Permutation resemblance

TL;DR

This paper introduces permutation resemblance as a measure of how close a function is to being a permutation, and explores its applications in constructing low differential uniformity bijections, especially in the context of planar functions.

Contribution

It defines permutation resemblance, establishes key properties, and demonstrates its use in designing low differential uniformity bijections, including for planar functions.

Findings

Permutation resemblance quantifies the proximity of a function to a permutation.

The concept helps in constructing bijections with low differential uniformity.

Analysis of permutation resemblance in planar functions reveals new insights.

Abstract

Motivated by the problem of constructing bijective maps with low differential uniformity, we introduce the notion of permutation resemblance of a function, which looks to measure the distance a given map is from being a permutation. We prove several results concerning permutation resemblance and show how it can be used to produce low differentially uniform bijections. We also study the permutation resemblance of planar functions, which over fields of odd characteristic are known not to be bijections and to have the optimal differential uniformity.