cc-differential uniformity, (almost) perfect cc-nonlinearity, and equivalences

TL;DR

This paper introduces new concepts related to $cc$-differential uniformity and spectrum, explores their invariance under certain equivalences, and characterizes these properties for vectorial Boolean functions and power functions.

Contribution

It defines and studies $cc$-differential uniformity, spectrum, and related equivalences, expanding understanding of cryptographic function properties.

Findings

$cc$-differential uniformity is invariant under $c$-1-equivalence.

$cc$-differential uniformity and spectrum are preserved under $c$-CCZ equivalence.

Characterization of $cc$-differential uniformity via Walsh transform.

Abstract

In this article, we introduce new notions $cc$-differential uniformity, $cc$-differential spectrum, PccN functions and APccN functions, and investigate their properties. We also introduce $c$-CCZ equivalence, $c$-EA equivalence, and $c1$-equivalence. We show that $c$-differential uniformity is invariant under $c1$-equivalence, and $cc$-differential uniformity and $cc$-differential spectrum are preserved under $c$-CCZ equivalence. We characterize $cc$-differential uniformity of vectorial Boolean functions in terms of the Walsh transformation. We investigate $cc$-differential uniformity of power functions $F(x)=x^d$. We also illustrate examples to prove that $c$-CCZ equivalence is strictly more general than $c$-EA equivalence.