Modular Transformations, Order-Chaos Transitions and Pseudo-Random Number Generation

Successive pairs of pseudo-random numbers generated by standard linear congruential transformations display ordered patterns of parallel lines. We study the ordered'' and chaotic’’ distribution of such pairs by solving the eigenvalue problem for two-dimensional modular transformations over integers. We conjecture that the optimal uniformity for pair distribution is obtained when the slope of linear modular eigenspaces takes the value $n_{opt} = maxint(p /\sqrt{p-1})$,where$p$ is a prime number. We then propose a new generator of pairs of independent pseudo-random numbers, which realizes an optimal uniform distribution (in the ``statistical’’ sense) of points on the unit square $(0,1]\times (0,1]$. The method can be easily generalized to the generation of $k$-tuples of random numbers (with $k>2$)