On

 ą1-representations of integers

János Demetrovics1, Attila Peth2 and Lajos Rónyai

This paper is dedicated to Professor Ferenc Gécseg on the occasion of his 60th birthday.

1. Introduction

 In public key cryptography cryptosystems employing elliptic curves are playing an important role. Such systems are based on the elliptic version of the discrete logarithm problem. Let \Bbb K be a finite field, E = E(\Bbb K) be an elliptic curve over \Bbb K and let P Î E. If the binary expansion of n Î \Bbb N is

 then one can compute P(n) = n P by using the following algorithm:

1.  P(n) Ź P

2.  for i Ź 1 to l do { P(n) Ź 2P(n), if bl-i = 1 then P(n) Ź P(n)+P. }

 This algorithm requires l doubling and ĺi = 0l-1bi addition steps. All operations are performed of course on the curve E. The idea is quite old. In a recipe for integer multiplication it appears in the Egyptian Rhind Papyrus dated from about 1650 B.C.

Footnotes:

  1 Computer and Automation Institute, Hungarian Academy of Sciences, Budapest. The support of OTKA grants 016503, 016526, EC Grant ALTEC-KIT and FKFP grants 0612/1997, 0206/1997 is gratefully acknowledged.

  2 Institute of Mathematics, Kossuth Lajos University, Debrecen. Research supported in part by the Hungarian Foundation for Scientific Research, Grant N0. 25157/98, and by FKFP grant 0612/1997.