New Perfect Nonlinear Functions and Their Semifields

TL;DR

This paper introduces two new classes of perfect nonlinear functions over finite fields, analyzes their associated semifields, and demonstrates their distinctness from known semifields and other functions.

Contribution

The paper proposes novel perfect nonlinear functions over finite fields and investigates their semifields, showing they are not isotopic to known semifields and are CCZ-inequivalent to existing classes.

Findings

New perfect nonlinear functions over _{p^{2m}}.

Semifields associated are not isotopic to known semifields.

Functions are CCZ-inequivalent to existing classes.

Abstract

In this paper, two new classes of perfect nonlinear functions over $\mathbb{F}_{p^{2m}}$ are proposed, where $p$ is an odd prime. Furthermore, we investigate the nucleus of the corresponding semifields of these functions and show that the semifields are not isotopic to all the known semifields. Particularly, the new perfect nonlinear functions are CCZ-inequivalent to other classes in general.