Стандартизація еліптичних кривих: аналіз та впровадження в криптографічні протоколи

Abstract

The purpose of the article is to consider the current state of elliptic cryptography, the prerequisites for its use, as well as the requirements of modern standards related to the use of elliptic cryptography The use of elliptic curves in cryptography is considered one of the most promising areas of development of modern security algorithms. This mathematical approach is based on the complexity of solving the discrete logarithm problem in a group of points of an elliptic curve over a finite field. The use of cryptography on elliptic curves allows you to ensure the security of data exchange using effective encryption algorithms and the creation of digital signatures (DI). This study examines elliptic curves for cryptographic purposes, and provides basic operations on the point group of elliptic curves. Special attention is paid to Elliptic curve Diffie-Hellman (ECDH) and Elliptic Curve Digital Signature Algorithm (ECDSA) key exchange algorithms. The standards regulating the use of elliptic curves in cryptographic systems are also analyzed, and the advantages of this cryptographic paradigm compared to the main asymmetric algorithms are considered. Potential threats and vulnerabilities of cryptographic algorithms based on elliptic curves are investigated. Examples of popular standardized curves recommended by relevant organizations, such as NIST, used in real-world cryptographic applications are also provided. Elliptic curve cryptography (ECC) is currently one of the foundations for the development of modern public-key cryptographic algorithms. ECC has gained recognition in cryptography for providing a high level of security with shorter key lengths (compared to other cryptographic approaches), high speed, resource savings, and versatility, giving it an advantage over other methods such as RSA and others. It provides a secure network connection, generates secret keys for TLS servers and their clients, and is also used to create digital signatures that guarantee the authenticity of transactions in cryptocurrency systems.