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

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.

💡 Research Summary

The paper investigates the Fourier spectra of infinite families of quadratic almost perfect nonlinear (APN) functions defined over the binary extension field (L = \mathrm{GF}(2^n)). It begins by recalling that for odd extension degree (n), a quadratic function is almost bent (AB) if and only if it is APN, and in this case the Fourier spectrum consists solely of the three values ({0, \pm 2^{(n+1)/2}}). For even (n) no such direct correspondence is known, and the relationship between the spectrum and the APN property becomes obscure.

The authors focus on the fifth family of quadratic APN functions, introduced in a previous work