Fourier Spectra of Binomial APN Functions

TL;DR

This paper computes the Fourier spectra of binomial APN functions, revealing their nonlinearity and connections to error-correcting codes, and offers an alternative proof of their APN property for odd-degree fields.

Contribution

It provides the first detailed Fourier spectra analysis of these binomial APN functions, linking their cryptographic and coding-theoretic properties.

Findings

Determined the nonlinearity of the functions.

Showed related error-correcting codes have BCH code weight distribution.

Provided an alternative proof of APN property for odd-degree fields.

Abstract

In this paper we compute the Fourier spectra of some recently discovered binomial APN functions. One consequence of this is the determination of the nonlinearity of the functions, which measures their resistance to linear cryptanalysis. Another consequence is that certain error-correcting codes related to these functions have the same weight distribution as the 2-error-correcting BCH code. Furthermore, for fields of odd degree, our results provide an alternative proof of the APN property of the functions.