Distribution of Nonlinear Congruential Pseudorandom Numbers Modulo Almost Squarefree Integers

Exponential sums for nonlinear recurring sequences

“…It seems that the approach of this paper may also lead to improvements on the bounds in [3,4,6,7,13]. The improved method was already used in [5] as well.…”

“…The improved method was already used in [5] as well. More precisely, the papers [3,4] deal with nonlinear recurrences y n+1 ≡ f (y n ) mod M with composite moduli M. The papers [6,7] contain extensions of (2) to nonlinear recurrences of higher orders y n+1 = f (y n , . .…”

On the Distribution of the Power Generator over a Residue Ring for Parts of the Period

“…Here we show that the original method of [20], and more recently also used in [8,9], combined with bounds for exponential sums with sparse polynomials from [12] allows us to study the distribution of the power generator of pseudorandom numbers over a residue ring. In particular, in [12] a distribution result for the sequence generated by (1) has been established for the sequence over the entire period.…”

“…Several other results about non-linear pseudorandom number generators have been obtained in [8,9,15]. However these apply to generators of the form u n ≡ f (u n−1 ) (mod M ) where f is a polynomial or a rational function, whilst [17,21,22] provide results for the inverse generator.…”

On the Distribution of Counter-Dependent Nonlinear Congruential Pseudorandom Number Generators in Residue Rings

“…In the case that f (X, Y ) = h(X) ∈ Z Z M [X] does not depend on the second variable we get the well-studied nonlinear congruential pseudorandom number generators, see [4,6,8,13] for the distribution of the elements and for the distribution of powers in prime fields see [15]. However, in this case the period t is at most M and it is possible that the generated sequences have unexpectedly short period as it is noted in [17].…”