MATHEMATICAL FUNDAMENTALS OF ALGEBRAIC LATTICIES AND THEIR APPLICATION IN QUANTUM CRYPTOLOGY

Abstract

The ongoing development of quantum computers threatens state-of-the-art public key cryptographic schemes, such as discrete logarithm factorization key generation schemes, digital signatures, and elliptic curve cryptography. It is necessary to develop new cryptographic algorithms capable of resisting the attacks of quantum computers. Post-quantum cryptography (PQC) aims to develop algorithms that can be used without significant modifications to existing networks. The US National Institute of Standards and Technology (NIST) organizes a competition for the selection and standardization of new algorithms. This article provides an overview and analysis of the evaluation and selection process of NIST algorithms based on lattice theory problems. It gives basic definitions, describes the main problems of algebraic lattice theory, and summarizes the advantages of this class of cryptography, including its resistance to quantum computing. The work contributes to the study and comparison of post-quantum cryptographic algorithms, and also provides recommendations for their further use and standardization to ensure their security in the development of quantum computers.