Abstract
We introduce statistical tests for pseudorandom number generations, which are based on functionals of sample paths of random walks. We consider the following pseudorandom number generatios : m-sequences, additive number generators, cellular automata generators. The results of random walk tests detect statistical biases of these generators. Moreover, from the results we can find some conjectures, for example, ● The multiples of a trinomial f over GF (2) are only those having the form such as f^2, f^4, and so on, while the degrees of multiples are not so large. This conjecture has been proved by Munemasa (1998) in case that the degree of multiples are less than or equal to the twice of the degree of f. ● A primitive polynomial over GF (2) and ite reciprocal polynomial must have different algebraic properties. This conjecture is assured by results of maximum tests, and sojourn time tests.
