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63b8f537f2
The more features and options I add to PrimeCandidateSource, the more cumbersome it will be to replicate each one in a command-line option to the ultimate primegen() function. So I'm moving to an API in which the client of primegen() constructs a PrimeCandidateSource themself, and passes it in to primegen(). Also, changed the API for pcs_new() so that you don't have to pass 'firstbits' unless you really want to. The net effect is that even though we've added flexibility, we've also simplified the call sites of primegen() in the simple case: if you want a 1234-bit prime, you just need to pass pcs_new(1234) as the argument to primegen, and you're done. The new declaration of primegen() lives in ssh_keygen.h, along with all the types it depends on. So I've had to #include that header in a few new files.
212 lines
6.7 KiB
C
212 lines
6.7 KiB
C
/*
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* Prime generation.
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*/
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#include <assert.h>
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#include "ssh.h"
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#include "mpint.h"
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#include "mpunsafe.h"
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#include "sshkeygen.h"
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/*
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* This prime generation algorithm is pretty much cribbed from
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* OpenSSL. The algorithm is:
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*
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* - invent a B-bit random number and ensure the top and bottom
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* bits are set (so it's definitely B-bit, and it's definitely
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* odd)
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*
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* - see if it's coprime to all primes below 2^16; increment it by
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* two until it is (this shouldn't take long in general)
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*
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* - perform the Miller-Rabin primality test enough times to
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* ensure the probability of it being composite is 2^-80 or
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* less
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*
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* - go back to square one if any M-R test fails.
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*/
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/*
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* The Miller-Rabin primality test is an extension to the Fermat
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* test. The Fermat test just checks that a^(p-1) == 1 mod p; this
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* is vulnerable to Carmichael numbers. Miller-Rabin considers how
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* that 1 is derived as well.
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*
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* Lemma: if a^2 == 1 (mod p), and p is prime, then either a == 1
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* or a == -1 (mod p).
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*
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* Proof: p divides a^2-1, i.e. p divides (a+1)(a-1). Hence,
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* since p is prime, either p divides (a+1) or p divides (a-1).
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* But this is the same as saying that either a is congruent to
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* -1 mod p or a is congruent to +1 mod p. []
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*
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* Comment: This fails when p is not prime. Consider p=mn, so
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* that mn divides (a+1)(a-1). Now we could have m dividing (a+1)
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* and n dividing (a-1), without the whole of mn dividing either.
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* For example, consider a=10 and p=99. 99 = 9 * 11; 9 divides
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* 10-1 and 11 divides 10+1, so a^2 is congruent to 1 mod p
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* without a having to be congruent to either 1 or -1.
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*
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* So the Miller-Rabin test, as well as considering a^(p-1),
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* considers a^((p-1)/2), a^((p-1)/4), and so on as far as it can
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* go. In other words. we write p-1 as q * 2^k, with k as large as
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* possible (i.e. q must be odd), and we consider the powers
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*
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* a^(q*2^0) a^(q*2^1) ... a^(q*2^(k-1)) a^(q*2^k)
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* i.e. a^((n-1)/2^k) a^((n-1)/2^(k-1)) ... a^((n-1)/2) a^(n-1)
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*
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* If p is to be prime, the last of these must be 1. Therefore, by
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* the above lemma, the one before it must be either 1 or -1. And
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* _if_ it's 1, then the one before that must be either 1 or -1,
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* and so on ... In other words, we expect to see a trailing chain
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* of 1s preceded by a -1. (If we're unlucky, our trailing chain of
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* 1s will be as long as the list so we'll never get to see what
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* lies before it. This doesn't count as a test failure because it
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* hasn't _proved_ that p is not prime.)
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*
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* For example, consider a=2 and p=1729. 1729 is a Carmichael
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* number: although it's not prime, it satisfies a^(p-1) == 1 mod p
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* for any a coprime to it. So the Fermat test wouldn't have a
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* problem with it at all, unless we happened to stumble on an a
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* which had a common factor.
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*
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* So. 1729 - 1 equals 27 * 2^6. So we look at
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*
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* 2^27 mod 1729 == 645
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* 2^108 mod 1729 == 1065
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* 2^216 mod 1729 == 1
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* 2^432 mod 1729 == 1
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* 2^864 mod 1729 == 1
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* 2^1728 mod 1729 == 1
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*
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* We do have a trailing string of 1s, so the Fermat test would
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* have been happy. But this trailing string of 1s is preceded by
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* 1065; whereas if 1729 were prime, we'd expect to see it preceded
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* by -1 (i.e. 1728.). Guards! Seize this impostor.
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*
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* (If we were unlucky, we might have tried a=16 instead of a=2;
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* now 16^27 mod 1729 == 1, so we would have seen a long string of
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* 1s and wouldn't have seen the thing _before_ the 1s. So, just
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* like the Fermat test, for a given p there may well exist values
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* of a which fail to show up its compositeness. So we try several,
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* just like the Fermat test. The difference is that Miller-Rabin
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* is not _in general_ fooled by Carmichael numbers.)
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*
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* Put simply, then, the Miller-Rabin test requires us to:
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*
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* 1. write p-1 as q * 2^k, with q odd
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* 2. compute z = (a^q) mod p.
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* 3. report success if z == 1 or z == -1.
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* 4. square z at most k-1 times, and report success if it becomes
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* -1 at any point.
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* 5. report failure otherwise.
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*
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* (We expect z to become -1 after at most k-1 squarings, because
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* if it became -1 after k squarings then a^(p-1) would fail to be
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* 1. And we don't need to investigate what happens after we see a
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* -1, because we _know_ that -1 squared is 1 modulo anything at
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* all, so after we've seen a -1 we can be sure of seeing nothing
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* but 1s.)
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*/
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mp_int *primegen(PrimeCandidateSource *pcs,
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int phase, progfn_t pfn, void *pfnparam)
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{
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pcs_ready(pcs);
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int progress = 0;
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STARTOVER:
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pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress);
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mp_int *p = pcs_generate(pcs);
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/*
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* Now apply the Miller-Rabin primality test a few times. First
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* work out how many checks are needed.
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*/
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unsigned checks =
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bits >= 1300 ? 2 : bits >= 850 ? 3 : bits >= 650 ? 4 :
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bits >= 550 ? 5 : bits >= 450 ? 6 : bits >= 400 ? 7 :
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bits >= 350 ? 8 : bits >= 300 ? 9 : bits >= 250 ? 12 :
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bits >= 200 ? 15 : bits >= 150 ? 18 : 27;
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/*
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* Next, write p-1 as q*2^k.
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*/
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size_t k;
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for (k = 0; mp_get_bit(p, k) == !k; k++)
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continue; /* find first 1 bit in p-1 */
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mp_int *q = mp_rshift_safe(p, k);
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/*
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* Set up stuff for the Miller-Rabin checks.
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*/
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mp_int *two = mp_from_integer(2);
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mp_int *pm1 = mp_copy(p);
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mp_sub_integer_into(pm1, pm1, 1);
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MontyContext *mc = monty_new(p);
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mp_int *m_pm1 = monty_import(mc, pm1);
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bool known_bad = false;
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/*
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* Now, for each check ...
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*/
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for (unsigned check = 0; check < checks && !known_bad; check++) {
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/*
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* Invent a random number between 1 and p-1.
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*/
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mp_int *w = mp_random_in_range(two, pm1);
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monty_import_into(mc, w, w);
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pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress);
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/*
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* Compute w^q mod p.
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*/
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mp_int *wqp = monty_pow(mc, w, q);
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mp_free(w);
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/*
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* See if this is 1, or if it is -1, or if it becomes -1
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* when squared at most k-1 times.
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*/
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bool passed = false;
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if (mp_cmp_eq(wqp, monty_identity(mc)) || mp_cmp_eq(wqp, m_pm1)) {
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passed = true;
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} else {
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for (size_t i = 0; i < k - 1; i++) {
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monty_mul_into(mc, wqp, wqp, wqp);
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if (mp_cmp_eq(wqp, m_pm1)) {
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passed = true;
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break;
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}
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}
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}
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if (!passed)
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known_bad = true;
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mp_free(wqp);
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}
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mp_free(q);
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mp_free(two);
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mp_free(pm1);
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monty_free(mc);
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mp_free(m_pm1);
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if (known_bad) {
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mp_free(p);
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goto STARTOVER;
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}
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/*
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* We have a prime!
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*/
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pcs_free(pcs);
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return p;
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}
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