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

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.