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

TL;DR

This work addresses the cryptographic properties of Boolean functions, focusing on weight, balancedness, and nonlinearity for splitting and cubic classes, and introduces a second‑order derivative–based parameter to characterize APN status. It develops weight relations under affine splitting, derives nonlinearity bounds for functions with disjoint degree blocks, and classifies weights for a special class of cubic functions, including an algorithm to compute cubic weights. Central to the APN analysis is the parameter M(f), which ties the fourth Walsh moment to second‑order derivatives via L4(F) = 2^{3n}(2^n−1) + 2^{2n} M(F) and yields an exact APN criterion when equality holds; for quadratic/cubic partially‑bent F, M(F) can be computed from the dimensions of linear structures V(F_λ). The results provide a practical and theoretically grounded framework to assess APN properties through lower‑dimensional analyses and second‑order derivative behavior, with implications for constructing high‑nonlinearity, APN‑type functions and APN permutations.

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.

Theorems & Definitions (14)

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