Algebraic properties of subquasigroups and construction of finite quasigroups

Many important properties are identified and criteria are developed for the existence of subquasigroups in finite quasigroups. Based on these results, we propose an effective method that concludes the nonexistence of proper subquasigroups in a given quasigroup, or finds all its proper subquasigroups. This has an important application in checking the cryptographic suitability of a quasigroup.
Using arithmetic of finite fields, we introduce a binary operation to construct quasigroups of order $p^r$. Criteria are developed under which the quasigroups mentioned have desirable cryptographic properties, such as polynomial completeness and absence of proper subquasigroups. Effective methods are given for constructing cryptographically suitable quasigroups. The efficiency of the methods is illustrated by some standard examples and by implementation of all proposed algorithms in the computer algebra system Singular.