An important feature of the post-quantum period in cryptography is the significant uncertainty about the input data for cryptanalysis and counteracting the capabilities of quantum computers, their mathematical and software, and the application of quantum cryptanalysis to existing cryptoprotocols and cryptotransformations. Mathematical electronic signature (ES) methods have been selected as the main methods in the work, which have undergone significant analysis and substantiation in the process of extensive research by cryptologists and mathematicians at the highest level. The article analyzes the existing electronic signature algorithms based on the lattices of stage 2 of the NIST competition. The possibility of using the post-quantum electronic signature mechanism based on algebraic lattices as the post-quantum national electronic signature standard is considered. It is proposed to use the post-quantum Сrystals-Dilithium algorithm as such electronic signature algorithm. The article considers this algorithm and substantiates the possibility of its application. The algorithm parameters and rules for their construction are considered. The differences and features of safe implementation of the algorithm in comparison with stage 1 are analyzed. The analysis is conducted and it is concluded that the Crystals-Dilithium algorithm can be taken as one of the candidates for the development of a national electronic signature standard using cryptographic algorithms, standardized in Ukraine, such as the hashing function described in DSTU 7564:2014. According to the authors of the article, the post-quantum period national standard of Ukraine should include at least 3 algorithms based on different types of mathematical transformations, which are recognized by the world cryptographic community as those that can provide the necessary level of stability in the conditions of quantum cryptanalysis.
An important feature of the post-quantum period in cryptography is the significant uncertainty about the input data for cryptanalysis and counteracting the capabilities of quantum computers, their mathematical and software, and the application of quantum cryptanalysis to existing cryptoprotocols and cryptotransformations. Mathematical electronic signature (ES) methods have been selected as the main methods in the work, which have undergone significant analysis and substantiation in the process of extensive research by cryptologists and mathematicians at the highest level. The article analyzes the existing electronic signature algorithms based on the lattices of stage 2 of the NIST competition. The possibility of using the post-quantum electronic signature mechanism based on algebraic lattices as the post-quantum national electronic signature standard is considered. It is proposed to use the post-quantum Сrystals-Dilithium algorithm as such electronic signature algorithm. The article considers this algorithm and substantiates the possibility of its application. The algorithm parameters and rules for their construction are considered. The differences and features of safe implementation of the algorithm in comparison with stage 1 are analyzed. The analysis is conducted and it is concluded that the Crystals-Dilithium algorithm can be taken as one of the candidates for the development of a national electronic signature standard using cryptographic algorithms, standardized in Ukraine, such as the hashing function described in DSTU 7564:2014. According to the authors of the article, the post-quantum period national standard of Ukraine should include at least 3 algorithms based on different types of mathematical transformations, which are recognized by the world cryptographic community as those that can provide the necessary level of stability in the conditions of quantum cryptanalysis.
An important feature of the transition and post-quantum period is the application of new mathematical methods to counteract quantum cryptanalysis. The world cryptographic community pays special attention to the open competition for the post-quantum standard of electronic signature. The problem is to prove the stability of new mathematical methods for the synthesis of transformations such as electronic signature, in particular with the use of algebraic lattices. The existing algorithms of electronic signature of the 2nd stage of the NIST competition are analyzed. Among the selected candidates for the EP 2 standard, 3 of the 3 algorithms are based on algebraic lattices, CRYSTALS-DILITHIUM and FALCON. The NIST has issued a statement saying that it is most likely that one of the algorithms will be chosen due to the same mathematical basis used in both algorithms. The main attacks on electronic signature algorithms based on the problem of learning with errors, as well as the parameters of the EP Dilithium algorithm, which affect the stability and complexity of transformations, are considered. Methods for generating system-wide parameters of stability levels of 512 bits of classical and 256 bits of quantum security, as well as the protection of the algorithm againstfrom attacks by third-party channels are considered. The dependence of the time of electronic signature production on the keys is analyzed. The results of calculations for the level of stability 512/256 are given, and also recommendations on the choice of system-wide parameters are given. The results of the 2nd stage of the NIST competition of post quantum cryptographic algorithms, as well as the prospects of standardization of transformations such as electronic signature at the 3rd stage are considered. Conclusions are made about the need for a more detailed study of attacks on algorithms based on the problem of learning with errors, as well as the importance of generating system-wide parameters of higher levels.
An important feature of the post-quantum period in cryptography is the significant uncertainty regarding the source data for cryptanalysis and counteraction in terms of the capabilities of quantum computers, their mathematical and software, as well as the application of quantum cryptanalysis to existing cryptotransformations and cryptoprotocols. Mathematical methods of digital signature (DS) have been chosen as the main methods of NIST USA, which have undergone significant analysis and substantiation in the process of extensive research by cryptographers and mathematicians at the highest level. They are described in detail and studied at the first stage of the US NIST International Competition. In the second round, a number of decisions were made to merge some candidates for the post-quantum DS standard. 9 candidates were left for further research at the 2nd round: Crystals-Dilithium, Falcon, GeMSS, LUOV, MQDSS, Picnic, qTESLA, Rainbow and SPHINCS+. Three of them (Dilithium, Falcon, qTeSLA) are based on the stability of algebraic lattices (Lattice-based), four (GeMSS, LUOV, MQDSS, Rainbow) are based on multivariate transformations (MQ-transformations), one (SPHINCS+) is based on the stability of hash-function, one (Picnic) is based on the stability of the hash-function and block stream ciphers. During the 2nd round of the US NIST Competition the following finalist algorithms and alternative algorithms were selected as digital signatures according to the results of research on promising post-quantum cryptographic algorithms. As finalists algorithms such DS algorithms as Crystals-Dilithium, Falcon and Rainbow. Alternative algorithms are GeMSS, Picnic and SPHINCS+ were selected. This paper studies the peculiarities of construction of the digital signature algorithm considered as a candidate for the promising post-quantum standard of the NIST PQC competition – Picnic, also it analyzes the protection of the algorithm from known attacks. Data from the comparison of post-quantum algorithms such as digital signature are given. The description of the Picnic algorithm and its parameters are given.
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 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.
The article analyzes the current state and prospects of the standardization of post-quantum algorithms for asymmetric cryptographic transformations at the national and international levels. The main reasons for the transition to post-quantum cryptography are discussed, particularly the potential threats posed by quantum computers to existing cryptographic algorithms (RSA, ECC, etc.). An overview of the NIST competition for selecting standards for post-quantum key encapsulation mechanisms (KEM) and electronic signatures is provided, with detailed descriptions of the algorithms chosen for standardization (CRYSTALS-Kyber, CRYSTALS-Dilithium, FALCON, SPHINCS+, and HQC). Additionally, the article examines the role of ISO/IEC, ETSI, and IETF in developing international standards and recommendations. Special attention is given to Ukrainian national standards and initiatives, particularly the algorithms “Skelya” and “Vershyna,” their features, and their prospects for integration into international standards. The article emphasizes the importance and complexity of transitioning to quantum-resistant algorithms and outlines further steps for the successful implementation of post-quantum cryptography. Keywords: post-quantum cryptography, key encapsulation mechanisms, electronic signatures, standardization.