Fractal magmas and public-key cryptography

In this paper, we deal with magmas — the simplest algebras with a single binary operation. The main result of our research is algorithms for generating chain of finite magmas based on the self-similarity principle of its Cayley tables. In this way the cardinality of a magma's domain is twice as large as the previous one for each magma in the chain, and its Cayley table has a block-like structure. As an example, we consider a cyclic semigroup of binary operations generated by a finite magma's operation with a low-cardinality domain, and a modify the Diffie–Hellman–Merkle key exchange protocol for this case.