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On some quadratic APN functions. (English) Zbl 1419.11138

Summary: A construction of APN functions using the bent function \(B(x,y)=xy\) is proposed in [C. Carlet, Des. Codes Cryptography 59, No. 1–3, 89–109 (2011; Zbl 1229.94041)]. At this time, two families of APN functions using this construction are known, that is, the family of Carlet (loc. cit.) and the family of Y. Zhou and A. Pott [Adv. Math. 234, 43–60 (2013; Zbl 1296.12007)]. In this note, we propose another family of APN functions with this construction, which are not CCZ equivalent to the former two families on \(\mathbb F_{2^8}\). We also propose a family of presemifields and determined the middle, left, right nuclei and the center of the associated semifields.

MSC:

11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
06E30 Boolean functions
12K10 Semifields

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References:

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