# Software to factor integers

In my previous post, I showed how changing one bit of a semiprime (i.

e.

the product of two primes) creates an integer that can be factored much faster.

I started writing that post using Python with SymPy, but moved to Mathematica because factoring took too long.

SymPy vs MathematicaWhen I’m working in Python, SymPy lets me stay in Python.

I’ll often use SymPy for a task that Mathematica could do better just so I can stay in one environment.

But sometimes efficiency is a problem.

SymPy is written in pure Python, for better and for worse.

When it comes to factoring large integers, it’s for worse.

I tried factoring a 140-bit integer with SymPy, and killed the process after over an hour.

Mathematica factored the same integer in 1/3 of a second.

Mathematica vs PARI/GPThe previous post factors 200-bit semiprimes.

The first example, N = pq wherep = 1078376712338123201911958185123 q = 1126171711601272883728179081277took 99.

94 seconds to factor using Mathematica.

A random sample of 13 products of 100-bit primes and they took an average of 99.

1 seconds to factor.

Using PARI/GP, factoring the value of N above took 11.

4 seconds to factor.

I then generated a sample of 10 products of 100-bit primes and on average they took 10.

4 seconds to factor using PARI/GP.

So in these examples, Mathematica is several orders of magnitude faster than Mathematica, and PARI/GP is one order of magnitude faster than PARI/GP.

It could be that the PARI/GP algorithms are relatively better at factoring semiprimes.

To compare the efficiency of PARI/GP and Mathematica on non-semiprimes, I repeated the exercise in the previous post, flipping each bit of N one at a time and factoring.

This took 240.

3 seconds with PARI/GP.

The same code in Mathematica took 994.

5 seconds.

So in this example, PARI/GP is about 4 times faster where as for semiprimes it was 10 times faster.

Python and PARIThere is a Python interface to PARI called cypari2.

It should offer the convenience of working in Python with the efficiency of PARI.

Unfortunately, the installation failed on my computer.

I think SageMath interfaces Python to PARI but I haven’t tried it.

Related postsRSA numbersFermat’s factoring trickHow long does it take to find large primes?.