Commutative encryption method based on hidden logarithm problem

A candidate for post-quantum commutative encryption algorithm is proposed, which is based on the hidden discrete logarithm problem defined in a new 6-dimensional finite non-commutative associative algebra. The properties of the algebra are investigated in detail and used in the design of the proposed commutative cipher. The formulas describing the set of $p^2$ different global right-sided units contained in the algebra and local left-sided units are derived. Homomorphisms of two different types are considered and used in the commutative cipher. The encrypted message is represented in the form of a locally invertible element $T$ of the algebra and encryption procedure includes performing the exponentiation operation and homomorphism map followed by the left-sided multiplication by a randomly selected local right-sided unit. The introduced commutative cipher is secure to the known-plaintext attacks and has been used to develop the post-quantum no-key encryption protocol providing possibility to send securely a secret message via a public channel without using any pre-agreed key. The proposed commutative encryption algorithm is characterized in using the single-use keys that are selected at random directly during the encryption process.