METHOD FOR CONSTRUCTING POST-QUANTUM ALGORITHMS OF EDS WITH TWO HIDDEN GROUPS
Alexei Petrenko · Voprosy kiberbezopasnosti · 2025
Purpose of work is to develop and substantiate a method for constructing post-quantum EDS algorithms based on finite noncommutative associative algebras, which provides enhanced signature randomization due to double groups and chaotic mappings, compact key sizes and high performance, as well as automated evolutionary design of the multiplication table structure. Research methods: algebraic modeling of noncommutative structures and computer verification of the associativity of multiplication tables, mathematical modeling of the signature process and probabilistic assessment of cryptographic strength during mass signature collection, evolutionary search methods (evolutionary algorithms, crossover and mutation) for adaptive optimization of the structure of Λ, numerical experiments with the generation of one-time exponentials b,n through logistic mapping and testing of the received cryptoprimitive based on Python and the NumPy library. Research results: a basic cryptographic asset has been formed that supports double randomization of the signature. It is shown that the chaotic generation of exponents (b,n) significantly complicates statistical cryptanalysis, even with mass collection of signatures. An adaptive evolutionary algorithm has been developed that allows for the orderly selection of the best tables without losing associativity. An experimental analysis was carried out, as a result of which the exponential complexity of attacks was confirmed with the correct choice of parameters, and the results of implementing the scheme on average hardware resources were demonstrated. The scientific novelty: a combination of noncommutative algebras with a double group and a chaotic generator is proposed, which increases the level of signature randomization. For the first time, the evolutionary search for table parameters was systematically applied to the task of constructing post-quantum EDS algorithms, which ensures associativity, speed, and theoretical cryptographic stability of the generated tables. The fundamental stability of such a scheme to quantum attacks is shown due to the lack of known polynomial algorithms for solving nonlinear systems in a noncommutative structure. The results were obtained with the financial support of the project «Technologies for countering previously unknown quantum cyber threats», implemented within the framework of the state program of the «Sirius» Federal Territory «Scientific and technological development of the «Sirius» Federal Territory (Agreement No. 23-03 dated September 27, 2024).