Cryptosystems based on isomorphic transformations of elliptic curve points.

Abstract

The article presents research in the field of development and improvement of cryptographic systems based on elliptic curves isomorphic transformations potentially resistant to quantum cryptanalysis. Analysis results of existing asymmetric cryptosystems disadvantages and advantages, including those based on isomorphic transformations, are presented. The approaches to the construction of cryptographic algorithms based on isogenies of elliptic curves, which can become the basis for creating cryptosystems resistant to quantum attacks, are investigated. In the course of the research, program functions were developed to implement operations on elliptic curves isogenies of different orders, which will ensure the security levels specified in the standard: 256, 384, 512. A software implementation of the operations of a curve point scalar multiplication and operations on elliptic curve isogenies has been developed, on the basis of which experimental values of the time to perform the scalar product using parallelization have been obtained. Experiments have been conducted to compare the classical multiplication of a curve point with the representation of the scalar k as a sequence of 4-bit words, which made it possible to speed up the scalar multiplication operation by 30 times, for 8-bit words the speedup was 18.8 times. The direction of further research is the development of methods for the generation and verification of a digital signature, based on transformations over the isogeny points of the elliptic curve using the parallelization of operations of scalar multiplication of the curve point.