Full Orbit Sequences in Affine Spaces via Fractional Jumps and Pseudorandom Number Generation

TL;DR

This paper introduces a new method for generating full orbit sequences in affine spaces over finite fields, generalizing inversive congruential generators and offering computational advantages while maintaining similar discrepancy bounds.

Contribution

The authors develop a general theory for full orbit sequence construction in affine spaces, extending ICG to higher dimensions with improved computational efficiency.

Findings

Sequences have the same discrepancy bounds as ICG sequences.

Construction is easier to compute than ICG for n>1.

Generalizes ICG to higher dimensions.

Abstract

Let $n$ be a positive integer. In this paper we provide a general theory to produce full orbit sequences in the affine $n$-dimensional space over a finite field. For $n=1$ our construction covers the case of the Inversive Congruential Generators (ICG). In addition, for $n>1$ we show that the sequences produced using our construction are easier to compute than ICG sequences. Furthermore, we prove that they have the same discrepancy bounds as the ones constructed using the ICG.