Decomposition of Quaternion-Like Algebras into a Set of Commutative Subalgebras

Algebraic digital signature algorithms with a hidden group, which are based on the computational difficulty of finding a solution to a system of many quadratic equations with many unknowns, are very attractive as candidates for practical post-quantum cryptoschemes suitable well for solving security problems in Internet of things and in other information technologies. Finite non-commutative associative algebras are used as supports of said algorithms, and the hidden group represents a commutative group of sufficiently large order. A class of algebras, called finite quaternion-like algebras, has emerged as candidates foralgebraic support. The decomposition of finite quaternion-like algebras into commutative subalgebras, multiplicative groups which can potentially be used as the hidden group, is studied. The performed research shows that exactly three types of subalgebras exist. Formulas describing the number of subalgebras of each type and the orders of their multiplicative groups are derived. The results obtained are similar to the results of previous studies of the structure of four-dimensional algebras set by sparse basis vector multiplication tables. The obtained formulas can be used to generate parameters of signature algorithms with a hidden group when using finite quaternion-like algebras as algebraic support.