Polynomial completeness and completeness of finite n-quasigroups

Finite quasigroups and n-quasigroups are currently extensively utilized to implement various cryptographic functions. Cryptographic requirements lead to constraints imposed on quasigroups and n-quasigroups. In particular, V. A. Artamonov proposed using polynomially complete quasigroups. Polynomial completeness can be decided with the help of a criterion of J. Hagemann and C. Herrmann: a quasigroup is polynomially complete if and only if it is simple and non-affine. In our paper we generalize this result to the case of n-quasigroups and give a proof based on I. G. Rosenberg’s description of maximal classes in k-valued logics. We also obtain a completeness criterion and show that completeness is a cryptographically reasonable requirement.