Cryptographic Transformations Research Articles

ДОСЛІДЖЕННЯ АЛГОРИТМІВ ПОСТКВАНТОВОГО ЦИФРОВОГО ПІДПИСУ

The continuous development of quantum computing requires the development of cryptographic algorithms resistant to quantum cryptanalysis. Therefore, the National Institute of Standards and Technology (NIST) is holding a competition to standardize algorithms in the post-quantum period, including Crystals-Dilithium, Falcon, and Sphincs+ for digital signatures. In order to determine the ways of development and ways to improve the existing methods of cryptographic protection and electronic digital signature (EDS), an urgent task arose to assess the effectiveness (stability and computational complexity) of existing EDS algorithms. In this paper, a comparative evaluation of EDS algorithms was carried out by key indicators. Research results show the disadvantages and advantages of each algorithm, taking into account the importance of context-specific use. The conclusions indicate that Dilithium has the lowest load on the system's computing resources, Falcon stands out for its signature verification speed, and Sphincs+ provides reliable security. The capabilities of existing cryptographic transformation algorithms based on elliptic curves do not meet the growing need of authentication algorithms resistance to quantum cryptanalysis, and the use of modern candidates for the post-quantum cryptography standard does not allow for a short-term transition to the usage of new cryptographic algorithms in information systems. The paper also points out the impossibility of simultaneous transition to potentially quantum cryptanalysis-resistant algorithms in existing information systems. The direction of further research is the development of new cryptoalgorithms and methods that will ensure the integrity and confidentiality of information during its transmission over unprotected channels in the transition period to the widespread use and implementation of post-quantum cryptography standards.

A novel deep learning technique with cryptographic transformation for enhancing data security in cloud environments

A novel deep learning technique with cryptographic transformation for enhancing data security in cloud environments

Методические рекомендации по реализации алгоритма шифрования при помощи кодового слова на Python

The article offers theoretical and practical material on encryption and decryption of text messages using a codeword cipher. Is shown an example of the implementation of the algorithm for cryptographic transformations of the specified cipher in the Python programming language. The development and execution of the Python program code was carried out in a free interactive cloud environment — Google Colab.

Technology of protection of authentication data of computer network users

The article is devoted to the problem of perfect protection of authentication data of users of computer systems, especially in the case of a large number of different rights and powers that are personally granted to users. The importance of this problem increases especially when the number of such users in the system is hundreds or thousands. At the same time, each of them must create conditions for access only to their data and ensure protection against any outside interference, both by other users and by staff. A typical example of such conditions are secret electronic voting systems. The software and technical solutions described in this work have passed many years of testing and continue to be practically used in the electronic voting system of the Kyiv National University of Civil Engineering and Architecture. This system is regularly used for conducting elections to student self-government bodies and for surveys among students regarding the quality of teaching subjects. With the help of this system, elections of leaders of the Red Cross Society of Ukraine were held during the pandemic, which was associated with the Covid-19 virus. The mathematical basis of the described authentication data protection technology is the theory of algebraic groups, namely, the problem of discrete logarithmization over large Galois fields. Thanks to the use of cryptographic transformations over these fields, instead of known hash functions, it was possible to get rid of the possibility of collisions and neutralize the disclosure of passwords by attackers who were able to do it with the help of specialized Internet resources. It was also possible to increase resistance to brute-force password cracking, since conversions to Galois fields take ten times more time than hash function calculations.

Подходы к оценке защищённости информационных систем с криптографическим преобразованием объектов

The file system is one of the components of an information system, especially vulnerable to attacks by intruders. Consequently, it is impossible to increase the security level of an information system without ensuring a sufficient level of protection for file system objects, attacks on which affect, among other things, the interests of users of information systems. In order to make decisions on the use of certain mechanisms for protecting file system objects, a regular assessment of their current level of security is necessary. The purpose of the study. Building a model for assessing the security of an information system with cryptographic transformation of file system objects in the context of an attacker using a wide range of attacks on the file system. Materials and methods. The analysis of scientific publications on the problem of assessing the security of information systems allowed us to propose a methodology for assessing the security of information system assets. The methodology is based on a combination of a probabilistic approach to the analysis of possible scenarios for the implementation of threats in information systems for file system objects, expert assessments of basic attack events and representations in the form of attack trees. The threat implementation representation in the form of attack trees is used to simulate the actions of an attacker. Results. A structural model of an information system with cryptographic transformation of objects has been developed. Its valuable assets have been allocated. The list of the main threats to information security relevant for such systems is defined. Possible variants of the ways of implementing attacks and possible scenarios of the development of events in the processes of transferring file system objects with cryptographic transformation are modeled. A set of measures for the protection of information system objects with cryptographic transformation of objects is proposed. The calculation of the probabilities of attacks on an asset along the path determined by the attack tree, the possibility of implementing an attack, the degree of effectiveness of the proposed countermeasures is proposed. Conclusion. The presented methodology for assessing the security of an information system with cryptographic transformation of objects makes it possible to use well-known approaches in a comprehensive manner, and will be useful to developers and researchers in practical and scientific activities to ensure information security.

Analysis of cryptographic authentication and manipulation detection methods for big data

The emergence of Big Data has made it possible to obtain more information about the customer and competitive base, as well as about market trends. Therefore, the desire of criminals to gain unauthorized access to this data also increases in direct proportion. However, not enough attention is paid to security issues of Big Data systems and their creation does not take into account the information security component. The purpose of the article is to develop approaches to the cyber protection of Big Data, which are stored and transmitted by telecommunication communication channels when responding to a request, namely their integrity and authentication. The main problems when working with Big Data are analyzed. Based on the results of the analysis of Big Data protection problems, a cryptographic transformation of data indexes when they are stored in the database and when transmitted in the form of hash codes is proposed to increase the efficiency of search by user requests. The expediency of using crypto-code constructions on elliptic curves of various types of modifications to form a pseudo-random substrate in order to increase the crypto-resistance of transmitted messages is proven. When forming hash codes, it is advisable to use modified elliptical codes, and under stricter conditions, the level of cryptographic resistance of authentication codes can be increased due to hybrid crypto-code constructions. A pseudo-random substrate can be represented by varieties that must equally ensure the necessary transformations and preservation of universality properties by the UMAC algorithm.

МЕТОД АДАПТИВНОГО БАГАТОШАРОВОГО ЗАХИСТУ ІНФОРМАЦІЇ НА ОСНОВІ СТЕГАНОГРАФІЇ ТА КРИПТОГРАФІЇ

An analysis of known solutions to the problem of steganographic and cryptographic methods combining to achieve multilayer information protection is presented. Results of the analysis allowed us to identify trends and prospects for the development of these kink of methods. Based on the results of the analysis, the research task was determined concerning the adaptation of the used cryptographic and steganographic methods in order to achieve the best level of protection, which is needed at critical systems. Metrics were defined, those allowed to choose the best combination of cryptographic and steganographic protection methods parameters. A method of multilayered information protection is presented, which combines cryptographic and steganographic approaches to ensure an increased level of information’s confidentiality and integrity protection. It is proposed to introduce criteria for the selection of cryptographic transformations in such a way that their combination together with steganographic transformations gives the best impact. An instance of the proposed method implementation is given in order to prove the concept. An algorithm has been developed that implements the proposed method of adaptive multilayer information protection. The perspective of further research is determined.

Analysis of DSTU 8961:2019 in the quantum random oracle model

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.

Ukrainian National Encryption Standards for FPGA Based Embedded Systems

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.

ANALYSIS OF QUANTUM SECURE DIRECT COMMUNICATION PROTOCOLS

The development of modern computer technologies endangers the confidentiality of information, which is usually ensured by traditional cryptographic means. This circumstance forces us to look for new methods of protection. In view of modern trends, quantum cryptography methods can become such alternatives, which allow solving a number of important cryptographic problems, for which the impossibility of solving using only classical (that is, non-quantum) communication has been proven. Quantum cryptography is a branch of quantum informatics that studies methods of protecting information by using quantum carriers. The possibility of such protection is ensured by the fundamental laws of quantum mechanics. One of the promising directions of quantum cryptography is Quantum Secure Direct Communication (QSDC) that offers secure communication without any shared key. A characteristic feature of this method is the absence of cryptographic transformations, accordingly, there is no key distribution problem. The purpose of this work is a general overview of quantum cryptography protocols, finding their weak points for further development and improvement, as well as identifying vulnerabilities to different attacks. The article analyzes new methods and protocols, as well as presents their advantages and disadvantages. Based on partial generalizations of theoretical provisions and practical achievements in the field of quantum cryptography, a generalized classification was developed. By comparing various factors of the protocols, and their resistance to certain cyberattacks, we have the opportunity to identify several problems in this field and expand the possibilities for choosing appropriate methods for building modern quantum information protection systems. In accordance with this, conclusions were presented regarding the use of protocols and increasing the level of their effectiveness.

Краткие неинтерактивные аргументы с нулевым разглашением на основе наборов полиномов

The paper discusses the basic principles of construction and the main types of zeroknowledge succinct non-interactive argument of knowledge (zk-SNARK) which is used in the model of a three-way insecure computing environment and based on sets of polynomials. A number of zk-SNARK cryptographic protocols with different algorithms for generating public parameters (Trusted Setup) are given, constructing succinct proofs of reliability calculations (Prover) and public/designated verification of proofs (Verifier). The cases of satisfying the feasibility of discrete functions (arithmetic/ Boolean circuits) using different polynomial sets are presented in quadratic arithmetic programs (QAP), square arithmetic programs (SAP), quadratic span programs (QSP), square span programs (SSP), quadratic polynomial programs (QPP), etc., also the use of authenticated data are described. The cryptographic transformations needed to build zk-SNARKs based on symmetric and asymmetric hash functions, exponential knowledge problems, digital signatures, homomorphic encryption, bilinear pairings based on elliptic curves, etc. are presented. Examples of multilateral verifiable calculations based on zk-SNARK are given.

Side-channel leak detection methods for cryptographic information security tools

The analysis of the parameters of hardware implementations of cryptographic information security tools is one of the promising areas for searching for vulnerabilities in information security tools. Cryptographic transformations underlying the applied software or hardware-software tools of information protection are either mathematically or computationally robustness. To guarantee the computational robustness of a cryptographic algorithm, it is fundamentally important to assess the possibility of including additional information about the ongoing cryptographic transformations to the attack model. Sources of such information about the operation of a cryptographic tool can be: the time of performing various cryptographic operations, the power consumption of devices during the encryption of various data blocks, electromagnetic, vibroacoustic and other emitting that occurs during the operation of hardware information security devices, specially created computational errors (faults). The article observes the approaches used to detect leaks through side-channels based on the analysis of the power consumption of cryptographic information protection tools during performing data encryption. The use of a hardware device emulator to search for statistical dependencies between the device’s power consumption and the encrypted data set that occurs in hardware implementations of cryptographic systems is described.

DEVELOPMENTOFMETHODSFORTESTINGTHELIBRARYOFCRYPTOGRAPHICTRANSFORMATIONSONTHEEXAMPLEOFTHEMST3CRYPTOSYSTEMBASEDONGENERALIZEDSUZUKI2-GROUPS

The article proposes a methodology for testing a library of cryptographic transformations with the implementation of an improved encryption scheme on generalized Suzuki 2-groups in the MST3 cryptosystem. The need to improve existing methods of cryptosystem creation is driven by progress in quantum computer development, which possess sufficient computational power to compromise many existing public key cryptosystems. This is especially true for systems based on factorization and discrete logarithm, such as RSA and ECC. Over the last nearly 20 years, there have been proposals for using non-commutative groups to develop quantum-resistant cryptosystems. The unsolved word problem, formulated by Wagner and Magyarik, uses permutation groups and is a promising direction in cryptosystem development. Magliveras proposed logarithmic signatures, a special type of factorization applied to finite groups, and the latest version of this technology is known as MST3, based on the Suzuki group. The first implementation of the cryptosystem on the generalized Suzuki 2-group had limitations in encryption and protection against brute force attacks. Over the past years, many proposals have been made to improve the basic design. The research conducted by the authors expanded the possibilities of using public cryptography by refining parameters based on non-Abelian groups. The article demonstrates the methodology for conducting tests of the practical implementation of the library of cryptographic transformations with the implementation of an improved encryption scheme on Suzuki 2-groups, confirming its functionality.

Grounds and Legal Mechanism of Digital Transformation of the Issuance of Official Documents Using NFT in Russia

The digital transformation of modern Russian society requires the introduction of new network interaction tools, the predominant among which are blockchain systems. Cryptographic transformation of information allows some blockchain systems to generate non-interchangeable tokens (NFT). Taking into account this possibility, cryptographically protected NFT tokens allows to create unique digital images of various objects that cannot be copied or shared. These technological grounds allow us to put forward the idea of translating the usual documents on physical media (paper passport, certificates) into the NFT form. Taking into account the goals of digital transformation, the principles of building blockchain systems, as well as the essential features of NFT, the authors put forward the idea that NFT can become a reliable way to store and protect official documents. At the same time, Russian law currently does not regulate the implementation of NFT documents. Therefore, the authors propose to consolidate the legal concept of an NFT document in the provisions defining the status of individual official documents and the procedure for their issuance. By an official NFT document certifying legally significant facts, the authors propose to understand such a document, which is created and issued by public authorities or authorized organizations, and represents a unique set of metadata transformed cryptographically. For the effective implementation of the mechanism for issuing NFT documents and its timely improvement, the authors propose to conduct a law-making experiment to introduce into circulation the procedure for issuing individual official documents in the form of NFT, and further scale the results of the experiment taking into account the analysis of its results.

Use of implementations of arbitrary bitness permutations for cryptographic transformations

Background. Cryptographic transformations have always aroused the interest of the educated part of humanity and are an integral part of modern communications. A lot of different cryptographic algorithms exist for different tasks and requirements. Permutation functions are useful for cases where transformation speed is more critical than theoretical secrecy. Hardware implementation of such substitutions is quite simple.
 Objective. Investigate the model and combinational circuits for hardware implementation. Investigate algorithms for permutation functions software implementation. Investigate attack algorithms and cracking of permutation functions for cryptanalysis.
 Methods. The paper reviews algorithms of cryptographic transformations and their cryptanalysis for bijective permutations implemented by means of regular combinational structures of linear complexity. The proposed algorithms provide the rate of processing up to gigabits per second. The paper clarifies the algorithm of formation of elements of regular structures of permutations, specifies volumes of public and private data, reviews data formats, methods of their transfer and hardware implementation of one of the methods. The paper reviews attack types and permutation regular structure schemes cracking algorithms with experimental calculation of necessary operations quantity. The software implementation of the proposed algorithms for results calculation was developed.
 Results. Numerical results of the number of keys, the amount of memory required for hardware implementation and the number of required operations for cryptanalysis were obtained.
 Conclusions. The results show that the proposed algorithms for cryptographic transformations have a sufficient level of protection with a high-speed encryption and decryption.

Classification and analysis of vulnerabilities of modern information systems from classical and quantum attacks

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.

CRYPTO-RESISTANT METHODS AND RANDOM NUMBER GENERATORS IN INTERNET OF THINGS (IOT) DEVICES

Subject of research: crypto-resistant methods and tools of generating random sequences and hardware support of cryptographic transformations in IoT devices. The aim of the article is to study crypto-resistant methods and tools for generating and testing random sequences suitable for use in IoT devices with limited resources; determination of circuit implementations of random sequences hardware generators; formation of conclusions on the use of random number generators (RNG) in cryptographic protection systems of the IoT network. The article solves the following tasks: analysis of methods and hardware for generating random sequences to protect IoT solutions with limited resources; identification of safe and effective technologies for the implementation of RNG; classification of RNG attacks; analysis of the shortcomings of the practical use of statistical test packages to assess the quality of random sequences of RNG; evaluation of the speed of cryptoaccelerators of hardware support for cryptographic transformations; providing practical guidance on RNG for use in resource-constrained IoT devices. Research methods: method of structural and functional analysis of RNG and IoT devices, cryptographic methods of information protection, methods of random sequence generation, method of stability analysis of systems, methods of construction of autonomous Boolean networks and Boolean chaos analysis, methods of quality assessment of random sequences. Results of work: the analysis of technologies and circuit decisions of hardware RNG on characteristics: quality of numbers’ randomness and unpredictability of sequences, speed, power consumption, miniaturization, possibility of integral execution; providing practical recommendations for the use of RNG in cryptographic protection systems of the IoT network. The novelty of the study is the analysis of methods and hardware to support technologies for generating random sequences in the system of cryptographic protection of IoT solutions; classification of attacks on RNG and features of protection against them; identification of effective RNG technologies and circuit solutions for use in low-power IoT devices with limited computing resources; providing practical recommendations for the use of RNG in cryptographic protection systems of the IoT network. The analysis of technologies and circuit solutions allowed to draw the following conclusions: protection of IoT solutions includes: security of IoT network nodes and their connection to the cloud using secure protocols, ensuring confidentiality, authenticity and integrity of IoT data by cryptographic methods, attack analysis and network cryptographic stability; the initial basis for the protection of IoT solutions is the true randomness of the formed RNG sequences and used in algorithms for cryptographic transformation of information to protect it; feature of IoT devices is their heterogeneity and geographical distribution, limited computing resources and power supply, small size; The most effective (reduce power consumption and increase the generation rate) for use in IoT devices are RNG exclusively on a digital basis, which implements a three-stage process: the initial digital circuit, normalizer and random number flow generator; Autonomous Boolean networks (ABN) allow to create RNG with unique characteristics: the received numbers are really random, high speed – the number can be received in one measure, the minimum power consumption, miniature, high (up to 3 GHz) throughput of Boolean chaos; a promising area of ABN development is the use of optical logic valves for the construction of optical ABN with a bandwidth of up to 14 GHz; the classification of known classes of RNG attacks includes: direct cryptanalytic attacks, attacks based on input data, attacks based on the disclosure of the internal state of RNG, correlation attacks and special attacks; statistical test packages to evaluate RNG sequences have some limitations or shortcomings and do not replace cryptanalysis; Comparison of cryptoaccelerators with cryptographic transformation software shows their significant advantages: for AES block encryption algorithm, speeds increase by 10-20 times in 8/16-bit cryptoaccelerators and 150 times in 32-bit, growth hashing of SHA-256 in 32-bit cryptoaccelerators more than 100 times, and for the NMAS algorithm - up to 500 times.

Study of a new cost function for generating random substitutions of symmetric ciphers

Cryptographic transformations with a secret key play an essential role in providing information and cyber security. Block and stream symmetric ciphers are used in various applications both as a separate cryptographic protection mechanism and as part of other applications (pseudo-random sequence generators, hashing algorithms, electronic signature protocols, etc.). Therefore, the design and study of individual components of symmetric ciphers is a relevant and important scientific task. In this paper we consider and investigates iterative algorithms for generating non-linear substitutions (substitutions, S-boxes), which are used in modern block and stream encryption algorithms with a symmetric key. Cryptographic resistance of symmetric ciphers to statistical, differential, linear and other methods of cryptanalysis is provided by the properties of substitutions. In addition, S-boxes must be random from the point of view of the possibility to use algebraic cryptanalysis. Therefore, the task of quickly generating random S-boxes with the desired cryptographic properties is an urgent, but extremely difficult task. For example, the best known generation algorithm requires more than 65 thousand iterations to find a random bijective 8-bit substitution with a non-linearity of 104. In this paper, we study an iterative algorithm for generating substitutions for hill climbing with different cost functions and propose a new cost function, the use of which can significantly reduce the number of search iterations. In particular, the search for a bijective S-box with nonlinearity 104 requires less than 50 thousand iterations.

Analysis of partial key recovery attack on multivariate cryptographic transformations using rank systems

The Rainbow signature scheme, proposed by Ding and Schmidt in 2005, is one of the oldest and most studied signature schemes in multidimensional cryptography. The Rainbow, based on the unbalanced Oil and Vinegar signature scheme, has the necessary cryptocurrency since 1999 with the right parameters. Interest in multivariate cryptography has increased in the last decade, as it is considered to be quantum-stable.
 Cryptanalysis of the Rainbow and its predecessors was actively developed in the early 2000s. Attacks from this era include the MinRank attack, the HighRank attack, the Bill-Gilbert attack, the UOV agreement attack, and the Rainbow bandwidth attack. After 2008, cryptanalysis seemed to have stopped, until the Rainbow's participation in the NIST PQC project, which motivated the continuation of cryptanalysis. During the second round of NIST, Bardett and others proposed a new algorithm for solving the MinRank problem. This dramatically increased the effectiveness of MinRank's attack, although not enough to threaten the parameters provided to NIST. A less memory-intensive version of this algorithm was suggested by Baena et al. Perlner and Smith-Tone analyzed the Rainbow bandwidth attack in depth, which showed that the attack was more effective than previously thought. This prompted the Rainbow team to increase slightly the parameters for the third round. During the third round, Bellens introduced a new attack that reduced the Rainbow's security by 220 times for SL 1. The Rainbow team claimed that despite the new attacks, the Rainbow's parameters still met NIST requirement.
 The purpose of this article is to present two new (partial) key recovery attacks on multivariate cryptographic transformations using rank systems.

Comparison of the quality of sampling algorithms from discrete normal distribution on NTRU lattices

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.