Semigroups of binary operations and magma-based cryptography

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