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Diffstat (limited to 'millerrabin.c')
| -rw-r--r-- | millerrabin.c | 214 |
1 files changed, 0 insertions, 214 deletions
diff --git a/millerrabin.c b/millerrabin.c deleted file mode 100644 index 3358bc51..00000000 --- a/millerrabin.c +++ /dev/null @@ -1,214 +0,0 @@ -/* - * millerrabin.c: Miller-Rabin probabilistic primality testing, as - * declared in sshkeygen.h. - */ - -#include <assert.h> -#include "ssh.h" -#include "sshkeygen.h" -#include "mpint.h" -#include "mpunsafe.h" - -/* - * The Miller-Rabin primality test is an extension to the Fermat - * test. The Fermat test just checks that a^(p-1) == 1 mod p; this - * is vulnerable to Carmichael numbers. Miller-Rabin considers how - * that 1 is derived as well. - * - * Lemma: if a^2 == 1 (mod p), and p is prime, then either a == 1 - * or a == -1 (mod p). - * - * Proof: p divides a^2-1, i.e. p divides (a+1)(a-1). Hence, - * since p is prime, either p divides (a+1) or p divides (a-1). - * But this is the same as saying that either a is congruent to - * -1 mod p or a is congruent to +1 mod p. [] - * - * Comment: This fails when p is not prime. Consider p=mn, so - * that mn divides (a+1)(a-1). Now we could have m dividing (a+1) - * and n dividing (a-1), without the whole of mn dividing either. - * For example, consider a=10 and p=99. 99 = 9 * 11; 9 divides - * 10-1 and 11 divides 10+1, so a^2 is congruent to 1 mod p - * without a having to be congruent to either 1 or -1. - * - * So the Miller-Rabin test, as well as considering a^(p-1), - * considers a^((p-1)/2), a^((p-1)/4), and so on as far as it can - * go. In other words. we write p-1 as q * 2^k, with k as large as - * possible (i.e. q must be odd), and we consider the powers - * - * a^(q*2^0) a^(q*2^1) ... a^(q*2^(k-1)) a^(q*2^k) - * i.e. a^((n-1)/2^k) a^((n-1)/2^(k-1)) ... a^((n-1)/2) a^(n-1) - * - * If p is to be prime, the last of these must be 1. Therefore, by - * the above lemma, the one before it must be either 1 or -1. And - * _if_ it's 1, then the one before that must be either 1 or -1, - * and so on ... In other words, we expect to see a trailing chain - * of 1s preceded by a -1. (If we're unlucky, our trailing chain of - * 1s will be as long as the list so we'll never get to see what - * lies before it. This doesn't count as a test failure because it - * hasn't _proved_ that p is not prime.) - * - * For example, consider a=2 and p=1729. 1729 is a Carmichael - * number: although it's not prime, it satisfies a^(p-1) == 1 mod p - * for any a coprime to it. So the Fermat test wouldn't have a - * problem with it at all, unless we happened to stumble on an a - * which had a common factor. - * - * So. 1729 - 1 equals 27 * 2^6. So we look at - * - * 2^27 mod 1729 == 645 - * 2^108 mod 1729 == 1065 - * 2^216 mod 1729 == 1 - * 2^432 mod 1729 == 1 - * 2^864 mod 1729 == 1 - * 2^1728 mod 1729 == 1 - * - * We do have a trailing string of 1s, so the Fermat test would - * have been happy. But this trailing string of 1s is preceded by - * 1065; whereas if 1729 were prime, we'd expect to see it preceded - * by -1 (i.e. 1728.). Guards! Seize this impostor. - * - * (If we were unlucky, we might have tried a=16 instead of a=2; - * now 16^27 mod 1729 == 1, so we would have seen a long string of - * 1s and wouldn't have seen the thing _before_ the 1s. So, just - * like the Fermat test, for a given p there may well exist values - * of a which fail to show up its compositeness. So we try several, - * just like the Fermat test. The difference is that Miller-Rabin - * is not _in general_ fooled by Carmichael numbers.) - * - * Put simply, then, the Miller-Rabin test requires us to: - * - * 1. write p-1 as q * 2^k, with q odd - * 2. compute z = (a^q) mod p. - * 3. report success if z == 1 or z == -1. - * 4. square z at most k-1 times, and report success if it becomes - * -1 at any point. - * 5. report failure otherwise. - * - * (We expect z to become -1 after at most k-1 squarings, because - * if it became -1 after k squarings then a^(p-1) would fail to be - * 1. And we don't need to investigate what happens after we see a - * -1, because we _know_ that -1 squared is 1 modulo anything at - * all, so after we've seen a -1 we can be sure of seeing nothing - * but 1s.) - */ - -struct MillerRabin { - MontyContext *mc; - - size_t k; - mp_int *q; - - mp_int *two, *pm1, *m_pm1; -}; - -MillerRabin *miller_rabin_new(mp_int *p) -{ - MillerRabin *mr = snew(MillerRabin); - - assert(mp_hs_integer(p, 2)); - assert(mp_get_bit(p, 0) == 1); - - mr->k = 1; - while (!mp_get_bit(p, mr->k)) - mr->k++; - mr->q = mp_rshift_safe(p, mr->k); - - mr->two = mp_from_integer(2); - - mr->pm1 = mp_unsafe_copy(p); - mp_sub_integer_into(mr->pm1, mr->pm1, 1); - - mr->mc = monty_new(p); - mr->m_pm1 = monty_import(mr->mc, mr->pm1); - - return mr; -} - -void miller_rabin_free(MillerRabin *mr) -{ - mp_free(mr->q); - mp_free(mr->two); - mp_free(mr->pm1); - mp_free(mr->m_pm1); - monty_free(mr->mc); - smemclr(mr, sizeof(*mr)); - sfree(mr); -} - -struct mr_result { - bool passed; - bool potential_primitive_root; -}; - -static struct mr_result miller_rabin_test_inner(MillerRabin *mr, mp_int *w) -{ - /* - * Compute w^q mod p. - */ - mp_int *wqp = monty_pow(mr->mc, w, mr->q); - - /* - * See if this is 1, or if it is -1, or if it becomes -1 - * when squared at most k-1 times. - */ - struct mr_result result; - result.passed = false; - result.potential_primitive_root = false; - - if (mp_cmp_eq(wqp, monty_identity(mr->mc))) { - result.passed = true; - } else { - for (size_t i = 0; i < mr->k; i++) { - if (mp_cmp_eq(wqp, mr->m_pm1)) { - result.passed = true; - result.potential_primitive_root = (i == mr->k - 1); - break; - } - if (i == mr->k - 1) - break; - monty_mul_into(mr->mc, wqp, wqp, wqp); - } - } - - mp_free(wqp); - - return result; -} - -bool miller_rabin_test_random(MillerRabin *mr) -{ - mp_int *mw = mp_random_in_range(mr->two, mr->pm1); - struct mr_result result = miller_rabin_test_inner(mr, mw); - mp_free(mw); - return result.passed; -} - -mp_int *miller_rabin_find_potential_primitive_root(MillerRabin *mr) -{ - while (true) { - mp_int *mw = mp_unsafe_shrink(mp_random_in_range(mr->two, mr->pm1)); - struct mr_result result = miller_rabin_test_inner(mr, mw); - - if (result.passed && result.potential_primitive_root) { - mp_int *pr = monty_export(mr->mc, mw); - mp_free(mw); - return pr; - } - - mp_free(mw); - - if (!result.passed) { - return NULL; - } - } -} - -unsigned miller_rabin_checks_needed(unsigned bits) -{ - /* Table 4.4 from Handbook of Applied Cryptography */ - return (bits >= 1300 ? 2 : bits >= 850 ? 3 : bits >= 650 ? 4 : - bits >= 550 ? 5 : bits >= 450 ? 6 : bits >= 400 ? 7 : - bits >= 350 ? 8 : bits >= 300 ? 9 : bits >= 250 ? 12 : - bits >= 200 ? 15 : bits >= 150 ? 18 : 27); -} - |