八变量下的数百万个不等价二次APN函数

A lower bound on the number of inequivalent APN functions

In this paper, we establish a lower bound on the total number of inequivalent APN functions on the finite field with $2^{2m}$ elements, where $m$ is even. We obtain this result by proving that the APN functions introduced by Pott and the second author, that depend on three parameters $k$, $s$ and $α$, are pairwise inequivalent for distinct choices of the parameters $k$ and $s$. Moreover, we determine the automorphism group of these APN functions.

双射影几乎完全非线性函数的等价性

Equivalences of biprojective almost perfect nonlinear functions

针对几乎完全非线性函数的计数与等价性问题，本文提出强群论方法，解决了所有已知双射影APN函数的等价性问题，提出新函数族并证明其可产生指数级新不等价函数，为该类函数研究提供通用方法。

关于二次APN函数的等价性

On the Equivalence of Quadratic APN Functions

建立一对APN函数的CCZ等价性通常相当困难。在某些情况下，当试图证明一个假定的APN函数的新无限族与一个已知的族是CCZ不等价的时，我们依赖于小n值的计算机计算。本文给出了一种证明二次APN函数与Gold函数不等价的方法。我们的主要结果是：一个二次函数是CCZ-等价于一个APN Gold函数当且仅当它是EA-等价于该Gold函数。作为这一结果的应用，我们证明了存在于2^n阶有限域（n = 2 mod 4）上的一个三叉族APN函数与Gold函数CCZ不等价.证明依赖于与这样的函数相关联的代码的自同构群的一些知识。

New Instances of Quadratic APN Functions

In a recent work, Beierle, Brinkmann and Leander presented a recursive tree search for finding APN permutations with linear self-equivalences in small dimensions. In this paper, we describe how this search can be adapted to find many new instances of quadratic APN functions. In particular, we found 12,921 new quadratic APN functions in dimension eight, 35 new quadratic APN functions in dimension nine and five new quadratic APN functions in dimension ten up to CCZ-equivalence. Remarkably, two of the 35 new APN functions in dimension nine are APN permutations. Among the 8-bit APN functions, ther…

关于一族双射影APN函数近期推广的研究

On a recent extension of a family of biprojective APN functions

本文研究Göloğlu双射影APN函数族的近期推广，证明其在3不整除m时等价于原族，在3整除m时给出不等价函数数量，并确定其最小像集大小和Walsh谱，解决了若干公开问题。

The number of almost perfect nonlinear functions grows exponentially

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 $\frac{φ(m)}{2}\left\lceil \frac{2^m+1}{3m} \right\rceil$ inequivalent APN functions on the finite field with ${2^{2m}}$ elements, where $φ$ denotes Euler's totient function. This is a great improvement of previous results: for even $m$, the best known lower …