Post-quantum Cryptoschemes: New Finite Non-commutative Algebras for Defining Hidden Logarithm Problem

In the article we present some properties of non-commutative finite algebras of four-dimension vectors with parameterized multiplication operation characterized in that different modifications of the multiplication operation are mutually associative. One of the introduced finite algebras represents ring. Other algebra contains no global unit element, its elements are invertible locally, and is characterized in that the multiplication operation possess compression property. Regarding the investigated ring, the detailed attention is paid to properties of the set of non-invertible elements of the ring. Formulas for zero-divisors and unit elements of different types are derived. The introduced finite algebras represent interest to define over them the hidden discrete logarithm problem that is a promising cryptographic primitive for post-quantum cryptography.