Concept of a birepresentation for the Moufang loops is elaborated. 2000 MSC: 20N05

It is surveyed known (published) possible application of binary and n-ary quasigroups in cryptology. Mathematics Subject Classification: 20N05, 94A60.

We construct quasigroup-based hybrid of a code and a cipher. 2000 Mathematics Subject Classification: 94A60, 20N05, 20N15.

Reductivity in the Ma’tsev algebras is inquired. This property relates the Mal’tsev algebras to the general Lie triple systems. 2000 MSC: 20N05, 17D10

We describe all constructions for loops of Bol-Moufang type analogous to the Chein construction M(G, ∗, g0) for Moufang loops. Mathematics subject classification: 20N05.

It is shown how integrability of the generalized Lie equations of continous Moufang transformatiosn is related to the reductivity conditions and Sagle-Yamaguti identity. 2000 MSC: 20N05, 17D10

Modifications of Markovski quasigroup based crypto-algorithm have been proposed. Some of these modifications are based on the systems of orthogonal n-ary groupoids. T -quasigroups based stream ciphers have been constructed. 2000 Mathematics Subject Classification: 94A60, 20N05, 20N15

It is shown how integrability of the generalized Lie equations of a local analytic Moufang loop is related to the reductivity conditions and Sagle-Yamaguti identity. 2000 MSC: 20N05, 17D10

The differential equations for a local analytic Moufang loop are established. The commutation relations for the infinitesimal translations of the analytic Moufang are found. These commutation relations can be seen as a (minimal) generalization of the Maurer-Cartan equations. 2000 MSC: 20N05, 17D10

Using Vojvoda approach [34] we demonstrate that cryptographical primitives proposed in [26] are vulnerable relative to chosen ciphertext attack and chosen plaintext attack. We develop proposed in [31] modifications and add some new modifications of known quasigroup based stream ciphers [19, 26]. Systems of orthogonal n-ary groupoids are used. 2000 Mathematics Subject Classification: 20N05, 20N1...