Title: On Upper Bounds for Algebraic Degrees of APN Functions
Abstract: We study the problem of existence of APN functions of algebraic degree n over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> n. We characterize such functions by means of derivatives and power moments of the Walsh transform. We deduce several non-existence results which imply, in particular, that for most of the known APN functions F over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> n. the function x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n-1</sup> + F(x) is not APN, and changing a value of F in a single point then results in non-APN functions. This leads us to conjectures that an APN function modified in one point cannot remain APN and that there exists no APN function of algebraic degree n.
Publication Year: 2018
Publication Date: 2018-06-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 22
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