Application of non-associative structures for construction of homomorphic cryptosystems

Homomorphic encoding allows to perform certain mathematical operations with the encoded text and to get the encoded outcome that corresponds to the result of operations processed with a plaintext. There exist both fully homomorphic and partially homomorphic options (with respect to one or more operations). For practical use of such an encoding it is necessary to have a homomorphism with respect for at least one operation. Using non-associative operations, we construct in this paper an example of a cryptosystem based on the El-Gamal system that is homomorphic with respect to two on-going operations: a group and a quasigroup ones.