On a quasigroup prolongation and its recursive differentiability

We consider a method of quasigroups prolongation by adding two new elements and by using two transversals that intersect in exactly one cell. In a Latin square L of order q, along with usual transversals we call "free transversals" any set of q cells taken by one from each row and each column of L. It is shown that a Latin square of order q can have at most q-2 (ordinary) transversals that intersect exactly in one cell, which is also the only common cell of every two transversals. It is proved that a latin square of order 4 has exactly 24 free transversals, 576 (ordinary) transversals, 96 pairs of free transversals that intersect exactly in one cell and 13824 pairs of ordinary transversals that intersect exactly in one cell. We also investigate the recursive 1-differentiability of considered prolongations of Latin squares of order 5, one of the transversals being on the main diagonal. The research done in this paper is applicable in cryptography and is important for data integrity and fault tolerance.