## Generating Random Factored Numbers, Easily

### April 17, 2018

Here’s our version of the function, which uses the native Chez Scheme random-number function:

```; composite number between lo and hi, with its prime factorization
(define (random-composite lo hi) ; j cryptology (2003) 16: 287-289
(let loop ((n hi) (ss (list)))
(if (< 1 n) (let ((t (+ (random n) 1))) (loop t (cons t ss)))
(let* ((rs (filter prime? ss)) (r (apply * rs)))
(if (and (< 1 (length rs)) (< lo r hi) (< (random 1.0) (/ r hi)))
(values r rs)
(random-composite lo hi))))))```

And here are some examples:

```> (random-composite #e1e49 #e1e50)
39031688906845405947838510008765610782589401947814
(2 3 3 3 23819 472116863
64276252279614685494838344056247653)
> (random-composite #e1e199 #e1e200)
54043199014242862074068285893135249110810396044949893213716035706818704365084807
10747697771435656989138804427933641862633849990995410369411218898581482115751406
0271041039310245242619801236797864316382
(2 32551107389
18297496992808352052439
6412310674737823761171618278424510901936500685202576340621797
70752052794845144427307913529578325615157382037282566979843347735524206867191
74656452808739750971865168793)```

You can run the program at https://ideone.com/106ySG, where you will also see our Baillie-Wagstaff primality checker. Beware that it gets slower as hi gets larger.

Pages: 1 2

### 5 Responses to “Generating Random Factored Numbers, Easily”

1. Paul said

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

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

A later and more correct version is on ideone.

4. Daniel said

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, 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
```