Low c-differential uniformity for functions modified on subfields

TL;DR

This paper constructs and analyzes functions with low c-differential uniformity, improving previous results and exploring concatenations that preserve low uniformity, which is valuable for cryptographic applications.

Contribution

It introduces new piecewise functions and concatenation methods that maintain low c-differential uniformity, advancing the understanding of function modifications on subfields.

Findings

Constructed functions with improved c-differential uniformity bounds.

Proved that concatenations of functions preserve low c-differential uniformity.

Established formulas for the c-differential uniformity of composite functions.

Abstract

In this paper, we construct some piecewise defined functions, and study their $c$-differential uniformity. As a by-product, we improve upon several prior results. Further, we look at concatenations of functions with low differential uniformity and show several results. For example, we prove that given $\beta_i$ (a basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$), some functions $f_i$ of $c$-differential uniformities $\delta_i$, and $L_i$ (specific linearized polynomials defined in terms of $\beta_i$), $1\leq i\leq n$, then $F(x)=\sum_{i=1}^n\beta_i f_i(L_i(x))$ has $c$-differential uniformity equal to $\prod_{i=1}^n \delta_i$.