On the relationship between nonlinear and differential properties of vectorial Boolean functions

A. V. Miloserdov

Prikladnaya Diskretnaya Matematika. Supplement, no. 12 (2019), pp. 60-62

Résumé

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.

Détail

Keywords: APN function, AB function, differential uniformity, nonlinearity.