Hanzia R. Models, methods and means for generating elliptic curves general system parameters for cryptographic applications

The thesis is devoted to solving an important scientific problem, which consists in the development of models and methods for cryptographic transformations in groups of points of elliptic curves (EC) strong general system parameters (GSP) generating, justification the possibilities of using such parameters in the post-quantum period, generating strong GSP for high and ultrahigh security levels cryptographic transformations in EC groups of points mean implementation for national asymmetric cryptographic transformation standards. The object of the research is generating strong GSP for cryptographic transformations in EC groups of points over a finite binary field process-es. Subject of research is generating strong GSP for using in modern cryptographic applications methods, includes using in the post-quantum period. In the thesis norm of element value from p-adic elements ring with the help of Sylvester matrix determinant which can be used to find the trace of the Frobenius endomorphism when computing the EC order for using in cryptographic transformations calculation technique was developed for the first time. This allows significantly reduce the EC order computation complexity because of the lack of necessity to switch between bases in certain system solutions. Cryptographic transformations in the group of EC points GSP features and conditions of their use in the post-quantum period were grounded for the first time. Such decision allowed changing generating strong GSP mathematical model along the lines of cryptographic transformation based on EC security level increasing in case of quantum computer with sufficient value register for modern sizes asymmetric cryptographic transformations cryptanalysis appearance. Norm computation method with the help of Sylvester matrix determinant for the large order base point EC GSP generating was improved. Proposed method differs from existing by using the features of the internal structure of the matrix for determinant computing. This allows curve order finding and GSP cortege generating forcing. Strong GSP for EC which determined over binary field mathematical model generating in the conditions of quantum cryptanalysis counteraction was improved and adapted to the national standards of electronic signature and asymmetric encryption. Proposed method differs from existing by using optimal (by time indices) computing EC order and further GSP generating methods. This permits to keep on using such cryptographic transformations without changing of existing mathematical base and software even after quantum computer which will be capable of quantum cryptanalysis algorithms performing appearance. The intruder and threat models for modern and prospective cryptographic transformations were improved which allowed to substantiate the require-ments for GSP size in the conditions of the quantum computer, which can be used for cryptanalysis appearance. The obtained practical results are follows. A software tool that is capable of generating parameters of high and ultra-high security levels for polynomial time was developed. It's based on using models and methods that have been developed and improved. Proposals for GSP cryptographic transformations based on the modifications for the national digital signature standard DSTU 4145-2002 currently in force were made. Modifications touch on the issue of additions the existing parameters with the sizes from 431 bits to 1031 bits. This is necessary for cryptographic transformations security level increasing. Because such type of cryptographic transformations are currently used for cryptographic security in Ukraine. Then quantum computer will have to have more than 7,000 qubits for cryptanalysis. Based on conducted analysis as well as computing the number of EC points models and methods research on conditions that user according to the cryptographic applications needs can independently generate GSP in the dissertation methods and means for generating elliptic curves general system parameters for the nationals standards of electronic signatures and asymmetric encryption are proposed The analytical relations which determine the conditions for ensuring the modern cryptosystems security against quantum cryptanalysis were obtained. Also areas of development of post-quantum cryptographic algorithms for information protection after the quantum computers appearance were proposed. Estimates (for example, the use of the modified norm computation method gives a gain of about 15% in the time complexity in comparison with the standard method) and analytical relations for implementing the norm computation method with the help of the Sylvester matrix determinants were obtained. Derived result can be applied in the GSP generating software. The obtained scientific and practical results are implemented at the performance of researches which are related with strong EC GSP generation. The developed proposals were used for creating software and in the education.