On the Lattice Structure of Pseudorandom Numbers Generated over Arbitrary Finite Fields

Abstract.

Marsaglia's lattice test for congruential pseudorandom number generators modulo a prime is extended to a test for generators over arbitrary finite fields. A congruential generator η₀,η₁,…, generated by η _n =g(n), n = 0, 1,…, passes Marsaglia's s-dimensional lattice test if and only if s≤ deg(g). It is investigated how far this conditin holds true for polynomials over arbitrary finite fields F _q , particularly for polynomials of the form g _d (x)=α(x+β)^d+γ, α, β, γ∈F _q , α≠ 0, 1 ≤d≤q− 1.