Number of associative triples of quasigroup plays an important role in development of quasigroup based cryptographic schemes. In this paper we present algebraic properties of highly non-associative quasigroups and derive the criteria for polynomial completeness based on their multiplicative groups. We develop an algorithm to check the polynomial completeness of the quasigroup Q from its Latin square representation, which is based on the criteria derived by using element of Mult(Q) with specific cycle structure. We also develop and implement an algorithm for deriving associative triples of finite quasigroups based on commutators of their Latin squares. Experimental results on quasigroups of different order and all quasigroups of order 4 of different classes are reported.