Lecture notes on Computer and network security: Lecture 14 - Avinash Kak

Lecture 14, elliptic curve cryptography and digital rights management. The goals of this chapter are: Introduction to elliptic curves, a group structure imposed on the points on an elliptic curve, geometric and algebraic interpretations of the group operator, elliptic curves on prime finite fields, Perl and Python implementations for elliptic curves on prime finite fields,. | Lecture 14: Elliptic Curve Cryptography and Digital Rights Management Lecture Notes on “Computer and Network Security” by Avi Kak (kak@) February 28, 2016 11:40pm c 2016 Avinash Kak, Purdue University Goals: • Introduction to elliptic curves • A group structure imposed on the points on an elliptic curve • Geometric and algebraic interpretations of the group operator • Elliptic curves on prime finite fields • Perl and Python implementations for elliptic curves on prime finite fields • Elliptic curves on Galois fields • Elliptic curve cryptography (EC Diffie-Hellman, EC Digital Signature Algorithm) • Security of Elliptic Curve Cryptography • ECC for Digital Rights Management (DRM) CONTENTS Section Title Page Why Elliptic Curve Cryptography 3 The Main Idea of ECC — In a Nutshell 9 What are Elliptic Curves? 12 A Group Operator Defined for Points on an Elliptic Curve 17 The Characteristic of the Underlying Field and the Singular Elliptic Curves 23 An Algebraic Expression for Adding Two Points on an Elliptic Curve 27 An Algebraic Expression for Calculating 2P from P 31 Elliptic Curves Over Zp for Prime p 34 Perl and Python Implementations of Elliptic Curves Over Finite Fields 37 Elliptic Curves Over Galois Fields GF (2m ) 50 Is b = 0 a Sufficient Condition for the Elliptic Curve y 2 + xy = x3 + ax2 + b to Not be Singular 60 Elliptic Curves Cryptography — The Basic Idea 63 Elliptic Curve Diffie-Hellman Secret Key Exchange 65 Elliptic Curve Digital Signature Algorithm (ECDSA) 69 Security of ECC 73 ECC for Digital Rights Management 75 Homework Problems 80 Computer and Network Security by Avi Kak Lecture 14 : WHY ELLIPTIC CURVE CRYPTOGRAPHY? • As you saw in Section of Lecture 12, the computational overhead of the RSA-based approach to public-key cryptography increases with the size of the keys. As algorithms .