On a recent extension of a family of biprojective APN functions

Abstract: APN functions play a big role as primitives in symmetric cryptography as building blocks that yield optimal resistance to differential attacks. In this note, we consider a recent extension of a biprojective APN family by G\"olo\u{g}lu defined on $\mathbb{F}_{2^{2m}}$. We show that this generalization yields functions equivalent to G\"olo\u{g}lu's original family if $3\nmid m$. If $3|m$ we show exactly how many inequivalent APN functions this new family contains. We also show that the family has the minimal image set size for an APN function and determine its Walsh spectrum, hereby settling some open problems. In our proofs, we leverage a group theoretic technique recently developed by G\"olo\u{g}lu and the author in conjunction with a group action on the set of projective polynomials.