**On values of vectorial Boolean functions and related problems in APN functions**

*George Shushuev*

**Abstract: **In this paper we prove that there are only differential 4-uniform functions which are on distance 1 from an APN function. Also we prove that there are no APN functions of distance 1 from another APN functions up to dimension 5. We determine some properties of
the set of values of an arbitrary vectorial Boolean function from F_n^2 to F_n^2 in connection to the set of values of its derivatives. These results are connected to several open question concerning metric properties of APN functions.

**Category / Keywords: **foundations / Vectorial Boolean function, APN function, differentially δ-uniform function

**Date: **received 31 Dec 2015

**Contact author: **shushuev at math nsc ru

**Available format(s): **PDF | BibTeX Citation

**Version: **20151231:164956 (All versions of this report)

**Short URL: **ia.cr/2015/1244

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