Modular transformations, order-chaos transitions and pseudo-random number generation

Successive pairs of pseudo-random numbers generated by standard linear cong ruential transformations display ordered patterns of parallel lines. We stu dy the "ordered" and "chaotic" distribution of such pairs by solving the ei genvalue problem for two-dimensional modular transformations over integers. We conjecture that the optimal uniformity for pair distribution is obtaine d when the slope of linear modular eigenspaces takes the value n(opt) = max int(p/root 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 u niform distribution tin the "statistical" sense) of points on the unit squa re (0, 1] x (0, 1]. The method can be easily generalized to the generation of k-tuples of random numbers (with k > 2).