The paper considers the concept of a threat model, presents the results of substantiation and development of proposals for building a threat model for asymmetric cryptotransformations such as a promising electronic signature (ES), which can be used in the post-quantum period. The generalized models of threats concerning perspective ES are stated in detail and their estimation is given. Threat models for promising ES using classical and quantum cryptanalysis methods and tools, threat models for synthesis and application of ES in general, as well as threat models for synthesis and application of ES in the post-quantum period are proposed. A list of threats is identified based on the results of the analysis of the methods of synthesis and application of known and promising ES. Proposals are formulated for a list of threats for which protection should be provided. The list of threats is determined using the IT-Grundschutz Catalogues of the German database, and based on this a threat model is formed. It is determined that the threats to the use of classical cryptanalysis in the synthesis and application of EP must be identified in detail unconditionally. The main threats (methods) of classical cryptanalysis that must be taken into account are identified. Possible variants of side channel attacks are considered. The main threats (attacks) using quantum mathematical methods that can be implemented on a quantum computer (of course, if it is built). A comparative analysis of the complexity of factorization for classical and quantum algorithms, as well as a comparative analysis of the complexity of the algorithm of discrete logarithm in a finite field based on the sieve of a numerical field and the Shore algorithm are given. Threats (attacks) are considered on the example of the problem of stability of cryptotransformations based on learning with errors (LWE). In general, attacks on LWE can be divided into 2 major classes – attacks based on bust and attacks based on lattice reduce. Preliminary analysis allows us to conclude that modern versions of LWE mechanisms are based on polynomial rings.

The paper presents the results of substantiation and development of proposals for building a threat model for asymmetric cryptotransformations such as a promising electronic signature (ES), which can be used in the post-quantum period. The generalized models of threats concerning perspective ES are stated in detail and their estimation is given. Threat models for promising ES using classical and quantum cryptanalysis methods and tools, threat models for synthesis and application of ES in general, as well as threat models for synthesis and application of ES in the post-quantum period are proposed. Proposals are formulated for a list of threats for which protection should be provided. The list of possible security threats to existing and future ES is formed from the number of threats available in IT-Grundschutz Catalogs, taking into account hardware, software and hardware-software resources, data processing technologies and cryptographic protection mechanisms in the use of ES, including requirements and conditions of synthesis of promising ES and application of ES in the post-quantum period. The concepts of EUF-CMA and SUF-CMA security are considered. Algorithms of work of each of these schemes are given. The concept of a comprehensive security model is introduced and its components are presented. The model of the violator and its essence are considered. The main threats (attacks) are given using quantum mathematical methods that can be implemented on a quantum computer (of course, if it is built and available for use). Attacks (threats) against a promising ES are presented and considered. The analysis of signature schemes for compliance with the required security models is performed. The terms "forward secrecy" and "perfect forward secrecy" are introduced and used. An analysis of signature schemes that are EUF-CMA and SUF-CMA secure is performed. Signature schemes, that are key-dependent, with evolving keys, are considered in terms of compliance with the EUF-CMA or SUF-CMA security model. The stateless signature algorithm is also considered. Algorithms of operation of such signature schemes are given.

Globally, the efforts of a significant number of crypto-theorists, mathematicians and cryptologists-practitioners are focused on the NIST PQC open competition. One of the main tasks of the competition consists in development and adoption of a post-quantum ES standard or standards. The finalists of the second stage of the NIST competition were three ES mechanisms – CRYSTALS-DILITHIUM, Falcon and Rainbow. In addition, three alternative candidates were identified that require more detailed research. In general, a comprehensive analysis of the finalists is an important task for cryptologists in the global cryptocommunity. Moreover, security, i.e. bringing the cryptographic stability of two finalist candidates, to the ES standard – CRYSTALS-DILITHIUM and Falcon, is based on problems in the theory and practice of algebraic lattices. The EP Dilithium method (scheme) is based on the approach called "Fiat-Shamir Interruptions". The essence of the CRYSTALS-DILITHIUM ES algorithm is considered in the article. A detailed analysis of possible attacks on the algorithm and the mechanisms of their implementation is also carried out. Models of violator, threats and security are considered and analyzed. The main definitions of ES security models are provided. The main design elements of the mechanism of perspective post-quantum ES Dilithium are described in the generalized form. General estimations of the ES Dilithium security level are given. The substantiation and essence of models of threats, violator and security are given. The stability of the ES Dilithium algorithm is investigated. The purpose of the article is to substantiate a security model, classification, primary analysis and assessment of known attacks on the CRYSTALS-DILITHIUM EP cryptosystem, to establish restrictions and develop practical algorithms for calculating (generating) system-wide parameters to ensure 128, 256, 384 and 512 bits of security relative to classical and 64, 128, 192 and 256 bits relative to quantum cryptanalysis.

The present paper considers national electronic voting system problems in Ukraine, principles of construction and development prospects. Electronic voting refers to a way to exercise will, in which the voting, counting, and publication of the results processes are carried out by electronic means and systems. Most existing voting systems are built on centralized principles and this allows providing certain advantages, for example, high controllability of the system, its reliability, and autonomy. However, hierarchical systems also have significant drawbacks, in particular, single decision center and centralized storage leads to vulnerability to cyberattacks on them. Also it should be noted, that in centralized systems due to the abuse of administrative resources distortions of the results of expression of will are possible. This is the biggest threat to the modern democratic information society. Research, development, and implementation of new technologies of electronic voting, which would make it impossible to intervene and distort the results of the will through decentralization while maintaining all the system qualities for safety and reliability are promising. This article proposes particular proposals for architecture substantiating as well as a basic model and interaction protocols of a decentralized electronic blockchain voting system. A two-level blockchain voting architecture is proposed, researched and verified through physical prototyping. Its implementation will increase confidence in information resources and services (which is especially important for government agencies) will reduce time and overhead costs; make it impossible for centralized institutions to intervene and possible corrupt practices; will increase the reliability of information storage and the quality of services provided.

The paper deals with the concept of homomorphic encryption and the possibility of its use in the mechanism of electronic voting. One of the problematic requirements for electronic voting systems is voter anonymity. On the one hand, each voter must be identified, and on the other, the content of his or her vote must be unknown. Currently, the methods and mechanisms used in real voting systems do not provide real anonymity. Therefore, both theoretical and practical content is an urgent and necessary problem of developing mechanisms for anonymous counting of votes with the protection of their distortion. The paper also provides a general analysis of the security level of prospective homomorphic encryption schemes. The essence of homomorphic encryption is that there is some set of operations whose result of executing over ciphertexts (with subsequent decryption) coincides with similar actions over plaintexts. Homomorphic encryption allows you to perform some calculations on information without having access to the information itself. However, there are a number of problems when trying to apply such calculations. The main ones are the choice of the method of asymmetric encryption, which provides the necessary cryptographic stability from both classical and quantum attacks, the identification of possible candidates for asymmetric cryptotransformations in homomorphic encryption, their evaluation of comparison with each other, and, of course, the choice of the most rational for a given multiple restrictions. The asymmetric schemes of homomorphic encryption are compared using the hierarchy analysis process. The method of asymmetric encryption with zero knowledge is substantiated. The objective of this article is to substantiate the possibilities, conditions, and constraints on the use of standardized asymmetric cryptotransformations in the creation of modern homomorphic encryption-type transformations, when anonymity of electronic voting and practical implementation of anonymous voting based on proof of zero knowledge must be guaranteed.

The problem of cryptographic protection against classical and potential crypto-analytic attacks with the use of quantum computer and quantum mathematics has become an urgent issue. Understanding this problem, technologically advanced states are making significant efforts to analyze the cryptographic stability of existing standards for cryptographic information security in the post-quantum period and are seeking to establish post-quantum standards for asymmetric cryptography. A practical solution to this problem is being pursued globally during the NIST USA international competition. As previous studies have shown, algebraic lattices are now considered to be a reliable mathematical basis on which post-quantum asymmetric encryptions and PIK can be created. NTRU-like algorithms are a class of crypto-transformations algorithms that satisfy basically the requirements of post-quantum cryptography. Algorithms for key generation direct and reverse cryptographic transformations are the basic components in NTRU-like algorithms for asymmetric crypto-transformations. A number of authors today focus on optimizing polynomial multiplication for these algorithms by the criterion of time complexity. A special requirement for them is the independence of the time of the multiplication operation from the polynomials themselves, which makes it impossible to attack by side channels. This paper proposes the use of the NTT and Toom-Kuk algorithms. It proposes a new solution to this problematic issue, which made it possible to obtain an acceleration of almost 2 times while providing a constant polynomial multiplication time. The objective of this article is to optimize the polynomial multiplication algorithm by the time complexity criterion, used to generate keys and perform direct and reverse cryptographic transformations of asymmetric encryptions and PIK on algebraic lattices.

The analysis of hashing functions that are applied or can be used in various blockchain systems is carried out. In particular, the most common national and international standards are considered, which contain specifications of world-famous cryptographic hashing algorithms, and various projects for the construction of decentralized blockchain systems where these functions can be applied are investigated.

The analysis of hashing functions that are applied or can be used in various blockchain systems is carried out. In particular, the most common national and international standards are considered, which contain specifications of world-famous cryptographic hashing algorithms, and various projects for the construction of decentralized blockchain systems where these functions can be applied are investigated.

The paper provides a rationale for the possibilities and the need to create blockchain-based public key infrastructure. An improved public key infrastructure model with blockchain-based certificate transparency as well as the main problematic issues of promising blockchain-based public key infrastructures are analyzed. A general assessment of the public key infrastructure based on the blockchain stability in conditions of well-known attacks is carried out.

The paper analyzes the basic principles of building decentralized technologies with the use of blockchain technology and their requirements in terms of security, as well as the analysis of the features and conditions of the use of secure blockchain technologies. Potential attacks are described and analyzed when the use of blockchain is a significant defense mechanism against them. The essence and suggestions concerning counteraction to attacks of a special kind are given.

The results of NTRU-similar algorithm analysis with respect to resistance against side-channel attacks are given. Proposals of defending these attacks are given. The evaluations of how many keys will be reduced with key space limitations of key data (which are needed to ensure the properties of random sequences) are made.

The results of the analysis of electronic signatures with one-time keys for the post-quantum period are given. The requirements are justified and an improved mechanism of electronic signature is presented for the criteria of spatial complexity of keys and signatures. A model of the violator and threats is proposed. The main types of attacks on the electronic signature with one-time keys are substantiated. The main relations of the post-quantum cryptanalysis complexity estimation are presented. The main recommendations and the main spheres of using the electronic signature in the post-quantum period are given.

Наведено оцінки стійкості сучасних популярних асиметричних та симетричних криптосистем проти квантового криптоаналізу на основі алгоритмів Шора та Гровера, а також проти класичних алгоритмів криптоаналізу. Окреслено основні напрями розвитку постквантової криптографії та проведено оцінку можливостей застосування на практиці криптосистем, що є стійкими, на цей момент, до квантового криптоаналізу. The article presents the evaluation of the resistance of modern popular asymmetric and symmetric cryptosystems against cryptanalysis based on quantum Shor’s algorithm and Grover’s algorithm and against classical algorithms of cryptanalysis. There we present basic directions of postquantum cryptography and possibilities of the practical application of cryptosystems that are stable at the moment against the quantum cryptanalysis.