Geodesic
Characterization of APN functions by means of subfunctions
Prikladnaya Diskretnaya Matematika. Supplement, no. 7 (2014), pp. 15-16.

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A vectorial Boolean function $F\colon\{0,1\}^n\to\{0,1\}^n$ is called an APN function if the equation $F(x)\oplus F(x\oplus a)=b$ has at most 2 solutions for any vectors $a,b$, where $a\neq0$. The complete characterization of APN functions by means of subfunctions is found. It is proved that $F$ is APN function if and only if each of its subfunctions in $n-1$ variables is an APN function or has the order of differential uniformity 4 and the admissibility conditions are hold. Some numerical results of this characterization for small number $n$ of variables are presented.
Keywords: vectorial Boolean function, differentially $\delta$-uniform function, APN function. 
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     author = {A. A. Gorodilova},
     title = {Characterization of {APN} functions by means of subfunctions},
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A. A. Gorodilova. Characterization of APN functions by means of subfunctions. Prikladnaya Diskretnaya Matematika. Supplement, no. 7 (2014), pp. 15-16. http://geodesic.mathdoc.fr/item/PDMA_2014_7_a4/

[1] Nyberg K., “Differentially uniform mappings for cryptography”, Eurocrypt 1993, LNCS, 765, 1994, 55–64 | MR | Zbl

[2] Tuzhilin M. E., “Pochti sovershennye nelineinye funktsii”, Prikladnaya diskretnaya matematika, 2009, no. 3, 14–20

[3] Frolova A. A., “Iterativnaya konstruktsiya APN-funktsii”, Prikladnaya diskretnaya matematika. Prilozhenie, 2013, no. 6, 24–25