Characterization of polynomially complete quasigroups based on Latin squares for cryptographic transformations

We characterize polynomially complete quasigroups of order 4 from their corresponding Latin squares. This class of quasigroups is the suitable choice for cryptographic applications from algebraic point of view. Towards this direction we establish some criteria of Latin squares related to row and column permutations and their cyclic decompositions. We develop and implement an algorithm to classify the quasigroups of order 4 into four classes based on these algebraic properties. We also develop criteria for isotopy under which the polynomial completeness remains invariant and present a method to construct this class of isotopies. Finally we carry out experiments for cryptographic transformation based on quasigroups of different classes and draw some important observations.