On the Fourier Spectra of the Infinite Families of Quadratic APN Functions

TL;DR

This paper computes the Fourier spectra of a new family of quadratic APN functions, providing a comprehensive understanding of their nonlinear properties and completing the spectral analysis for all known infinite APN families.

Contribution

It introduces the Fourier spectrum of a new quadranomial family of APN functions, expanding the spectral data available for these cryptographically important functions.

Findings

Fourier spectra of the new APN family are explicitly computed.

All known infinite APN families now have their spectra and nonlinearities determined.

The results enhance understanding of the spectral properties of quadratic APN functions.

Abstract

It is well known that a quadratic function defined on a finite field of odd degree is almost bent (AB) if and only if it is almost perfect nonlinear (APN). For the even degree case there is no apparent relationship between the values in the Fourier spectrum of a function and the APN property. In this article we compute the Fourier spectrum of the new quadranomial family of APN functions. With this result, all known infinite families of APN functions now have their Fourier spectra and hence their nonlinearities computed.