Polynomial Completeness and Strong Non-affinity of Almost All n-Quasigroups

In the framework of cryptographic applications of finite quasigroups and n -quasigroups, there exist a number of properties imposed in order to provide cryptographic strength. In particular, V.A. Artamonov proposed using polynomially complete structures, or, equivalently, simple and non-affine n -quasigroups. An amplification of non-affinity is strong non-affinity, i.e., non-affinity of all isotopes. In our paper, we investigate generic nature of these properties. Specifically, we prove that almost all n -quasigroups are strongly non-affine (i.e., the fraction of strongly non-affine n -quasigroups tends to $$1$$ as the order tends to infinity). Additionally, we focus on n -quasigroups of the order $$4$$ . We obtain the exact number of simple, affine and simultaneously simple and affine n -quasigroups of the order $$4$$ . These results directly imply polynomial completeness and strong non-affinity of almost all n -quasigroups of the order $$4$$ .