The Reed - Muller code square and equivalence classes of McEliece - Sidelnikov cryptosystem private keys
Equivalence classes of McEliece - Sidelnikov cryptosystem private keys are studied in the work. The structure of the classes is described in the case, when the square of the code with the generator matrix (R|HR), where R is a generator matrix of the Reed - Muller code RM(r, m) of order r and length 2m, equals the Cartesian square of the code of order 2r and the same length. In this case, there exists a bijection between an equivalence class and the Cartesian square of automorphism group of the code RM(r,m). Moreover, it is shown that the ratio of matrices H causing other cases approaches zero when the code dimension approaches infinity.