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) 23. - 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. Language: English