Application of Partial Discrete Logarithms for Discrete Logarithm Computation
Dina B. Shaltykova, Yelizaveta Sergeevna Vitulyova, Kaisarali Kadyrzhan, Ibragim Esenovich Suleimenov · Computers · 2025
A novel approach to constructing an algorithm for computing discrete logarithms, which holds significant interest for advancing cryptographic methods and the applied use of multivalued logic, is proposed. The method is based on the algebraic delta function, which allows the computation of a discrete logarithm to be reduced to the decomposition of known periodic functions into Fourier–Galois series. The concept of the “partial discrete logarithm”, grounded in the existence of a relationship between Galois fields and their complementary finite algebraic rings, is introduced. It is demonstrated that the use of partial discrete logarithms significantly reduces the number of operations required to compute the discrete logarithm of a given element in a Galois field. Illustrative examples are provided to demonstrate the advantages of the proposed approach. Potential practical applications are discussed, particularly for enhancing methods for low-altitude diagnostics of agricultural objects, utilizing groups of unmanned aerial vehicles, and radio geolocation techniques.