Cryptographically properties of random S-boxes

The results of studies of the properties of random permutations carried out with the participation of the authors are generalized. It is shown that random substitutions overwhelmingly have good cryptographic, and in particular, algebraic properties. The prospects of using random S-blocks to build block symmetric ciphers with improved dynamic rates of arrival to random substitution are substantiated. A refined model of random substitution and the corresponding criteria are proposed, with the help of which one can verify the suitability of substitutions generated randomly for use in modern cipher designs. It is a check, since with a very high probability the checked substitutions will be suitable.