NASA Ames Research Center
NASA Advanced Supercomputing Division
M/S T27A-2
Moffett Field, CA 94035-1000
San.C.Vo@nasa.gov

NAS-03-012
August 2003

Abstract

The theory of elliptic curves is a classical topic in many branches of algebra and number theory, but recently, it is receiving more attention in cryptography. An elliptic curve is a two-dimensional (planar) curve defined by an equation involving a cubic power of coordinate x, and a square power of coordinate y. One class of these curves is elliptic curves over finite fields, also called Galois fields. These elliptic curves are finite groups with special structures, which can play naturally, and even more flexibly, the roles of the modulus groups in the discrete logarithm problems. This document attempts to provide clear, intuitive, and elementary explanations to guide a typical technical reader into the world of elliptic curve cryptography. Basic knowledge of cryptography and abstract algebra, including group theory and number theory would be helpful for readers in several technical areas. Part II of this survey (to be developed), intends to focus more on practical implementations. It is hoped that this survey will provide readers with a good background on elliptic curve cryptography, which is attracting more attention from cryptographers, computer scientists, and researchers all over the world.

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