Multidimensional spectral criterion for testing hypotheses on random permutations

Let $N$ random identically distributed pairs $(x,y)\in\mathbb{X}^2$ are observed, where $x$ has the uniform distribution on the finite set $\mathbb{X}$. We test the hypothesis that the matrix $Q=\|\mathsf{P}\{y=b\mid x=a\}\|_{a,b\in\mathbb{X}}$ equals $\|\frac1{|\mathbb{X}|}\|$ against the hypothesis $Q=\mathbb{P}^R$, where doubly stochastic matrix $\mathbb{P}$ and degree $R$ are known. A multidimensional tests based on eigenvectors of $\mathbb{P}$ are proposed. They are used to calculate the characteristics of differential distinguishing attacks on random permutations generated by ciphers of SmallPresent family with block lengths $n\in\{8,12,16\}$ and $4\le R\le 9$ rounds.