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的基本原语，在提供抵抗差分攻击的最优抗性方面发挥着重要作用。本文研究Göloğlu在$\mathbb{F}_{2^{2m}}$上定义的双射影APN函数族的一个最新扩展。我们证明：若$3\nmid m$，该推广产生的函数与Göloğlu原始族等价；若$3|m$，我们精确刻画了新族中包含的不等价APN函数数量。此外，我们证明该族具有APN函数的最小像集大小，并确定了其Walsh谱，从而解决了若干开放问题。证明过程中，我们利用了Göloğlu与作者近期发展的群论方法，结合射影多项式集合上的群作用。