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.
Galatenko et al. (Mon,) studied this question.