论文标题
几乎完美的非线性功能的数量呈指数增长
The number of almost perfect nonlinear functions grows exponentially
论文作者
论文摘要
几乎完美的非线性（APN）函数在块密码的设计中起着重要作用，因为它们具有对差分密码分析的最强阻力。尽管研究了25年以上，但仅知道有限的APN功能。在本文中，我们表明Taniguchi的最新建筑至少提供$ \ frac {φ（m）} {2} {2} \ left \ lceil \ lceil \ frac {2^m+1} {3m} {3m} \ right \ right \ rceil $ nogure $ indequil $ nogival apn在$ {2^2^2^2^2m} $ eiment protiment protight基本功能。这是对先前结果的一个很好的改进：即使对于$ m $，最著名的下限是$ \ frac {φ（m）} {2} {2} \ left（\ lfloor \ frac \ frac {m} {4} {4} \ rfloor +1 \ right）$，对于$ $ m $，没有这样的下限。此外，我们确定了Taniguchi APN功能的自动形态组。
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 bound has been $\frac{φ(m)}{2}\left(\lfloor \frac{m}{4}\rfloor +1\right)$, for odd $m$, there has been no such lower bound at all. Moreover, we determine the automorphism group of Taniguchi's APN functions.
