Diffusion rates of linear medium in AES-like ciphers

We model AES-like algorithms by XSLP-ciphers having uniformly diffusive permutations $P$ and fixed linear (differential) branch number $\rho$ of blocks of transformation matrices $L$. For all admissible values of $\rho$ and arbitrary number of rounds the exact lower bound for the number of active $\mathrm{S}$-boxes in systems of linear (differential) probabilistic relations is obtained.