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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 | int rsa_generate(struct RSAKey *key, int bits, progfn_t pfn, |
10 | void *pfnparam) |
11 | { |
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12 | Bignum pm1, qm1, phi_n; |
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13 | unsigned pfirst, qfirst; |
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14 | |
15 | /* |
16 | * Set up the phase limits for the progress report. We do this |
17 | * by passing minus the phase number. |
18 | * |
19 | * For prime generation: our initial filter finds things |
20 | * coprime to everything below 2^16. Computing the product of |
21 | * (p-1)/p for all prime p below 2^16 gives about 20.33; so |
22 | * among B-bit integers, one in every 20.33 will get through |
23 | * the initial filter to be a candidate prime. |
24 | * |
25 | * Meanwhile, we are searching for primes in the region of 2^B; |
26 | * since pi(x) ~ x/log(x), when x is in the region of 2^B, the |
27 | * prime density will be d/dx pi(x) ~ 1/log(B), i.e. about |
28 | * 1/0.6931B. So the chance of any given candidate being prime |
29 | * is 20.33/0.6931B, which is roughly 29.34 divided by B. |
30 | * |
31 | * So now we have this probability P, we're looking at an |
32 | * exponential distribution with parameter P: we will manage in |
33 | * one attempt with probability P, in two with probability |
34 | * P(1-P), in three with probability P(1-P)^2, etc. The |
35 | * probability that we have still not managed to find a prime |
36 | * after N attempts is (1-P)^N. |
37 | * |
38 | * We therefore inform the progress indicator of the number B |
39 | * (29.34/B), so that it knows how much to increment by each |
40 | * time. We do this in 16-bit fixed point, so 29.34 becomes |
41 | * 0x1D.57C4. |
42 | */ |
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43 | pfn(pfnparam, PROGFN_PHASE_EXTENT, 1, 0x10000); |
44 | pfn(pfnparam, PROGFN_EXP_PHASE, 1, -0x1D57C4 / (bits / 2)); |
45 | pfn(pfnparam, PROGFN_PHASE_EXTENT, 2, 0x10000); |
46 | pfn(pfnparam, PROGFN_EXP_PHASE, 2, -0x1D57C4 / (bits - bits / 2)); |
47 | pfn(pfnparam, PROGFN_PHASE_EXTENT, 3, 0x4000); |
48 | pfn(pfnparam, PROGFN_LIN_PHASE, 3, 5); |
49 | pfn(pfnparam, PROGFN_READY, 0, 0); |
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50 | |
51 | /* |
52 | * We don't generate e; we just use a standard one always. |
53 | */ |
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54 | key->exponent = bignum_from_long(RSA_EXPONENT); |
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55 | |
56 | /* |
57 | * Generate p and q: primes with combined length `bits', not |
58 | * congruent to 1 modulo e. (Strictly speaking, we wanted (p-1) |
59 | * and e to be coprime, and (q-1) and e to be coprime, but in |
60 | * general that's slightly more fiddly to arrange. By choosing |
61 | * a prime e, we can simplify the criterion.) |
62 | */ |
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63 | invent_firstbits(&pfirst, &qfirst); |
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64 | key->p = primegen(bits / 2, RSA_EXPONENT, 1, NULL, |
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65 | 1, pfn, pfnparam, pfirst); |
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66 | key->q = primegen(bits - bits / 2, RSA_EXPONENT, 1, NULL, |
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67 | 2, pfn, pfnparam, qfirst); |
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68 | |
69 | /* |
70 | * Ensure p > q, by swapping them if not. |
71 | */ |
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72 | if (bignum_cmp(key->p, key->q) < 0) { |
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73 | Bignum t = key->p; |
74 | key->p = key->q; |
75 | key->q = t; |
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76 | } |
77 | |
78 | /* |
79 | * Now we have p, q and e. All we need to do now is work out |
80 | * the other helpful quantities: n=pq, d=e^-1 mod (p-1)(q-1), |
81 | * and (q^-1 mod p). |
82 | */ |
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83 | pfn(pfnparam, PROGFN_PROGRESS, 3, 1); |
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84 | key->modulus = bigmul(key->p, key->q); |
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85 | pfn(pfnparam, PROGFN_PROGRESS, 3, 2); |
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86 | pm1 = copybn(key->p); |
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87 | decbn(pm1); |
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88 | qm1 = copybn(key->q); |
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89 | decbn(qm1); |
90 | phi_n = bigmul(pm1, qm1); |
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91 | pfn(pfnparam, PROGFN_PROGRESS, 3, 3); |
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92 | freebn(pm1); |
93 | freebn(qm1); |
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94 | key->private_exponent = modinv(key->exponent, phi_n); |
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95 | pfn(pfnparam, PROGFN_PROGRESS, 3, 4); |
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96 | key->iqmp = modinv(key->q, key->p); |
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97 | pfn(pfnparam, PROGFN_PROGRESS, 3, 5); |
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98 | |
99 | /* |
100 | * Clean up temporary numbers. |
101 | */ |
102 | freebn(phi_n); |
103 | |
104 | return 1; |
105 | } |