Розробка крипто-кодових конструкцій на гіпереліптичних кривих

The subject of this study is the algebraic processes of data formation, encoding and syndrome decoding in cryptographic code constructions based on hyperelliptic curves of higher genera over Galois fields. The problem addressed lies in the excessive computational complexity and energy consumption of post-quantum asymmetric cryptosystems, which precludes their use on hardware platforms with resource constraints. The results obtained include the construction of a mathematical model of an algebraic-geometric code in projective coordinates, the development of a point mapping algorithm, and the generation of sparse matrices. The implementation of hyperelliptic structures into the symmetric Rao–Nama scheme has enabled a reduction in energy consumption of 20–60% compared to the asymmetric McElys scheme. The increase in the overall efficiency of the system is explained by the complete elimination of the resource-intensive operation of inverting Galois field elements thanks to an isomorphic transition to projective space, as well as the complete avoidance of solving matrix equations during the process of decrypting cryptograms. The distinctive features of the results obtained, which enabled the solution of the problem under investigation, lie in the fact that the targeted selection of row vectors for the evaluation matrices allowed the Hamming weight of the code to be artificially minimized. At the same time, the synergy of multidimensional algebraic geometry with symmetric architecture ensured an approximation to linear time complexity of decoding. The scope and conditions for the practical application of the results obtained cover the use of synthesized cryptographic code constructs in nodes of modern cyber-physical systems and Internet of Things (IoT) networks under conditions of severe computational resource constraints and autonomous power supply limits.