Two new infinite classes of APN functions

TL;DR

This paper introduces two new infinite classes of APN functions over specific finite fields, expanding the landscape of APN functions with proven inequivalence to known families and connections to existing constructions.

Contribution

The paper presents two novel infinite classes of APN functions over GF(2^{2m}) and GF(2^{3m}), with detailed constructions and inequivalence proofs.

Findings

Two new infinite classes of APN functions are constructed.

The new classes are CCZ-inequivalent to all known APN families.

One class relates to a previously known APN function via code isomorphism.

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].