Geodesic
On properties of the set of values of an arbitrary vector Boolean function
Prikladnaya Diskretnaya Matematika. Supplement, no. 8 (2015), pp. 51-53

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For an arbitrary vector Boolean function $F\colon\mathbb F_2^n\to\mathbb F_2^n$, some sufficient conditions that $\{F(x)\oplus F(x\oplus a)\colon x,a\in\mathbb F_2^n\}=\mathbb F_2^n$ are stated. This result is applied to researching metrical properties of APN functions.
Keywords: vector Boolean function, differentially $\delta$-uniform function, APN function. 
@article{PDMA_2015_8_a18,
     author = {G. I. Shushuev},
     title = {On properties of the set of values of an arbitrary vector {Boolean} function},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {51--53},
     publisher = {mathdoc},
     number = {8},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2015_8_a18/}
}
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G. I. Shushuev. On properties of the set of values of an arbitrary vector Boolean function. Prikladnaya Diskretnaya Matematika. Supplement, no. 8 (2015), pp. 51-53. http://geodesic.mathdoc.fr/item/PDMA_2015_8_a18/