Digital Signature Schemes Based on Lucas Functions Digital Signature Schemes Based on Lucas Functions

Limited distribution notes: This report has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher its distribution outside the University of Technology Chemnitz-Zwickau prior to publication should be limited to peer communications and speciic requests. After outside publication, requests should be lled only by reprints or legally obtained copies of the article. Abstract In 1993 Lennon and Smith proposed to use Lucas functions instead of the exponen-tiation function as a one-way function in cryptographic mechanisms. Recently Smith and Skinner presented an ElGamal signature scheme based on Lucas functions. In this paper we point out the weakness in this approach and present our version of an ElGamal signature scheme based on Lucas functions. Furthermore we outline how to apply the ideas of the Meta-ElGamal signature scheme to Lucas functions. As a result we get various new signature schemes. Unfortunately the new schemes are slightly less eecient than the schemes in nite elds and additionally { in contradiction to a conjecture by Smith and Skinner { the security of the schemes isn't increased: It can be proved that a variant of the signature schemes based on Lucas functions can be universally forged ii a related signature scheme in GF(p) can be universally forged.