$\otimes_{\mathbf{W}}$-markovianity of XSL-block ciphers connected with properties of their round functions

Earlier we have described conditions for the existence of permutation homomorphisms for some class of a $\otimes_{\mathbf{W}}$-markovian block ciphers. In this paper we give conditions on $S$-boxes and on linear transformations which guaranteing the $\otimes_{\mathbf{W}}$-markovian property. In particular, we provide sufficient conditions on these transformations under which the corresponding XSL-block cipher is $\otimes_{\mathbf{W}}$-markovian.