Period Distribution of Inversive Pseudorandom Number Generators Over Finite Fields

TL;DR

This paper analyzes the period distribution of inversive pseudorandom number generators over finite fields, providing an analytical method to determine periods and guidance for parameter selection to achieve desired cycle lengths.

Contribution

It introduces an analytical approach using generating functions and finite field theory to characterize period distributions of IPRNGs over prime fields, aiding in optimal parameter choice.

Findings

Many small periods occur if parameters are not properly chosen

Theoretical analysis matches experimental results

Guidelines for selecting parameters for specific periods

Abstract

In this paper, we focus on analyzing the period distribution of the inversive pseudorandom number generators (IPRNGs) over finite field $({\rm Z}_{N},+,\times)$, where $N>3$ is a prime. The sequences generated by the IPRNGs are transformed to 2-dimensional linear feedback shift register (LFSR) sequences. By employing the generating function method and the finite field theory, the period distribution is obtained analytically. The analysis process also indicates how to choose the parameters and the initial values such that the IPRNGs fit specific periods. The analysis results show that there are many small periods if $N$ is not chosen properly. The experimental examples show the effectiveness of the theoretical analysis.