Almost perfect nonlinear (APN) functions play an important role in the design of block ciphers as they offer the strongest resistance against differential cryptanalysis. Despite more than 25 years of research, only a limited number of APN functions are known. In this paper, we show that a recent construction by Taniguchi provides at least $$\frac{\varphi (m)}{2}\left\lceil \frac{2^m+1}{3m} \right\rceil $$ φ ( m ) 2 2 m + 1 3 m inequivalent APN functions on the finite field with $${2^{2m}}$$ 2 2 m elements, where $$\varphi $$ φ denotes Euler’s totient function. This is a great improvement of previous results: for even m , the best known lower bound has been $$\frac{\varphi (m)}{2}\left( \lfloor \frac{m}{4}\rfloor +1\right) $$ φ ( m ) 2 ⌊ m 4 ⌋ + 1 ; for odd m , there has been no such lower bound at all. Moreover, we determine the automorphism group of Taniguchi’s APN functions.

中文翻译：

---

几乎完美的非线性函数的数量呈指数增长

几乎完美的非线性 (APN) 函数在分组密码的设计中发挥着重要作用，因为它们提供了对差分密码分析的最强抵抗力。尽管进行了 25 年以上的研究，但已知的 APN 功能数量有限。在本文中，我们证明了谷口最近的构造提供了至少 $$\frac{\varphi (m)}{2}\left\lceil \frac{2^m+1}{3m} \right\rceil $ $ φ ( m ) 2 2 m + 1 3 m 不等价APN 函数在具有$${2^{2m}}$$ 2 2 m 元素的有限域上，其中$$\varphi $$ φ 表示欧拉的totient 函数。这是对先前结果的巨大改进：即使是 m ，最著名的下限是 $$\frac{\varphi (m)}{2}\left( \lfloor \frac{m}{4}\rfloor + 1\right) $$ φ ( m ) 2 ⌊ m 4 ⌋ + 1 ; 对于奇数 m ，根本没有这样的下限。而且，