Usage of Hyperelliptic Curves in the Digital Signature Protocol

Abstract

During the last years in many countries the standards of a digital signature based on elliptic curves are accepted. Their stability is based on large computing complexity of discrete logarithm problem in group of points of an elliptic curve. For today cryptographic transformations on elliptic curves rather satisfy required level of secrecy. However increasing of computer technology power and development of cryptoanalysis methods in the near future may result in reducing stability of such transformations. In 1989 N. Koblitz offered using in cryptography hyperelliptic curves. In contrast to elliptic curves, points of the hyperelliptic curve will not derivate group. However, the additive Abelian group can be constructed from divisors. Order of such a group considerably exceeds the quantity of the points of a curve, which allows to reach acceptable stability at a smaller size of a base field. Direct and return cryptography transformations in this case are more composite in compare to elliptic curves, however key length is diminished till 50-80 bit depending on the curve kind. It makes hyperelliptic curves specially attractive for hardware application in devices with restricted size of resources. Until recently, the cryptography transformations on hyperelliptic curves were considered so complicated that their speed can not reach practically acceptable level. However analysis of the last works [1,2,3] has shown, that application of modern calculation methods in finite fields allows to improve their speed values in a considerable way. At present time all around the world intensive works on learning stability of transformations on hyperelliptic curves are conducted. Probably, this mathematical basis will be used in new standards of a digital signature and directional encoding. The purpose of the work is to demonstrate the protocol of a digital signature on hyperelliptic curves.