Geodesic
On the relationship between nonlinear and differential properties of vectorial Boolean functions
Prikladnaya Diskretnaya Matematika. Supplement, no. 12 (2019), pp. 60-62.

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The relations between the linear approximation table (LAT) and the differences distribution table (DDT) of the vectorial Boolean functions are investigated. Let $F$ be a function from $\mathbb{F}_2^n$ into $\mathbb{F}_2^n$. DDT of $F$ is a $2^n\times 2^n$ table defined by DDT$(a, b) = |\{x\in\mathbb{F}_2^n | F(x) \oplus F(x\oplus a) = b \}|$ for each $a,b\in \mathbb{F}_2^n$. LAT of $F$ is a $2^n \times 2^n$ table, in the cell $(v, u)$ of which the squared Walsh — Hadamard coefficient is stored. It is proved that the presence of coinciding rows in DDT and LAT is an invariant under affine equivalence as well as under EA-equivalence for normalized DDT and LAT. It is hypothesized that if all rows in the LAT (DDT) of a vectorial Boolean function $F$ are pairwise different, then all rows in its DDT (LAT) are also pairwise different. This hypothesis is checked for functions in a small number of variables and for known APN functions in not more than 10 variables.
Keywords: APN function, AB function, differential uniformity, nonlinearity. 
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     author = {A. V. Miloserdov},
     title = {On the relationship between nonlinear and differential properties of vectorial {Boolean} functions},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {60--62},
     publisher = {mathdoc},
     number = {12},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2019_12_a17/}
}
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A. V. Miloserdov. On the relationship between nonlinear and differential properties of vectorial Boolean functions. Prikladnaya Diskretnaya Matematika. Supplement, no. 12 (2019), pp. 60-62. http://geodesic.mathdoc.fr/item/PDMA_2019_12_a17/

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