On the Fourier Spectra of New APN Functions

Almost perfect nonlinear (APN) functions on F2n are functions achieving the lowest possible differential uniformity. All APN functions discovered until now are either power or quadratic ones, except for one sporadic multinomial nonquadratic example on F26 due to Edel and Pott. It is well known that certain binary codes with good properties can be obtained from APN functions, and determining their (Hamming) weight distribution is equivalent to determining the Fourier spectra of the corresponding functions. The Fourier spectra of all known infinite families of quadratic APN functions discovered through 2010 have been determined, and it was found that they are the same as the ones of the Gold APN functions, i.e., a 5-valued set when n is even and a 3-valued set when n is odd, while a sporadic example on F26 found by Dillon has a 7-valued Fourier spectrum. In 2011, two new generic constructions of APN functions were presented in [Y. Zhou and A. Pott, Adv. Math., 234 (2013), pp. 43–60] and [C. Carlet, Des. Codes Cryptogr., 59 (2011), pp. 89–109]. In this paper, we determine the Fourier spectra of the APN functions obtained from them and show that their Fourier spectra are again the same as those of the Gold APN functions. Moreover, since the APN functions in [C. Bracken, C. H. Tan, and Y. Tan, On a Class of Quadratic Polynomials with No Zeros and Its Applications to APN Functions, preprint, arXiv:1110.3177v1, 2011], which are demonstrated to exist when n ≡ 0 mod 4 and 3 n, are covered by the construction in [C. Carlet, Des. Codes Cryptogr., 59 (2011), pp. 89–109], a positive answer to the conjecture proposed in the former paper on determining their Fourier spectrum is given in this paper.