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arXiv 2004.118962020-04math.COcs.ITmath.IT

The number of almost perfect nonlinear functions grows exponentially

Christian Kaspers, Yue Zhou

备注：: 38 pages. arXiv admin note: text overlap with arXiv:2002.00673

英文摘要：

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.

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