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POHow can I improve this Pollard's rho algorithm to handle products of semi-large primes?
 

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  1. POHow can I improve this Pollard's rho algorithm to handle products of semi-large primes?
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    <p>Below is my implementation of Pollard's rho algorithm for prime factorization:</p> <pre class="lang-c++ prettyprint-override"><code>#include &lt;vector&gt; #include &lt;queue&gt; #include &lt;gmpxx.h&gt; // Interface to the GMP random number functions. gmp_randclass rng(gmp_randinit_default); // Returns a divisor of N using Pollard's rho method. mpz_class getDivisor(const mpz_class &amp;N) { mpz_class c = rng.get_z_range(N); mpz_class x = rng.get_z_range(N); mpz_class y = x; mpz_class d = 1; mpz_class z; while (d == 1) { x = (x*x + c) % N; y = (y*y + c) % N; y = (y*y + c) % N; z = x - y; mpz_gcd(d.get_mpz_t(), z.get_mpz_t(), N.get_mpz_t()); } return d; } // Adds the prime factors of N to the given vector. void factor(const mpz_class &amp;N, std::vector&lt;mpz_class&gt; &amp;factors) { std::queue&lt;mpz_class&gt; to_factor; to_factor.push(N); while (!to_factor.empty()) { mpz_class n = to_factor.front(); to_factor.pop(); if (n == 1) { continue; // Trivial factor. } else if (mpz_probab_prime_p(n.get_mpz_t(), 5)) { // n is a prime. factors.push_back(n); } else { // n is a composite, so push its factors on the queue. mpz_class d = getDivisor(n); to_factor.push(d); to_factor.push(n/d); } } } </code></pre> <p>It's essentially a straight translation of the <a href="https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm" rel="nofollow">pseudocode on Wikipedia</a>, and relies on GMP for big numbers and for primality testing. The implementation works well and can factor primes such as</p> <pre><code>1000036317378699858851366323 = 1000014599 * 1000003357 * 1000018361 </code></pre> <p>but will choke on e.g.</p> <pre><code>1000000000002322140000000048599822299 = 1000000000002301019 * 1000000000000021121 </code></pre> <p>My question is: Is there anything I can do to improve on this, short of switching to a more complex factorization algorithm such as <a href="https://en.wikipedia.org/wiki/Quadratic_sieve" rel="nofollow">Quadratic sieve</a>?</p> <p>I know one improvement could be to first do some trial divisions by pre-computed primes, but that would not help for products of a few large primes such as the above.</p> <p>I'm interested in any tips on improvements to the basic Pollard's rho method to get it to handle larger composites of only a few prime factors. Of course if you find any stupidities in the code above, I'm interested in those as well.</p> <p>For full disclosure: This is a homework assignment, so general tips and pointers are better than fully coded solutions. With this very simple approach I already get a passing grade on the assignment, but would of course like to improve.</p> <p>Thanks in advance.</p>
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      1. POHow can I improve this Pollard's rho algorithm to handle products of semi-large primes?
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      1. POHow can I improve this Pollard's rho algorithm to handle products of semi-large primes?
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    1. COThe wikipedia-article seem to hint that changing your `f(x)` will change it's behaviour. Maybe that could help you progress faster on larger primes?
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      1. POHow can I improve this Pollard's rho algorithm to handle products of semi-large primes?
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    2. COYour specific example of 1000000000002322140000000048599822299 = 1000000000002301019 * 1000000000000021121 should be easily handled by [Fermat factorization](https://en.wikipedia.org/wiki/Fermat%27s_factorization_method), since the difference of the factors is only 2279936.
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      1. POHow can I improve this Pollard's rho algorithm to handle products of semi-large primes?
      1. USIlmari Karonen
 

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