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Prikl. Diskr. Mat. Suppl., 2016, Issue 9, Pages 39–40 (Mi pdma295)  

Discrete Functions

Functions on distance one from APN functions in small number of variables

G. I. Shushuev

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: In this paper, we deal with vectorial Boolean functions $F\colon\mathbb F_2^n\to\mathbb F_2^n$ of dimension $n\geq1$. Functions $F$ and $G$ are EA-nonequivalent if $G\neq A_1\circ F\circ A_2\oplus A$ for any affine functions $A_1$, $A_2$ and $A$, where $A_1$ and $A_2$ are permutations. A function $F$ is called APN if for any $a,b\in\mathbb F_2^n$, where $a$ is nonzero, the equation $F(x)\oplus F(x\oplus a)=b$ has at most two solutions. We prove that there are no APN functions on the distance one from an APN functions up to dimension $5$, from all quadratic APN functions of dimension $6$, and from all known EA-nonequivalent APN functions of dimensions $7$ and $8$.

Keywords: vectorial Boolean function, APN function.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-07-01328


DOI: https://doi.org/10.17223/2226308X/9/16

Full text: PDF file (580 kB)
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Document Type: Article
UDC: 519.7

Citation: G. I. Shushuev, “Functions on distance one from APN functions in small number of variables”, Prikl. Diskr. Mat. Suppl., 2016, no. 9, 39–40

Citation in format AMSBIB
\Bibitem{Shu16}
\by G.~I.~Shushuev
\paper Functions on distance one from APN functions in small number of variables
\jour Prikl. Diskr. Mat. Suppl.
\yr 2016
\issue 9
\pages 39--40
\mathnet{http://mi.mathnet.ru/pdma295}
\crossref{https://doi.org/10.17223/2226308X/9/16}


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