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, $\sum (a_i{x}^{2^i}{y}^{2^i},b_i{x}^{2^i}{y}^{2^i})$, to a known class of APN functions by {G{ö}lo{ǧ}lu} over $\gf_{2^m}^2$. The second one is of the form $L(z{)}^{2^m+1}+v{z}^{2^m+1}$ 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].

References in corpus (1)

• A complete characterization of the APN property of a class of quadrinomials