Involutions, Trace Maps, and Pseudorandom Numbers

TL;DR

This paper explores how involutions and trace maps in finite fields can be used to generate pseudorandom numbers, leveraging the uniform distribution of trace elements for improved randomness properties.

Contribution

It introduces a novel approach using involutions of a specific form combined with trace maps to produce pseudorandom numbers in finite fields.

Findings

Involutions of the form u/z help generate pseudorandom sequences.

The distribution of zero-trace elements is uniform among non-zero trace elements.

The method enhances pseudorandomness based on finite field properties.

Abstract

Interesting properties of the partitions of a finite field $\mathbb F_q$ induced by the combination of involutions and trace maps are studied. The special features of involutions of the form $\frac{u}{z}$, $u$ being a fixed element of $\mathbb F_q$, are exploited to generate pseudorandom numbers, the randomness resting on the uniform distribution of the images of zero-trace elements among the sets of non-zero trace elements of $\mathbb F_q$.