Geodesic
Functions on distance one from APN functions in small number of variables
Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 39-40.

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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. 
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     author = {G. I. Shushuev},
     title = {Functions on distance one from {APN} functions in small number of variables},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {39--40},
     publisher = {mathdoc},
     number = {9},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2016_9_a15/}
}
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G. I. Shushuev. Functions on distance one from APN functions in small number of variables. Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 39-40. http://geodesic.mathdoc.fr/item/PDMA_2016_9_a15/

[1] Nyberg K., “Differentially uniform mappings for cryptography”, LNCS, 765, 1994, 55–64 | MR | Zbl

[2] Biham E., Shamir A., “Differential cryptoanalysis of DES-like cryptosystems”, J. Cryptology, 1991, no. 4, 3–72 | DOI | MR | Zbl

[3] Shushuev G. I., “Vektornye bulevy funktsii na rasstoyanii odin ot APN-funktsii”, Prikladnaya diskretnaya matematika. Prilozhenie, 2014, no. 7, 36–37

[4] Budagyan L., Carlet C., Helleseth T., Li N., On the (non-)existence of APN $(n,n)$-functions of algebraic degree $n$, http://ia.cr/2016/143

[5] Brinkmann M., Leander G., “On the classification of APN functions up to dimension five”, Des. Codes Cryptogr., 49 (2008), 273–288 | DOI | MR | Zbl

[6] Yu Y., Wang M., Li Y., “A matrix approach for constructing quadratic APN functions”, Des. Codes Cryptogr., 73 (2014), 587–600 | DOI | MR | Zbl