Inversive congruential pseudorandom numbers avoid the planes

Abstract

Nonlinear congruential pseudorandom number generators based on inversions have recently been introduced and analyzed. These generators do not show the lattice structure of the widely used linear congruential method. In the present paper it is proved that the points formed by d consecutive pseudorandom numbers of an inversive congruential generator with prime modulus possess an even stronger property: Any hyperplane in d-space contains at most d of these points, that is to say, the hyperplane spanned by d arbitrary points of an inversive congruential generator contains no further points. This feature makes the inversive congruential method particularly attractive for simulation problems where linear structures within the generated points should be avoided.