Generating Random Factored Numbers, Easily

April 17, 2018

Sometimes it is convenient to have large random composite numbers with known factorization, particularly for testing prime number programs. Eric Bach gives a fast but complicated method. Adam Kalai gives a simpler method that’s not so fast:

Input: Integer n > 0.

Output: A uniformly random number 1 ≤ r ≤ n.

  1. Generate a seqneuce n ≥ s1 ≥ s2 ≥ … ≥ si = 1 by choosing s1 ∈ {1, 2, …, n} and si+1 ∈ {1, 2, … si}, until reaching 1.
  2. Let r be the product of the prime si‘s.
  3. If r ≤ n, output r with probability r / n.
  4. Otherwise, RESTART.

Your task is to implement Kalai’s method of generating random composite numbers with their factorization. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Posted by programmingpraxis
Filed in Exercises
5 Comments »

5 Responses to “Generating Random Factored Numbers, Easily”

  1. Paul said
    April 17, 2018 at 12:16 PM

    In Python.
    from random import randrange
    from ma.primee import is_prime

    def kalai(n):
    while True:
    s, r = n, 1
    while s > 1:
    s = randrange(1, s)
    if is_prime(s):
    r *= s
    if r <= n and randrange(1, n+1) <= r:
    return r

  2. Paul said
    April 17, 2018 at 12:18 PM

    Oops, forgot to format.

    from random import randrange
from ma.primee import is_prime

def kalai(n):
    while True:
        s, r = n, 1
        while s > 1:
            s = randrange(1, s)
            if is_prime(s):
                r *= s
        if r <= n and randrange(1, n+1) <= r:
                return r
  3. Paul said
    April 19, 2018 at 7:14 AM

    A later and more correct version is on ideone.

  4. Daniel said
    May 11, 2018 at 6:15 AM

    Here’s a solution in C++11 using GMP. Prime checking is probabilistic and can return false positives with some non-zero probability (and consequently a factorization that can include composites).

    /* random_factored.cpp */
  
// Build
//   $ g++ -std=c++11 \
//         random_factored.cpp \
//         -o random_factored \
//         -lgmpxx -lgmp

#include <cstdlib>
#include <chrono>
#include <vector>

#include <gmpxx.h>

using std::vector;

void random_factored(const mpz_class& max,
                     mpz_class* rand,
                     vector<mpz_class>* factors,
                     int reps = 50) {
  using namespace std::chrono;
  gmp_randclass rng(gmp_randinit_default);
  auto now = system_clock::now();
  auto ms = time_point_cast<milliseconds>(now);
  long seed = ms.time_since_epoch().count();
  rng.seed(seed);
  mpz_class s;
  while (true) {
    s = max;
    *rand = 1;
    factors->clear();
    while (s > 1) {
      s = rng.get_z_range(s) + 1;
      if (mpz_probab_prime_p(s.get_mpz_t(), reps)) {
        *rand *= s;
        if (*rand > max) goto contin;
        factors->push_back(s);
      }
    }
    if ((rng.get_z_range(max) + 1) <= *rand) return;
contin:;
  }
}

int main(int argc, char* argv[]) {
  if (argc != 2) {
    fprintf(stderr, "Usage: random_factored <MAX>\n");
    return EXIT_FAILURE;
  }
  mpz_class max(argv[1], 10);
  mpz_class rand;
  vector<mpz_class> factors;
  random_factored(max, &rand, &factors);
  gmp_printf("%Zd\n\n", rand.get_mpz_t());
  for (const mpz_class factor : factors) {
    gmp_printf("%Zd\n", factor.get_mpz_t());
  }
  return EXIT_SUCCESS;
}

    Example:

    $ ./random_factored 1000
690

23
5
3
2

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