Non-associative structures in homomorphic encryption

In this paper, we obtain a classification of quasigroup rings by the quantity of elements with null left annihilator for different quasigroups. This classification becomes possible due to a criterion of being an element with null left annihilator in a quasigroup ring. By virtue of this criterion, we make a calculation to find regularities using various fields and quasigroups with order $4$. This outcome helps us to obtain two results where any two quasigroup rings have the same number of elements with null left annihilator and the element of the quasigroup ring $\mathrm{GF}(p)Q$ with fixed quasigroup $Q$ has null left annihilator in the quasigroup ring $\mathrm{GF}(p^n)Q$.