Construction of $(n,n)$-functions with low differential-linear uniformity

TL;DR

This paper introduces new classes of $(n,n)$-functions with low differential-linear uniformity, enhancing cryptographic resistance by refining exponential sums and constructing functions with optimal or near-optimal DLU.

Contribution

It proposes novel power functions and polynomial constructions, including generalized cyclotomic mappings, with improved differential-linear uniformity properties.

Findings

Constructed classes of $(n,n)$-functions with low DLU

Achieved functions with optimal or near-optimal DLU

Enhanced understanding of differential-linear properties of polynomials

Abstract

The differential-linear connectivity table (DLCT), introduced by Bar-On et al. at EUROCRYPT'19, is a novel tool that captures the dependency between the two subciphers involved in differential-linear attacks. This paper is devoted to exploring the differential-linear properties of $(n,n)$-functions. First, by refining specific exponential sums, we propose two classes of power functions over $\mathbb{F}_{2^n}$ with low differential-linear uniformity (DLU). Next, we further investigate the differential-linear properties of $(n,n)$-functions that are polynomials by utilizing power functions with known DLU. Specifically, by combining a cubic function with quadratic functions, and employing generalized cyclotomic mappings, we construct several classes of $(n,n)$-functions with low DLU, including some that achieve optimal or near-optimal DLU compared to existing results.