Two important problems on almost perfect nonlinear (APN) functions are the enumeration and equivalence problems. In this paper, we solve these two problems for any biprojective APN function family by introducing a strong group theoretic method for those functions. Roughly half of the known APN families of functions on even dimensions are biprojective. By our method, we settle the equivalence problem for all known biprojective APN functions. Furthermore, we give a new family of biprojective APN functions. Using our method, we count the number of inequivalent APN functions in all known biprojective APN families and show that the new family found in this paper gives exponentially many new inequivalent APN functions. Quite recently, the Taniguchi family of APN functions was shown to contain an exponential number of inequivalent APN functions by Kaspers and Zhou (J. Cryptol. 34 (1), 2021) which improved their previous count (J. Comb. Th. A 186, 2022) for the Zhou-Pott family. Our group theoretic method substantially simplifies the work required for proving those results and provides a generic natural method for every family in the large super-class of biprojective APN functions that contains these two family along with many others.

翻译：关于几乎完美非线性（APN）函数的两个重要问题是枚举与等价性问题。本文通过引入一种针对双射射影APN函数的强群论方法，解决了任意双射射影APN函数族的这两个问题。已知偶数维APN函数族中约有一半属于双射射影类型。利用我们的方法，我们解决了所有已知双射射影APN函数的等价性问题。此外，我们提出了一个新的双射射影APN函数族，并通过该方法统计了所有已知双射射影APN函数族中非等价APN函数的数量，证明本文发现的新函数族可生成指数级数量的新非等价APN函数。最近，Kaspers 与 Zhou（J. Cryptol. 34 (1), 2021）证明了Taniguchi APN函数族包含指数级数量的非等价APN函数，这改进了他们此前对Zhou-Pott函数族的结果（J. Comb. Th. A 186, 2022）。我们的群论方法大幅简化了证明这些结果所需的工作，并为包含这两个族及其他多个族在内的双射射影APN函数大超类中的每个族提供了通用的自然方法。