Methods of construction and properties of logariphmic signatures

Abstract

Development and promising areas of research in the construction of practical models of quantum computers contributes to the search and development of effective cryptographic primitives. Along with the growth of the practical possibilities of using quantum computing, the threat to classical encryption and electronic signature schemes using classical mathematical problems as a basis, being overcome by the computational capabilities of quantum computers. This fact motivates the study of fundamental theorems concerning the mathematical and computational aspects of candidate post-quantum cryptosystems. Development of a new quantum-resistant asymmetric cryptosystem is one of the urgent problems. The use of logarithmic signatures and coverings of finite groups a promising direction in the development of asymmetric cryptosystems. The current state of this area and the work of recent years suggest that the problem of factorizing an element of a finite group in the theory of constructing cryptosystems based on non-Abelian groups using logarithmic signatures is computationally complex; it potentially provides the necessary level of cryptographic protection against attacks using the capabilities of quantum calculations. The paper presents logarithmic signatures as a special type of factorization in finite groups; it also considers their properties and construction methods.