Abstract
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 \(4\) .
