Computational work for some $TU$-based permutations

The problem of evaluating the computational complexity of certain classes of substitutions with a $TU$-representation is considered. The metrics used include combinatorial complexity and the depth of the function that defines the substitution. To obtain these evaluations, the representation of field elements in various bases is investigated, including polynomial, normal, mixed, as well as PRR and RRB representations. The primary focus is on analyzing different representations of field elements and their impact on computational complexity. The combinatorial complexity is assessed based on the number of elementary operations required to implement the substitution, while the function depth is determined by the maximum number of logical levels in the circuit. The use of different bases allows us to identify the most effective representation methods that help minimize computational complexity. As an example, we provide an evaluation of the specified characteristics for the substitution used in Russian standardized symmetric algorithms. The lowest known estimate of combinatorial complexity has been obtained, which equals $169$.