NEW BASE POINT ALGORITHMS FOR EDWARDS ELLIPTIC CURVES

Abstract

Transformations on  elliptic  curves which are  used in the national digital signature standard DSTU 4145 2002, satisfy  modern requirements. However, the fast development of computer technologies and a significant interest in cryptology worldwide have led to an increase in research, the constant emergence of new powerful cryptanalysis methods and, as a consequence, to the possible shortening of the lifetime of existing and new algorithms. This article addresses the current scientific and practical problem of investigating the properties of elliptic curves in the Edwards form over a finite field , p ≠ 2, suitable for use in asymmetric cryptosystem, in particular, digital signature algorithms. Based on the research completed, new ways of determination of a base point on Edwards curves were outlined and described.  Three new algorithms were proposed  for determination of the base point for constructing a cryptosystem on the full and twisted Edwards  curves. In this work the comparative analysis of the performance of the developed algorithms of the Edwards curves base point determination and the performance of crypto-algorithms on the non-perpendicular elliptic curves in the Weierstrass form over the fields of characteristic 2 was carried out. The analysis shows that proposed algorithms are faster than the standard Weirstrass digital signature curve algorithm – respectively, the first algorithm – 180 times, the second–times, and the third algorithm– (times. It is proved that the use of elliptic curves in the form of Edwards over finite  field , instead of Weierstrass curves, can increase the speed of operations of adding  points in asymmetric cryptosystems. The results of the work can be applied to the analysis of existing problems and creation of new algorithms and standards of asymmetric cryptography.