The Matrix Power Function And Its Application To Block Cipher S-Box Construction

In this paper, we are continuing the security investigation of the new S-box function based on so-called matrix power function proposed earlier by the same authors. Security is considered against algebraic cryptanalysis. We claim that constructed S-box function is a candidate to be a one-way function since we showed that its inversion is related with one known NP-complete problem, namely with a solution of the system of multivariate quadratic (MQ) polynomial equations. This problem is known as an MQ problem. We show that the total break of the cipher providing key recovery having one or several plaintext-ciphertext pairs and using algebraic cryptanalysis corresponds to the solution of the large system of MQ polynomial equations. This system is underdefined and not sparse. Hence, we show that proposed S-box has a provable security property since the solution of such system of equations is intractable. The guess and determine attack also fails since the guessing and computing sets of solutions are too large to be solvable effectively due to the underdefined nature of MQ equations.