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. 
@article{PDMA_2014_7_a4,
     author = {A. A. Gorodilova},
     title = {Characterization of {APN} functions by means of subfunctions},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {15--16},
     publisher = {mathdoc},
     number = {7},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2014_7_a4/}
}
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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/