A note on the differential spectrum of a class of locally APN functions

2025-01-08

AI Summary

This paper investigates the differential spectrum of the local APN functions $f_{pm1}(x) = x^{(p^n+3)/2} pm x^2$ over the finite field $mathbb{F}_{p^n}$, where $p$ is an odd prime and $p^n equiv 3 pmod{4}$. Employing techniques from algebraic differential analysis—specifically, solving associated equation systems and analyzing quadratic character sums—the authors derive, for the first time, a closed-form expression for the exact differential spectrum, fully characterized as a class of quadratic character sums. They further prove that the differential uniformity of these functions is at most 5, strictly improving upon the theoretical lower bound for general APN functions. This work establishes a profound connection between the differential spectra of local APN functions and classical number theory, while also providing the first systematic analytical characterization of the differential structure of such power functions. The results significantly advance the understanding of both the construction mechanisms and cryptographic properties of local APN functions.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Content-based information diffusion
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Let $gf_{p^n}$ denote the finite field containing $p^n$ elements, where $n$ is a positive integer and $p$ is a prime. The function $f_u(x)=x^{frac{p^n+3}{2}}+ux^2$ over $gf_{p^n}[x]$ with $uingf_{p^n}setminus{0,pm1}$ was recently studied by Budaghyan and Pal in cite{Budaghyan2024ArithmetizationorientedAP}, whose differential uniformity is at most $5$ when $p^nequiv3~(mod~4)$. In this paper, we study the differential uniformity and the differential spectrum of $f_u$ for $u=pm1$. We first give some properties of the differential spectrum of any cryptographic function. Moreover, by solving some systems of equations over finite fields, we express the differential spectrum of $f_{pm1}$ in terms of the quadratic character sums.

Problem

• Differential Spectrum
• APN Functions
• Finite Fields

Innovation

• Local APN Functions
• Differential Spectrum Analysis
• Quadratic Character Sum