Question

By best I mean the one that:

• passes all statistical tests
• behaves well even at very high dimensions
• has an extremely large period

I can think of Mersenne Twister, but which variant is the best? Is there any better PRNG?

Answer 1

Try MT’s successor: SFMT ( http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/SFMT/index.html ). The acronym stands for SIMD-oriented Fast Mersenne Twister.
It uses vector instructions,
like SSE or AltiVec, to quick up random numbers generation.

Moreover it displays larger periods than the original MT: SFMT can be configured to use periods up to 2²¹⁶⁰⁹¹ -1.

Finally, MT had some problems when badly initialized: it tended to draw lots of 0, leading to bad quality random numbers. This problem could last up to 700000 draws before being compensated by the recurrence of the algorithm. As a consequence, SFMT has also been designed to leave this zero-excess state much quicker than its elder.

Check the link I’ve given at the beginning of this post to find the source code and the scientific publications describing this algorithm.

In order to definitely convince you, you can see here http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/SFMT/speed.html a table comparing generation speeds of both MT and SFMT. In any case, SFMT is quicker while laying out better qualities than MT.

— edit following commentaries —

More generally, when you’re choosing a PRNG, you need to take into account the application you’re developing. Indeed, some PRNGs fit better to some applications constraints. MT and WELL generators for instance aren’t well suited for cryptographic applications, whereas they are the best choice when dealing with Monte Carlo Simulations.

In our case, WELL may seem ideal thanks to its better equidistribution properties than SFMT. Nonetheless, WELL is also far slower and he’s not able to display periods as large as SFMT.

As a conclusion, a PRNG cannot be stated as best for all the applications, but for a particular domain and in particular circumstances withal.