Elliptic Curve Public Key Cryptosystems

Elliptic curves have been intensively studied in algebraic geometry and number theory. In recent years they have been used in devising efficient algorithms for factoring integers and primality proving, and in the construction of public key cryptosystems.
Elliptic Curve Public Key Cryptosystems provides an up-to-date and self-contained treatment of elliptic curve-based public key cryptology. Elliptic curve cryptosystems potentially provide equivalent security to the existing public key schemes, but with shorter key lengths. Having short key lengths means smaller bandwidth and memory requirements and can be a crucial factor in some applications, for example the design of smart card systems. The book examines various issues which arise in the secure and efficient implementation of elliptic curve systems.
Elliptic Curve Public Key Cryptosystems is a valuable reference resource for researchers in academia, government and industry who are concerned with issues of data security. Because of the comprehensive treatment, the book is also suitable for use as a text for advanced courses on the subject.

1 Introduction to Public Key Cryptography.- 1.1 Private Key Cryptography.- 1.2 Diffie-Hellman Key Exchange.- 1.3 Public Key Cryptography.- 1.4 Trapdoor One-Way Functions Based on Groups.- 1.5 NIST Digital Signature Standard.- 1.6 Elliptic Curve Cryptosystems.- 1.7 Notes.- 2 Introduction to Elliptic Curves.- 2.1 Definitions.- 2.2 Group Law.- 2.3 The Discriminant and j-Invariant.- 2.4 Curves over K, char(K) # 2,3.- 2.5 Curves over K, char(K) = 2.- 2.6 Group Structure.- 2.7 Divisor Theory.- 2.8 Elliptic Curves over ?n.- 2.9 Notes.- 3 Isomorphism Classes of Elliptic Curves over Finite Fields.- 3.1 Introduction.- 3.2 Isomorphism Classes of Curves over Fq, char(Fq) 2, 3..- 3.3 Isomorphism Classes of Non-Supersingular Curves over F2m.- 3.4 Isomorphism Classes of Supersingular Curves over F2m, m odd.- 3.5 Isomorphism Classes of Supersingular Curves over F2m, m even.- 3.6 Number of Points.- 3.7 Notes.- 4 The Discrete Logarithm Problem.- 4.1 Algorithms.- 4.2 Reducing Some Logarithm Problems to Logarithms in a Finite Field.- 4.3 Notes.- 5 The Elliptic Curve Logarithm Problem.- 5.1 The Weil Pairing.- 5.2 Reducing Elliptic Curve Logarithms to Logarithms in a Finite Field.- 5.3 Cryptographic Implications.- 5.4 Finding the Group Structure.- 5.5 Notes.- 6 Implementation of Elliptic Curve Cryptosystems.- 6.1 Field Arithmetic in F2m.- 6.2 Selecting a Curve and Field K.- 6.3 Projective Coordinates.- 6.4 ElGamal Cryptosystem.- 6.5 Performance.- 6.6 Using Supersingular Curves.- 6.7 Elliptic Curve Cryptosystems over ?n.- 6.8 Implementations.- 6.9 Notes.- 7 Counting Points on Elliptic Curves Over F2m.- 7.1 Some Basics.- 7.2 Outline of Schoof’s Algorithm.- 7.3 Some Heuristics.- 7.4 Implementation and Results.- 7.5 Recent Work.- 7.6 Notes.