Cryptanalysis of the system based on word problems using logarithmic signatures

Rapid development and advances of quantum computers are contributing to the development of public key cryptosystems based on mathematically complex or difficult problems, as the threat of using quantum algorithms to hack modern traditional cryptosystems is becoming much more real every day. It should be noted that the classical mathematically complex problems of factorization of integers and discrete logarithms are no longer considered complex for quantum calculations. Dozens of cryptosystems were considered and proposed on various complex problems of group theory in the 2000s. One of such complex problems is the problem of the word. One of the first implementations of the cryptosystem based on the word problem was proposed by Magliveras using logarithmic signatures for finite permutation groups and further proposed by Lempken et al. for asymmetric cryptography with random covers. The innovation of this idea is to extend the difficult problem of the word to a large number of groups. The article summarizes the known results of cryptanalysis of the basic structures of the cryptosystem and defines recommendations for ways to improve the cryptographic properties of structures and the use of non-commutative groups as basic structures.

Methods of construction and properties of logariphmic signatures

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.

Construction of strongly aperiodic logarithmic signatures

The aim of the paper is to present a general construction of strongly aperiodic logarithmic signatures (SALS) for elementary abelianp-groups. Their existence significantly extends the classes of tame logarithmic signatures which are used for the cryptosystem{\mathrm{MST}_{3}}. They have particular characteristics that do not share with the well-known classes of transversal or fused transversal logarithmic signatures, and therefore they will play a vital role for logarithmic signature based cryptosystems in practice. In theory, the construction of SALS is interesting in its own right as well.

Tame logarithmic signatures of abelian groups

The security of the asymmetric cryptosystem MST