Some conditions for the applicability of the integral cryptanalysis to $4$-rounds of AES-like ciphers

A number of necessary conditions and one sufficient condition are obtained so that the integral cryptanalysis can be applied to block algorithms constructed similarly to the AES algorithm (i.e., SQUARE, Rijndael, Crypton) with a reduced number of rounds to four, which are denoted as $f_1,f_2,f_3,f_4$. For example, it was proved that if we consider the multiset $\{y_{j}(x)\in V_{8}:x\in I_{i}\}$, where $I_{i} =\{(B_{0},\ldots,B_{i-1},z,B_{i+1},\ldots,B_{15}):z\in V_{8}\}$, the subvector $B_{i} =z$, $i\in\{0,\ldots,15\}$, takes all possible values from $V_{8}$, and the other data block subvectors are fixed, $(y_{0}(x),\ldots ,y_{15}(x))=f_{4} \circ f_{3} \circ f_{2} \circ f_{1}(x,k_{0}^{*})$, $k_{0}^{*} $ is the true key, then a necessary condition for obtaining information about the fourth round key $k_{4, j} $ by the integral method is: the subset $ Y_{j} ^{*} =\{\alpha \in V_{8} :|\{x\in I_{i} :y_{j}(x)=\alpha\}| =2k-1,k\in\mathbb{N}\}$ is not empty. The experimental data on the application of the integral method to the Rijndael algorithm are presented.