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.

The paper deals with the concept of homomorphic encryption and the possibility of its use in the mechanism of electronic voting. One of the problematic requirements for electronic voting systems is voter anonymity. On the one hand, each voter must be identified, and on the other, the content of his or her vote must be unknown. Currently, the methods and mechanisms used in real voting systems do not provide real anonymity. Therefore, both theoretical and practical content is an urgent and necessary problem of developing mechanisms for anonymous counting of votes with the protection of their distortion. The paper also provides a general analysis of the security level of prospective homomorphic encryption schemes. The essence of homomorphic encryption is that there is some set of operations whose result of executing over ciphertexts (with subsequent decryption) coincides with similar actions over plaintexts. Homomorphic encryption allows you to perform some calculations on information without having access to the information itself. However, there are a number of problems when trying to apply such calculations. The main ones are the choice of the method of asymmetric encryption, which provides the necessary cryptographic stability from both classical and quantum attacks, the identification of possible candidates for asymmetric cryptotransformations in homomorphic encryption, their evaluation of comparison with each other, and, of course, the choice of the most rational for a given multiple restrictions. The asymmetric schemes of homomorphic encryption are compared using the hierarchy analysis process. The method of asymmetric encryption with zero knowledge is substantiated. The objective of this article is to substantiate the possibilities, conditions, and constraints on the use of standardized asymmetric cryptotransformations in the creation of modern homomorphic encryption-type transformations, when anonymity of electronic voting and practical implementation of anonymous voting based on proof of zero knowledge must be guaranteed.

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.