Eecient Algorithms for Elliptic Curve Cryptosystems

Elliptic curves are the basis for a relative new class of public-key schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow eecient implementations of elliptic curve crypto systems. This thesis deals with such algorithms. EEcient algorithms for elliptic curves can be classiied into low-level algorithms , which deal with arithmetic in the underlying nite eld and high-level algorithms , which operate with the group operation. This thesis describes three new algorithms for eecient implementations of elliptic curve cryptosystems. The rst algorithm describes the application of the Karatsuba-Ofman Algorithm to multiplication in composite elds GF ((2 n) m). The second algorithm deals with eecient inversion in composite Galois elds of the form GF ((2 n) m). The third algorithm is an entirely new approach which accelerates the multiplication of points which is the core operation in elliptic curve public-key systems. The algorithm explores computational advantages by computing repeated point doublings directly through closed formulae rather than from individual point doublings. Finally we apply all three algorithms to an implementation of an elliptic curve system over GF ((2 16) 11). We provide absolute performance measures for the eld operations and for an entire point multiplication. We also show the improvements gained by the new point multiplication algorithm in conjunction with the k-ary and improved k-ary methods for exponentiation. iii Preface This thesis describes the research that I conducted while completing my Master's degree at Worcester Polytechnic Institute. I have attempted to compile as much information related to elliptic curves as it is adequate and relevant to this thesis. I have also tried to be as explicit as possible in the derivation of formulae and theorems. Finally, I hope this work to be of use in both university and industry settings in which elliptic curve cryptosystems are being developed. I would like to thank the following people for their support throughout the months that I worked on this thesis. First, I would like to thank Prof. Paar for all the help, advise, and support that he gave me throughout my graduate work at WPI. I would also like to thank Prof. Paar for the atmosphere of friendship and camaraderie that he developed while working with him. I am grateful to Prof. David Cyganski and Prof. Gabor Sarkozy from Worces-ter Polytechnic Institute and Dr. Robert Dulude …