On some cryptographic properties of Boolean functions and their second-order derivatives

TL;DR

This paper investigates cryptographic properties of Boolean functions, especially second-order derivatives, to determine conditions under which quadratic or cubic functions are APN, enhancing understanding of their security features.

Contribution

It introduces new quantities based on second-order derivatives to identify APN properties in quadratic and cubic Boolean functions, advancing cryptographic analysis methods.

Findings

Derived quantities from second-order derivatives indicate APN status.

Characterized conditions for quadratic and cubic functions to be APN.

Enhanced criteria for evaluating Boolean function cryptographic strength.

Abstract

In this paper some cryptographic properties of Boolean functions, including weight, balancedness and nonlinearity, are studied, particularly focusing on splitting functions and cubic Boolean functions. Moreover, we present some quantities derived from the behaviour of second-order derivatives which allow us to determine whether a quadratic or cubic function is APN.