A novel version of the hidden logarithm problem for post-quantum signature algorithms

Abstract

In the article we propose a novel version of the hidden logarithm problem (HLP), the peculiarity of which is using the local units instead of the global unit in the HLP proposed earlier for designing post-quantum signature schemes. Two different 4-dimensional finite associative algebras with non-commutative multiplication operation are proposed as carriers of the introduced version of the HLP. One of the proposed algebras contains the global two-sided unit. The second algebra contains only local units (single-sided and two-sided ones). In the first algebra, every non-invertible element, which is not a zero-divisor, defines many local single-sided (the right-sided and the left-sided) and many local two-sided units. The both types of the single-sided units play an important role for defining the HLP of the proposed version. Formulas describing the local units of different types are derived and used to generate the public key of a developed HLP-based signature algorithm that is resistant to quantum attacks. Design criterion and security discussion of the introduced signature scheme are presented.