Elliptic Curve Public Key Cryptography

In this paper we mainly introduce public key cryptography based on the elliptic curve discrete logarithm problem.In Chapters 1 and 2 we introduce the background of cryptography,the application of public key cryptography,the idea of public-key cryptography,and the earliest implementations,leading to cryptosystems based on difficulty to factorize large integers(RSA),cryptosystems based on the discrete logarithmic problems(DLP),and Elliptic Curves Cryptosystems(ECC).We also talk about the challenge of quantum computers to public key cryptography.In Chapter 3 we talk about the application of discrete logarithm problem in public key cryptography:Diffie-Hellman key exchange,Elgamal cryptography,digital signatures based on discrete logarithmic problems.After that,the Shanks algorithm and the Pohlig-Hellman algorithm are introduced to attack discrete logarithmic problems,and the Pohlig-Hellman algorithm is used to illustrate some criteria for selecting groups of good discrete logarithmic problems.In Chapter 4 we talk about the content of elliptic curves.The definition of the additive group on an elliptic curve and the coordinate formula for addition(and multiplication)are given.Then important definitions and properties of elliptic curve on finite fields are given,the famous Hasse theorem is given by using these definitions and properties to prepare for the introduction of related algorithms for elliptic curves later.In Chapter 5 we introduce Elliptic Curve Cryptography(ECC).Firstly,the elliptic curve discrete logarithm problem is introduced,and then the public-key cryptography(Menezes-Vanstones cryptosystem,Diffie-Hellman key exchange,digital signature)constructed by applying the elliptic curve discrete logarithm problem is introduced.We then introduce the Schoof algorithm to calculate the number of rational points of elliptic curves over finite fields.Finally,two attack methods(Xedni attack and MOV attack)of elliptic curve discrete logarithm problem on special domain are introduced.