Fourier Spectra of Binomial APN Functions

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.

💡 Research Summary

The paper investigates the Fourier spectra of two recently discovered families of binary quadratic APN (Almost Perfect Nonlinear) functions. After a brief introduction to the concepts of APN, AB (Almost Bent) functions, and their relevance to cryptographic resistance against differential and linear attacks, the authors recall that the Fourier transform of a function (f: \mathrm{GF}(2^n)\to\mathrm{GF}(2^n)) is defined by
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