Symmetric and Asymmetric Cryptography on the Special Linear Cracovian Quasigroup

We propose symmetric and public key cryptographic protocols on the non-associative special linear Cracovian quasigroup. The SLn(Z) Cracovian quasigroup is the set of n x n matrices with determinant one and entries in the ring of integers Z with the non-associative multiplication. The strength of the proposed symmetric cipher is based on the NP-hardness of the non-negative matrix factorization problem. We define the totient function for a given matrix from the special linear group over the ring Z/mZ of integers modulo m as the order of a cyclic subgroup generated by this matrix. Defined in this way totient function allows us to construct the public key cryptographic protocol on the special linear group SLn(Z(m)) and generalize it to the Cracovian quasigroup.