Algorithms for computing cryptographic characteristics of vectorial Boolean functions

There are presented algorithms for calculating the cryptographic characteristics of vectorial Boolean functions, such as the order of correlation immunity, nonlinearity, component algebraic immunity, and differential uniformity order. In these algorithms, the components of a vectorial Boolean function are enumerated according to the Gray code. Experimental results are given for random vectorial Boolean functions, permutations, and two known classes $\mathcal{K}_{n}$ and $\mathcal{S}_{n,k}$ of invertible vectorial Boolean functions in $n$ variables with coordinates essentially depending on all variables and on $k$ variables, $k<n$, respectively. Some properties of differential uniformity are theoretically proved for functions in $\mathcal{K}_{n}$ and $\mathcal{S}_{n,k} $, namely, the differential uniformity order $\delta_F$ equals $2^n$ for any $F\in\mathcal{S}_{n,k}$, and the inequality $2^n-4(n-1)\leq\delta_F\leq 2^n-4$ holds for any $F\in\mathcal{K}_{n}$.