This article is devoted to the newly developed methods for obtaining pseudorandom (PRS) and «conditionally» random DNA-based sequences (RS) using the Ukrainian national standard of block symmetric encryption (BSE) DSTU 7624:2014. The paper discusses the representation of DNA sequences in binary form for the purpose of further usage, and estimates the properties of such "raw" sequences and, most importantly, demonstrates the use of DNA sequence fragments to obtain PRS and RS by encrypting them using block symmetric cipher in the CTR mode and performs their statistical analysis.

The work is devoted to the influence of the algebraic structure of q-ary lattices on the complexity of cryptanalysis of cryptographic problems on lattices. The root-mean-square errors for existing lattice reduction models are obtained. It is shown that the deterministic Albrecht-Lee simulator shows the smallest mean square deviations on small lattices. Collapsibility estimates for nesting and decoding attacks on the algebraic structure of q-ary lattices have been found. A new method for selecting attack parameters has been proposed for the decoding attack. It is shown that a decoding attack with such a choice of parameters can be at the expense of nesting attacks. To distribute the power of the secret vector in decoding attacks, which differ from the normal one, the approximation method is used by the normal division, which minimizes the Kolmogorov-Smirnov equation and calculates the optimal values for some distributions. There are no parameters for any divisions. For the SIS problem, a new approach to safety assessment has been proposed, which takes into account the possibility of various Euclidean norms. Based on the new approach, estimates of the SIS problem for Crystals-Dilithium electronic signature schemes have been calculated.

The object of this study is the methods and algorithms for computing, evaluating, and comparing DNA-based pseudorandom sequences (PRSs) and random sequences (RSs). This paper addresses the task of extracting (P)RSs with the required stochastic and statistical properties from a DNA noise source, experimentally validating these properties in compliance with the requirements of current international standards, as well as evaluating and comparing the sequences obtained for different DNA samples. The results are the developed algorithms for generating DNA-based (P)RSs, improved algorithms for their comparison, and proposals for the process of evaluating their properties. For the output of implemented algorithms, more than 96 % of the bit streams of sequences successfully pass all statistical tests, and the entropy per bit for RSs is close to 1. A special feature of the developed generation algorithms is the use of validated conditioning component – block cipher in CTR mode – which explains the possibility of obtaining unique random data with required properties. The peculiarity of the proposed evaluation algorithm is the complexity of the tests and checks used: complete assessment of statistical properties and entropy values. Special features of the improved comparison algorithms are resource saving and the ability to evaluate much larger data sets. This is due to the use of structures and data types that are better in terms of memory usage and flexible cryptographic primitives with different modes. The results could be practically applied to constructs where randomness is required: for computing the keys and system-wide parameters, when performing transformations as part of hash functions, while obtaining sequences of an arbitrary alphabet, in zero-knowledge protocols, etc.

Randomness is an integral component of most special software and hardware CIP tools. Work and development towards improving the process of randomness generation is actively underway and requires special attention. This paper is devoted to the research into algorithms for generating (pseudo)random sequences over an arbitrary alphabet, estimating their complexity (time and capacity), as well as statistical characteristics. This study is important because modern post-quantum algorithms use such sequences to generate keys and random components of digital signatures and key encapsulation. Obtained results can serve as a motivation for choosing an algorithm for generating sequences of an arbitrary alphabet for use in certain cryptographic transformations or algorithms.

Although the mathematical problems used in post-quantum cryptography algorithms appear to be mathematically secure, a class of attacks known as side-channel attacks may prove to be a threat to the security of such algorithms. Side-channel attacks affect the hardware on which the cryptographic algorithm runs, they are not attacks on the algorithm itself. The good news is that side-channel analysis on new post-quantum cryptographic algorithms started early, even before the algorithms were standardized, given that older algorithms still face side-channel problems. Kyber is a lattice-based post-quantum algorithm based on the complexity of the M-LWE problem. Kyber offers a secure public key encryption (PKE) scheme against a chosen plaintext attack (CPA) and a secure key encapsulation mechanism against a chosen ciphertext attack (CCA). This paper provides a study of side-channel and fault-injection attacks on lattice-based schemes, with focus on the Kyber (KEM). Considering the wide range of known attacks, the protection of the algorithm requires the implementation of individual countermeasures. The paper presents and tests a number of countermeasures capable of providing/improving protection against existing SCA/FIA for Kyber KEM. The obtained results show that the presented countermeasures incur a reasonable performance cost. Therefore, the use of special countermeasures in real implementations of lattice-based schemes, either alone or as an augmentation of general countermeasures, is necessary.

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.

Currently, world civilization is taking significant steps in science and practice related to quantum calculations. Significant steps are being taken to achieve the competitive advantage of countries in the field of quantum information science and the practice of introducing quantum technologies. Scientific and practical research is first aimed at reducing the risks related to quantum computers on cybersecurity, economic and national security. Although the full range of quantum computers is still unknown, it is obvious that further technological and scientific leadership of states will at least partially depend on the country's ability to maintain a competitive advantage in quantum computing and quantum information science. However, along with the potential advantages, quantum calculations are likely to cause significant risks about economic and national security. Specific actions are determined that technologically developed states that begin a long-term process of transferring vulnerable computer systems to quantum-resistant cryptography. An important problem in cryptology is to analyze ways of reducing risks for vulnerable cryptographic systems and the state of their development, adoption and implementation at the international and national levels of post-quantum standards of asymmetric cryptotransformations of electronic signatures (ES), asymmetric ciphers (AC) and key encapsulation protocols (KEP). Therefore, the processes of reducing risks for vulnerable existing standardized cryptographic systems and determining the directions of development of mathematical methods and the study of the prospects for their application in the creation of standardized AC, KEP and ES are significantly significant. They are reduced to justification and definition of mathematical methods and mechanisms that will create promising (post-quantum) standardized AC, KEP and ES. The course of analysis, evaluation and results of comparison of existing and post-quantum asymmetric cryptotransformations of AC, KEP and ES, and standardization at the international and national levels, including for transitional and post-quantum periods, are the main objective of this article.

The paper presents the hardware implementation based on FPGA of the main cryptographic transformations of the symmetric transformation algorithm of DSTU 7624:2014 and the stream cipher of DSTU 8845:2019, which are the national encryption standards of Ukraine. In the case of DSTU 7624: 2014 developed and implemented a hardware implementation for multiplication of two polynomials modulo x8+x4+x3+x2+1 in the form of a combinational circuit that allows to execute the MixColumn transformation by one cycle. SubBytes transformation is implemented based on asynchronous read-only memory. For stream cipher, DSTU 8845:2019 the nonlinear function T are implemented as subtitution byte operation in the form of precalculated cells of ROM memory. The multiplication function by α and α-1 in Galois field arithmetic GF (2 64) is realized based on ROM and combinational logic. The control of the modes of operation of the shift register with linear feedback is performed based on a FSM. Both hardware implementations of encryption standards have been verified by the authors according to the specified data in the standard, and their HDL code can be provided by the authors for further research to interested parties.

Modern cryptographic transformations require provable security against a relatively wide class of threats. Typically, such evidentiary security is achieved through formal analysis within the chosen security model. The development of quantum computers led to the emergence of new attack vectors to which classical cryptography was vulnerable. However, there are cryptographic systems that are considered resistant to quantum attacks and some of them are even standardized. The formal analysis of such systems has faced difficulties for a long time, which were associated with the impossibility of applying classical methods of proof to formal models that take into account quantum effects. However, in recent years, many new results have appeared that allow obtaining formal security proofs for quite complex cryptographic transformations, and most of the existing post-quantum asymmetric encryption and key encapsulation schemes currently have corresponding formal proofs within the quantum random oracle model, the most widespread security model for of post-quantum cryptography. DSTU 8961:2019 is the Ukrainian post-quantum standard for asymmetric encryption and key encapsulation. However, security proofs in the quantum random oracle model have not yet been published for it. As part of this work, security evidence was obtained for the design of the key encapsulation mechanism described in DSTU 8961:209. The obtained result is generalized for an arbitrary asymmetric encryption scheme, which may contain decryption errors and can be used to assess the security of not only DSTU 8961:2019, but also other similar asymmetric transformations.

Virtually all asymmetric cryptographic schemes currently in use are threatened by the potential development of powerful quantum computers. Although there is currently no definite answer and it is very unclear when or even if CRQC will ever be built and the gap between modern quantum computers and the envisioned CRQC is huge, the risk of creating CRQC means that currently deployed public key cryptography must be replaced by quantum-resistant ones alternatives. For example, information encrypted using modern public key cryptography can be recorded by cryptanalysts and then attacked if a QRQC can be created. The potential harm that CRQC could cause is the basis of the motivation to seek countermeasures, even though we have uncertainties about when and if these computers can be built. Deployed systems that use public key cryptography can also take years to update. Post-quantum cryptography is one way to combat quantum computer threats. Its security is based on the complexity of mathematical problems that are currently considered unsolvable efficiently – even with the help of quantum computers. Post-quantum cryptography deals with the development and research of asymmetric cryptosystems, which, according to current knowledge, cannot be broken even by powerful quantum computers. These methods are based on mathematical problems for the solution of which neither efficient classical algorithms nor efficient quantum algorithms are known today. Various approaches to the implementation of post-quantum cryptography are used in modern research, including: code-based cryptography, lattice-based cryptography, hashing-based cryptography, isogeny-based cryptography, and multidimensional cryptography. The purpose of this work is to review the computational model of quantum computers; quantum algorithms, which have the greatest impact on modern cryptography; the risk of creating cryptographically relevant quantum computers (CRQC); security of symmetric cryptography and public key cryptography in the presence of CRQC; NIST PQC standardization efforts; transition to quantum-resistant public-key cryptography; relevance, views and current state of development of quantum-resistant cryptography in the European Union. It also highlights the progress of the most important effort in the field: NIST's standardization of post-quantum cryptography.

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 is the development and adoption of 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 have been identified who need 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. proving 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. Studies show that among the ES schemes on lattices it differs slightly from other candidates and has prospects for the adoption as a standard the Falcon algorithm. The main and dominant approach to the design of the Falcon ES mechanism is the use of the Fiat-Shamir transformation with interruptions. For the safe use of the Falcon ES, the sets of system-wide parameters that ensure resistance to all known and potential attacks must be found. In the process of forming the requirements for ES NIST within the competition was interested only in sets of system-wide parameters up to 256 bits of classical security inclusive. However, according to the authors of this work, in the future it is advisable to provide at least 384 and 512 bits of security for classical cryptanalysis and at least 192 and 256 bits of security for quantum cryptanalysis. The article brie

Modern cryptographic applications use cryptographic algorithms with a symmetric key. They provide high conversion rates and resistance to crypto-graphic attacks. To complicate the plaintext – cipher-text ratio, symmetric ciphers usually use nonlinear substitutions (S-boxes). S-boxes cryptographic metrics play a crucial role in ensuring resilience to most known attacks (differential, linear, algebraic, and other cryptanalysis methods). However, generating efficient s-boxes is a challenge. Even for small input/output sizes, there are an extremely large number of possible solutions. Usually, the substitution is represented as a set of Boolean functions. This allows you to apply discrete transformations, for example, Walsh-Hadamard, to evaluate cryptographic indicators. However, methods for generating s-boxes by selecting suitable Boolean functions are extremely complex. Therefore, it is necessary to study new mathematical methods for representing nonlinear substitutions, studying their cryptographic properties, and developing generation algorithms. In this paper, we propose applying factorial number systems to represent nonlinear substitutions. Each substitution can be represented in a unique way through a set of inversions, which, in turn, can be transformed into a factorial number. That is, we can naturally arrange all substitutions by numbering them in the factorial number system. We give examples of such numbering and investigate the cryptographic characteristics of S-boxes with their initial numbers. In particular, we show how the variable functions used in heuristic algorithms for generating non-linear substitutions change. The results obtained can be used to simplify heuristic methods in order to speed up the generation of non-linear substitutions.

It is well known that quantum algorithms offer exponential speedup in solving the integer factorization and discrete logarithm problems that existing public-key systems rely on. Thus, post-quantum cryptography seeks alternative classical algorithms that can withstand quantum cryptanalysis. Growing concern about the quantum threat has prompted the National Institute of Standards and Technology (NIST) to invite and evaluate applications for a post-quantum cryptography standard, an ongoing process scheduled to be completed by 2023. Falcon is an electronic signature algorithm based on the mathematics of algebraic lattices. The disadvantage of this algorithm is the small number of studies of resistance against special attacks, as well as attacks through side channels. This material examines existing attacks on the implementation, and also analyzes the speed with applying countermeasures that would prevent such attacks. Although the Falcon scheme sampler, as well as certain mathematical transformations, are still vulnerable to attacks (which in turn allow the private key to be obtained), the efficiency of the components and mathematics of this signature algorithm make it competitive with other schemes, even with countermeasures against these attacks. The work will also consider the attack by side channels on the Falcon. Such an attack is a known-plaintext attack that uses the device's electromagnetic radiation to derive secret signature keys, which can then be used to forge signatures in arbitrary messages. The obtained results show that Falcon is quite vulnerable to side-channel attacks and does not yet have protection against such attacks in the proposed implementation. Because of this, standardization or implementation should consider the possibility of physical attacks, as well as options for countering such attacks.

Post-quantum cryptography is a field of research that studies cryptographic transformations protected against attacks using quantum computers. For many years, lattice-based cryptography has become one of the most promising solutions to protect against the threat of quantum computing. An important feature of the post-quantum period in cryptography is the significant uncertainty about the source data for cryptanalysis and countermeasures in the capabilities of quantum computers, their mathematical support and software, as well as the application of quantum cryptanalysis to existing cryptocurrencies and cryptoprotocol. The main methods are mathematical methods of electronic signature, which have undergone significant analysis and justification in the process of extensive research by cryptologists and mathematicians at the highest level. The security of signature schemes depends strongly on the standard deviation of the discrete Gaussian distribution, which has a sampling algorithm. In this paper, the most common variants of sampling algorithms were considered and analyzed, because the quality of all algorithms depends significantly on the structure of the lattice for which sampling takes place. A comparison of the quality of lattice sampling algorithms is highlighted. In particular, the paper considers Klein's algorithms (its modification is the Thomas Prest and Dukas algorithm), Peikert's algorithm and the floating-point sampling algorithm. Klein's sampling algorithm, in particular its modification, namely, the Dukas-Prest algorithm, gives the smallest vectors. Theoretically, it is much better than Klein's algorithm on NTRU lattices, but it requires the use of floating-point arithmetic, which complicates greatly the analysis of its security and tocreation of software or hardware implementation.

To ensure the security of information technology, cryptographic information protection tools are used, in particular block and stream encryption algorithms with a symmetric key. Reliability and cryptographic strength of cryptoalgorithms is provided by the properties of the applied primitives. For example, non-linear substitutions (S-boxes) are used as the main component of modern symmetric ciphers. Therefore, generation of substitutions is an important scientific task directly related to the security of information technology and improvement of modern symmetric ciphers. The paper investigates the properties of iterative algorithms for generating non-linear substitutions and special cost functions, which play a decisive role in the heuristic search for S-boxes with the required properties. We consider the cost function of the WCF (Cost Function of the content of the Walsh-Hadamard spectrum) and optimize its parameters. The obtained optimization results in combination with the Hill Climbing iterative search algorithm can reduce significantly the number of iterations. In particular, we show that for a substitution search with a non-linearity of 104, on average, we reduce the computational complexity of generation by more than 20%. In addition, it is possible to increase the success rate of the heuristic search. In particular, for the selected settings, in 100% of cases, a beaktive S-box with a non-linearity of 104 was found.

To date, in all economically developed countries of the world, multibillion-dollar scientific and exploratory research is being carried out on the problems of building quantum-resistant cryptographic systems. In particular, work is being carried out on the analysis, development and research of models, methods and computational algorithms for post-quantum cryptographic transformations, their possible standardization and implementation. At the same time, it is necessary to have an objective, scientifically based approach for assessing the level of information security, which can be guaranteed when implementing one or another protection mechanism. This paper presents formulated scientific approaches, derived expressions making it possible to evaluate the strength of MAC codes as mechanisms for ensuring the authenticity, integrity, and authenticity of messages. It is shown that it is necessary to use the statistics of joint message distributions for accurate calculation of the MAC codes simulation and collision stability. It is proved that the lower limits for the probabilities of imitation and substitution ignore the statistical properties of authenticators arrays. They are based on the pseudo-randomness model of f(x) function and determine the minimum requirements for the key space size and the MAC values space. The upper bounds for the imitation and substitution probabilities are related to combinatorial properties of the MAC arrays and evaluate the value of collisions in the space of MAC values f(x) and messages for the worst case choice of keys and messages. Collision characteristics of MAC codes are considered. The derived equations make it possible to solve accurately the problem of determining the number of experiments indispensable to create a collision with a certain probability on a set of MAC code values. The MAS stability estimates for one of the types of hash functions are obtained using the derived equations.

Ensuring cyber and information security for critical infrastructure is achieved through the implementation of an appropriate set of information security management measures, which can be provided in the form of software policies, methods, procedures, organizational structures and functions. Information security requirements are determined, in particular, by systematic risk assessment of information security, which can be one of the elements of the predicted approach to identifying hazards in the provision of services to service participants in the information interaction of the information system. The paper presents conceptual provisions for assessing and managing cybersecurity risks of the critical infrastructure information system. The proposed concept involves the definition of: areas of security threats to the information system; involved information assets and calculation of their value; assessment of the probability of attacks on the information system; assessment of the probability of success of attacks on the information system and more. Risk assessment methods are proposed that take into account the probability of success of an attack and the probability of an attack occurring, which makes it possible to eliminate the shortcomings inherent in known approaches and provide more accurate identification of attack methods associated with the attacker's behavior. The concept of cybersecurity risk assessment and the methodology for analyzing and assessing security threats that are presented in the work correspond to approaches to building risk-oriented information security management systems and can become the basis for developing an information security system in the information system of a critical infrastructure object.

Recent advances in quantum technology and the potential that practical quantum computers may become a reality in the future have led to renewed interest in developing cryptographic technologies that are secure against conventional and quantum attacks. Currently, virtually all asymmetric cryptographic schemes in use are threatened by the potential development of powerful quantum computers. Post-quantum cryptography is one of main the ways to combat this threat. Its security is based on the complexity of mathematical problems that are currently considered unsolvable efficiently, even with the help of quantum computers. The security of information systems is ensured through protection against various threats that use system vulnerabilities. Security protocols are the building blocks of secure communication. They implement security mechanisms to provide security services. Security protocols are considered abstract when analyzed, but may have additional vulnerabilities in implementation. This work contains a holistic study of security protocols. Basics of security protocols, taxonomy of attacks on security protocols and their implementation are considered, as well as various methods and models of protocol security analysis. In particular, the differences between information-theoretic and computational security, computational and symbolic models are specified. In addition, an overview of the computational security models for Authenticated Key Exchange (AKE) and Password Authentication Key Exchange (PAKE) protocols is provided. The most important security models for the AKE and PAKE protocols were also described. With the emergence of new technologies that may have different security requirements, as well as with increased opportunities for competition, there is always a need to develop new protocols. Thus, the purpose of this article is to review, classify, analyze, and research the vulnerabilities of information systems from classical, quantum, and special attacks, performed taking into account the forecast regarding the possibilities of attacks on post-quantum cryptographic transformations; studying security assessment models for existing cryptographic protocols, as well as reviewing and benchmarking security models and providing suggestions for protection against existing potential attacks.

In recent years, there has been steady progress in the creation of quantum computers. If large-scale quantum computers are implemented, they will threaten the security of many widely used public-key cryptosystems. Key-establishment schemes and digital signatures based on factorization, discrete logarithms, and elliptic curve cryptography will be most affected. Symmetric cryptographic primitives such as block ciphers and hash functions will be broken only slightly. As a result, there has been an intensification of research on finding public-key cryptosystems that would be secure against cryptanalysts with both quantum and classical computers. This area is often called post-quantum cryptography (PQC), or sometimes quantum-resistant cryptography. The goal is to design schemes that can be deployed in existing communication networks and protocols without significant changes. The National Institute of Standards and Technology is in the process of selecting one or more public-key cryptographic algorithms through an open competition. New public-key cryptography standards will define one or more additional digital signatures, public-key encryption, and key-establishment algorithms. It is assumed that these algorithms will be able to protect confidential information well in the near future, including after the advent of quantum computers. After three rounds of evaluation and analysis, NIST has selected the first algorithms that will be standardized as a result of the PQC standardization process. The purpose of this article is to review and analyze the state of NIST's post-quantum cryptography standardization evaluation and selection process. The article summarizes each of the 15 candidate algorithms from the third round and identifies the algorithms selected for standardization, as well as those that will continue to be evaluated in the fourth round of analysis. Although the third round is coming to an end and NIST will begin developing the first PQC standards, standardization efforts in this area will continue for some time. This should not be interpreted as meaning that users should wait to adopt post-quantum algorithms. NIST looks forward to the rapid implementation of these first standardized algorithms and will issue future guidance on the transition. The transition will undoubtedly have many complexities, and there will be challenges for some use cases such as IoT devices or certificate transparency.