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.

翻译：APN函数作为对称密码学中原语，在提供最优差分攻击抵抗性的构造模块中起着重要作用。本文研究了Göloğlu在$\mathbb{F}_{2^{2m}}$上定义的双射影APN函数族的一个新扩展。我们证明，当$3\nmid m$时，该推广所得函数与Göloğlu原始函数族等价；当$3|m$时，我们精确刻画了该新族包含的非等价APN函数数量。同时证明该函数族具有APN函数的最小像集规模，并确定了其Walsh谱，从而解决了一些开放问题。在证明中，我们利用了Göloğlu与作者近期发展的群论技术，结合射影多项式集合上的群作用。