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 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.
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.
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 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.
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. brining 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 the Falcon algorithm standard. The main and dominant approach to the design of the Falcon ES mechanism is the use of the Fiat-Shamir transformation with interruptions. The sets of system-wide parameters that ensure resistance to all known and potential attacks should be found for the safe use of the Falcon ES. In the process of forming the requirements for ES within the competition, the NIST 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 briefly considers the essence of the Falcon electronic signature (ES) algorithm. An analysis of possible attacks on the algorithm and the mechanisms of their implementation is also performed. The process of generating system-wide parameters for 256, 384, 512 stability bits is considered. Conclusions and recommendations are given. The objective of the work is the classification and initial analysis of known attacks on the ES Falcon cryptosystem, setting limits and developing practical algorithms for calculating (generating) system-wide parameters to provide not less than 256, 384 and 512 security bits for classical and not less than 128, 192 and 256 security bits for quantum cryptanalysis.
Background. The work examines the development and promotion of post-quantum standards, as well as the analysis and improvement of existing algorithms, on the basis of which the functioning of standards in the field of electronic digital signature can be based. One of the main reasons was also the migration of classical cryptography to post-quantum cryptography. The robot is aligning three popular post-quantum standards: CRYSTALS-Dilithium, Falcon and SPHINCS+. As a result of the investigation, the most optimal standards were selected before implementation of classical electronic signature schemes. The article is dedicated to the development and promotion of post-quantum standards in the field of electronic digital signature. It is also necessary to analyze and level up existing algorithms, on the basis of which the functioning of such standards can be based. The research is considered relevant due to the growing interest in quantum technologies and the need for secure electronic communications in the upcoming quantum world. Мethods. Мigrating classical cryptography to post-quantum cryptography. However, power is important, because the power of quantum computers can be affected by certain cryptographic algorithms. An analysis of the feasibility of transitioning to new standards and their existing security capabilities is also carried out. In addition, three popular post-quantum standards are being updated: CRYSTALS-Dilithium, Falcon and SPHINCS+. This allows us to determine the most optimal and reliable standard for implementation of classical electronic signature schemes. The authors of the work carry out the selection of the optimal standard, ensuring the safety and security of its authorities. Results. Сontains important results from the study of post-quantum standards for electronic digital signatures, which may be useful for developers of cryptographic software and information security engineers. The power supply to the influences and changes in the advanced post-quantum standards, such as the complexity of implementation, the flexibility of infrastructure and power standardization, are thoroughly examined. The prospects for the future development of post quantum cryptography and its influx into modern electronic signature systems have also been highlighted. This will help readers to take away the more common sense and context of the importance and potential contributions in this area. Conclusions. Falcon and CRYSTALS-Dilithium have a high signing fluidity and a larger key size, making them practical for rich stagnation. SPHINCS+, regardless of its viscosity, has less fluidity and requires a larger key size. The choice between CRYSTALS-Dilithium, Falcon and SPHINCS+ will depend on the specific drying needs, as well as compromises between fluidity, key size and viscosity. This demonstrates the high speed of subscription, and this is one of its key advantages. It is intended for use in quick operations, for example, on servers and industrial systems.
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