D.T.Muhamediyeva, F.A.Tagayev, Yu.Sh.Yuldoshev
National Research University "Tashkent Institute of Irrigation and Agricultural Mechanization Engineers", Mamun University

Abstract. This article analyzes the main algorithms of post-quantum cryptography and their mathematical foundations. The weakening of traditional cryptographic systems as a result of the development of quantum computers requires the development of new, quantum-resistant algorithms. The study deeply studies lattice-based, code-based, multivariable, and hash-function-based cryptographic approaches. The principle of operation, mathematical model, and security level of each algorithm are comparatively evaluated. The advantages and disadvantages of these algorithms are also identified and their practical application prospects are discussed. The results obtained are of great importance in choosing optimal approaches in the field of post-quantum cryptography.

Keywords: Post-quantum cryptography, quantum computers, lattice-based algorithms, LWE, code-based cryptography, multivariate systems, hash functions, cryptographic security

Introduction. 
     Ensuring information security is one of the most important issues in the modern digital society. Currently widely used cryptographic algorithms, including RSA, Diffie–Hellman, and systems based on elliptic curves, rely on mathematically complex problems. However, since the development of quantum computers allows for effective solutions to these problems, existing cryptosystems may lose their relevance. In this regard, post-quantum cryptography is emerging as a new scientific direction. This direction is aimed at developing cryptographic algorithms that remain secure even in the conditions of quantum computers. The basis of post-quantum cryptography is mathematical problems that are complex for classical computers, but for which effective solutions have not been found for quantum computers. Currently, there are several main approaches to post-quantum cryptography. Among them, lattice-based cryptography, code-based systems, multi-variable quadratic equations, and signature systems based on hash functions occupy a special place. Each approach has its own mathematical model, and their level of security and practical effectiveness vary [1-4].
​​​​    This article systematically analyzes the main algorithms of post-quantum cryptography, their mathematical foundations and principles of operation. Also, the advantages and disadvantages of various approaches are compared, and their practical application prospects are highlighted. This serves as an important scientific basis for the selection and development of secure cryptographic systems in the future. Scientific research in the field of post-quantum cryptography has been dramatically activated with the development of quantum computers. The security of classical cryptosystems is mainly based on factorization and discrete logarithm problems, which can be effectively solved using quantum algorithms. Therefore, new cryptographic approaches are based on complex mathematical problems. An analysis of the literature shows that post-quantum cryptography is divided into several main areas, and each area has its own mathematical model [5- 8]...