log(B), i.e. about
49 * 1/0.6931B. So the chance of any given candidate being prime
50 * is 20.33/0.6931B, which is roughly 29.34 divided by B.
52 * So now we have this probability P, we're looking at an
53 * exponential distribution with parameter P: we will manage in
54 * one attempt with probability P, in two with probability
55 * P(1-P), in three with probability P(1-P)^2, etc. The
56 * probability that we have still not managed to find a prime
57 * after N attempts is (1-P)^N.
59 * We therefore inform the progress indicator of the number B
60 * (29.34/B), so that it knows how much to increment by each
61 * time. We do this in 16-bit fixed point, so 29.34 becomes
64 pfn(pfnparam, -1, -0x1D57C4 / (bits / 2));
65 pfn(pfnparam, -2, -0x1D57C4 / (bits - bits / 2));
69 * We don't generate e; we just use a standard one always.
71 key->exponent = bignum_from_short(RSA_EXPONENT);
74 * Generate p and q: primes with combined length `bits', not
75 * congruent to 1 modulo e. (Strictly speaking, we wanted (p-1)
76 * and e to be coprime, and (q-1) and e to be coprime, but in
77 * general that's slightly more fiddly to arrange. By choosing
78 * a prime e, we can simplify the criterion.)
80 key->p = primegen(bits / 2, RSA_EXPONENT, 1, 1, pfn, pfnparam);
81 key->q = primegen(bits - bits / 2, RSA_EXPONENT, 1, 2, pfn, pfnparam);
84 * Ensure p > q, by swapping them if not.
86 if (bignum_cmp(key->p, key->q) < 0) {
93 * Now we have p, q and e. All we need to do now is work out
94 * the other helpful quantities: n=pq, d=e^-1 mod (p-1)(q-1),
98 key->modulus = bigmul(key->p, key->q);
100 pm1 = copybn(key->p);
102 qm1 = copybn(key->q);
104 phi_n = bigmul(pm1, qm1);
108 key->private_exponent = modinv(key->exponent, phi_n);
110 key->iqmp = modinv(key->q, key->p);
114 * Clean up temporary numbers.