Hash-based signatures are one of the most promising classes of cryptographic schemes considered quantum resistant ones. The strength of cryptographic hash functions is one of the most important aspects of ensuring the security of hash-based schemes. Since classical hash-based signatures require tracking the number of signatures used, they were considered to be stateful for a long time. The SPHINCS scheme overcome this limitation, subsequently refined to SPHINCS+. The paper provides an assessment of the security of ES based on hash functions relative to side channel attacks.It also gives an analysis of recommendations for the use of one of the candidates of the NIST competition, based on hash cryptography - SPHINCS+, and conclusions about the feasibility of its use.

У даній роботі надається комплексне дослідження атак сторонніми каналами (SCAside-channel attacks) і атак помилками (FIAfault injection attacks) на схеми на основі решітки, з акцентом на схемі підпису Dilithium, яка є провідним кандидатом в процесі стандартизації Націолального Інституту Стандартизації NIST для постквантової криптографії.Також у роботі перевіряється низка індивідуальних контрзаходів, здатних забезпечити

At present and in the future, mathematical methods, mechanisms and algorithms of standardized asymmetric cryptotransformations such as electronic signature (ES) are and will be used for information cryptographic protection. Electronic signature is the main and essential component of cybersecurity, in terms of providing quality information security services such as integrity, irresistibility and authenticity of information and data being processed. However, there are well-founded suspicions that in the post-quantum period the existing ES standards will be broken and compromised using classical and quantum cryptanalytic systems with appropriate mathematical, software and hardware-software. An analysis was performed, which confirms that quantum computers have already been developed, manufactured and used. This work is devoted to the analysis of methods and algorithms for generating key data for Falcon-like algorithms of electronic signature. Some of the basic algorithms for Falcon-shaped algorithms of electronic signature are considered, namely the algorithm of key data generation and algorithm of random polynomials f, g generation, which satisfy the Gauss distribution. The Falcon algorithm itself is the finalist of the post-quantum electronic signature contest due to the satisfactory value of the public key and signature lengths, but the key data generation algorithm uses many methods and difficult to implement. The Falcon authors use this algorithm for polynomials n=512, 1024. To increase the sixth level of cryptostability, this algorithm can be expanded for n=2048. This work is devoted to study the Falcon algorithm, taking into account its expansion for n=512, 1024, 2048 in terms of generating key data. Also, the paper considers the results of justifying the choice of a mathematical apparatus for implementing a software package for generating a key pair of a cryptographic algorithm for an electronic signature in order to create reliable electronic signatures.

The paper considers post-quantum projects of the Falcon and Dilithium electronic signature standards (ES), which are finalists of the NIST USA competition. The mathematical apparatus of algebraic lattices and appropriate methods are used in their construction. In further study and comparison of these post-quantum ES draft standards, both from a theoretical and practical standpoint, it is fundamental to substantiate the requirements for parameters and keys and in general to calculate the main indicators according to the accepted conditional and unconditional criteria. In such studies, it is important to determine the sufficiency of ensuring the guarantee of their security against classical, quantum, special and error-based attacks. This can be ensured, inter alia, through a reasonable choice of the sizes of common parameters and keys, and their practical construction in accordance with the adopted security model. However, when choosing the sizes of common parameters and keys, a significant contradiction arises between the properties of the draft of the Falcon and Dilithium ES standards, So increasing the size of the general parameters and keys leads to an increase in the complexity of transformations, and vice versa. The purpose of this article consists in analysis of problematic issues of choosing the size of parameter and keys for post-quantum ES projects based on mathematical methods of Falcon and Dilithium, and features of their implementation, including implementation according to the adopted security model. Comparative analysis of the stability and complexity of the Falcon and Dilithium ES draft standards depending on the size of the parameters and keys, including for 6 and 7 security levels. Development of proposals for decisions on the adoption of national post-quantum ES standards based on the mathematical methods Falcon and Dilithium. Determining the influence of unconditional, conditional and pragmatic criteria on the advantages when deciding on the ES standardization based on Falcon and Dilithium mathematical methods, including taking into account the availability of patents and the need to obtain licenses, etc.

At present and in the future, mathematical methods, mechanisms and algorithms of standardized asymmetric cryptotransformations such as electronic signature (ES) are and will be used for information cryptographic protection. Electronic signature is the main and essential component of cybersecurity, in terms of providing quality information security services such as integrity, irresistibility and authenticity of information and data processed. However, there are well-founded suspicions that in the post-quantum period the existing ES standards will be broken and compromised using classical and quantum cryptanalytic systems with appropriate mathematical, software and hardware-software. An analysis was performed, which confirms that quantum computers have already been developed, manufactured and used. It is believed that the actual state of development and use of powerful quantum computers and their mathematical and software is obviously strictly confidential and secure, and only publicly known data on quantum computers and their applications in cryptology are disclosed. A preliminary analysis has been carried out showing that in Ukraine there is an understanding of the existence of threats to cybersecurity and information security in the case of using available standardized ES in the transition and post-quantum periods. Currently, development and adoption of post-quantum ES standards is also one of the main issues in ensuring the necessary levels of security in the transition and post-quantum periods. The objective of this article is to substantiate, compare alternatives and develop proposals for the selection and standardization of post-quantum ES standards at the international and national levels, taking into account the results of the 2nd and 3rd rounds of the NIST US competition and national researches.

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 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 asymmetric encryption and keys encapsulation in polynomial rings (algebraic lattices) built algorithms are presented and considered, the essence of used asymmetric encryption transformations and key encapsulation protocols are analyzed. Encryption and encapsulation mechanisms with different sets of parameters that determine stability are considered.

The asymmetric encryption and keys encapsulation in polynomial rings (algebraic lattices) built algorithms are presented and considered, the essence of used asymmetric encryption transformations and key encapsulation protocols are analyzed. Encryption and encapsulation mechanisms with different sets of parameters that determine stability are considered.

The research was carried out and the development of an effective practical algorithm for the construction of system-wide parameters and keys of cryptographic transformations such as asymmetric ciphers for a special form of setting the Product Form polynomial was performed. The experimental confirmation of the built-in system-wide parameters for 5–7 stability levels NTRU PRIME UKRAINE, taking into account a combined attack is given.

The research was carried out and the development of an effective practical algorithm for the construction of system-wide parameters and keys for cryptographic transformations such as asymmetric ciphers and the key encapsulation protocol was performed. The experimental confirmation of the built-in system-wide parameters and keys of cryptographic transformations such as asymmetric cipher and the key encapsulation protocol of 6 – 7 stability levels based on transformations in the ring of polynomials over the finite fields is presented. The types of attacks that are possible with respect to the specified cryptographic transformations are also presented in this work.

The research was carried out and the development of an effective practical algorithm for the construction of system-wide parameters and keys of cryptographic transformations such as asymmetric ciphers for a special form of setting the Product Form polynomial was performed. The experimental confirmation of the built-in system-wide parameters for 5 – 7 stability levels NTRU PRIME UKRAINE, taking into account a combined attack. is 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 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.

Modern cryptographic transformations of the asymmetric encryption type, namely – NTRU-like cryptosystems are considered. A new cryptographic system NTRU PRIME IIT UKRAINE was created based on existing cryptographic transformations of this type. A brief description of this cryptosystem is presented and an analysis of its resistance to known attacks is carried out, conclusions are made and recommendations are given.