SEMIGROUPS OF BINARY OPERATIONS AND MAGMA-BASED CRYPTOGRAPHY

In this article, algebras of binary operations as a special case of finitary homogeneous relations algebrasare investigated. The tools of our study are based on unary and associative binary operations acting on theset of ternary relations. These operations are generated by the converse operation and the left-composition ofbinary relations. Using these tools, we are going to define special kinds of ternary relations that correspondto functions, injections, right- and left-total binary relations. Then we obtain criteria for these properties interms of ordered semigroups. Note, that there is an embedding of the semigroup of quasigroups operationsin the semigroup of magmas operation and further in the semigroup of ternary relations. This is similar toembedding the semigroup of bijections in the semigroup of functions and then in the semigroup of binaryrelations. Taking a binary operation as the generator of a cyclic semigroup, we can apply an exponentialsquaring method for the fast computation of its positive integer powers. Given that this is the main methodof public key cryptography, we are adapting the Diffie-Hellman-Merkle key exchange algorithm for magmasas a result.