Two new infinite classes of APN functions

Abstract

In this paper, we present two new infinite classes of APN functions over $\gf_{{2^{2m}}}$ and $\gf_{{2^{3m}}}$, respectively. The first one is with bivariate form and obtained by adding special terms, $\sum(a_ix^{2^i}y^{2^i},b_ix^{2^i}y^{2^i})$, to a known class of APN functions by {G{\"{o}}lo{\v{g}}lu} over $\gf_{{2^m}}^2$. The second one is of the form $L(z)^{2^m+1}+vz^{2^m+1}$ over $\gf_{{2^{3m}}}$, which is a generalization of one family of APN functions by Bracken et al. [Cryptogr. Commun. 3 (1): 43-53, 2011]. The calculation of the CCZ-invariants $\Gamma$-ranks of our APN classes over $\gf_{{2^8}}$ or $\gf_{{2^9}}$ indicates that they are CCZ-inequivalent to all known infinite families of APN functions. Moreover, by using the code isomorphism, we see that our first APN family covers an APN function over $\gf_{{2^8}}$ obtained through the switching method by Edel and Pott in [Adv. Math. Commun. 3 (1): 59-81, 2009].