Matrix formula for the spectrum of output distribution of block cipher scheme and statistical criterion based on this formula

Arbitrary block iterative cipher scheme with random independent binary input and output vectors is considered. A matrix formula for the spectrum of the scheme output distribution is obtained by means of the pseudo-Boolean linear representation of the iterative vector-function. Based on this formula, a statistical criterion of the hypothesis testing that binary vectors are obtained as an output of the scheme against the hypothesis of their uniform distribution is given. Asymptotic type I and type II errors are calculated. An experimental comparison of the criterion with the “Bookstack” test (and its proposed modification) is done during the construction of a distinguishing attack on the mini-models of the block cipher PRESENT (with block size 12 bits and the number of rounds $R\le10$).