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1 | /* |
2 | * RSA key generation. |
3 | */ |
4 | |
5 | #include "ssh.h" |
6 | |
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7 | #define RSA_EXPONENT 37 /* we like this prime */ |
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8 | |
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9 | #if 0 /* bignum diagnostic function */ |
10 | static void diagbn(char *prefix, Bignum md) |
11 | { |
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12 | int i, nibbles, morenibbles; |
13 | static const char hex[] = "0123456789ABCDEF"; |
14 | |
15 | printf("%s0x", prefix ? prefix : ""); |
16 | |
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17 | nibbles = (3 + bignum_bitcount(md)) / 4; |
18 | if (nibbles < 1) |
19 | nibbles = 1; |
20 | morenibbles = 4 * md[0] - nibbles; |
21 | for (i = 0; i < morenibbles; i++) |
22 | putchar('-'); |
23 | for (i = nibbles; i--;) |
24 | putchar(hex[(bignum_byte(md, i / 2) >> (4 * (i % 2))) & 0xF]); |
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25 | |
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26 | if (prefix) |
27 | putchar('\n'); |
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28 | } |
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29 | #endif |
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30 | |
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31 | int rsa_generate(struct RSAKey *key, int bits, progfn_t pfn, |
32 | void *pfnparam) |
33 | { |
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34 | Bignum pm1, qm1, phi_n; |
35 | |
36 | /* |
37 | * Set up the phase limits for the progress report. We do this |
38 | * by passing minus the phase number. |
39 | * |
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. |
45 | * |
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. |
51 | * |
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. |
58 | * |
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 |
62 | * 0x1D.57C4. |
63 | */ |
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64 | pfn(pfnparam, -1, -0x1D57C4 / (bits / 2)); |
65 | pfn(pfnparam, -2, -0x1D57C4 / (bits - bits / 2)); |
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66 | pfn(pfnparam, -3, 5); |
67 | |
68 | /* |
69 | * We don't generate e; we just use a standard one always. |
70 | */ |
71 | key->exponent = bignum_from_short(RSA_EXPONENT); |
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72 | |
73 | /* |
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.) |
79 | */ |
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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); |
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82 | |
83 | /* |
84 | * Ensure p > q, by swapping them if not. |
85 | */ |
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86 | if (bignum_cmp(key->p, key->q) < 0) { |
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87 | Bignum t = key->p; |
88 | key->p = key->q; |
89 | key->q = t; |
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90 | } |
91 | |
92 | /* |
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), |
95 | * and (q^-1 mod p). |
96 | */ |
97 | pfn(pfnparam, 3, 1); |
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98 | key->modulus = bigmul(key->p, key->q); |
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99 | pfn(pfnparam, 3, 2); |
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100 | pm1 = copybn(key->p); |
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101 | decbn(pm1); |
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102 | qm1 = copybn(key->q); |
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103 | decbn(qm1); |
104 | phi_n = bigmul(pm1, qm1); |
105 | pfn(pfnparam, 3, 3); |
106 | freebn(pm1); |
107 | freebn(qm1); |
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108 | key->private_exponent = modinv(key->exponent, phi_n); |
109 | pfn(pfnparam, 3, 4); |
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110 | key->iqmp = modinv(key->q, key->p); |
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111 | pfn(pfnparam, 3, 5); |
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112 | |
113 | /* |
114 | * Clean up temporary numbers. |
115 | */ |
116 | freebn(phi_n); |
117 | |
118 | return 1; |
119 | } |