Lecture 5, finite fields (Part 2: Modular arithmetic theoretical underpinnings of modern cryptography). This chapter include objectives: To review modular arithmetic, to present Euclid’s GCD algorithms, to present the prime finite field Zp, to show how Euclid’s GCD algorithm can be extended to find multiplicative inverses, Perl and Python implementations for calculating GCD and multiplicative inverses. | Lecture 5: Finite Fields (PART 2) PART 2: Modular Arithmetic Theoretical Underpinnings of Modern Cryptography Lecture Notes on “Computer and Network Security” by Avi Kak (kak@) February 28, 2016 11:25pm c 2016 Avinash Kak, Purdue University Goals: • To review modular arithmetic • To present Euclid’s GCD algorithms • To present the prime finite field Zp • To show how Euclid’s GCD algorithm can be extended to find multiplicative inverses • Perl and Python implementations for calculating GCD and multiplicative inverses CONTENTS Section Title Page Modular Arithmetic Notation 3 Examples of Congruences 5 Modular Arithmetic Operations 6 The Set Zn and Its Properties 8 So What is Zn ? 10 Asymmetries Between Modulo Addition and Modulo Multiplication Over Zn 11 Euclid’s Method for Finding the Greatest Common Divisor of Two Integers 14 Steps in a Recursive Invocation of Euclid’s GCD Algorithm 16 An Example of Euclid’s GCD Algorithm in Action 17 Proof of Euclid’s GCD Algorithm 19 Implementing the GCD Algorithm in Perl and Python 20 Prime Finite Fields What Happened to the Main Reason for Why Zn Could Not be an Integral Domain Finding Multiplicative Inverses for the Elements of Zp 27 29 30 Proof of Bezout’s Identity 32 Finding Multiplicative Inverses Using Bezout’s Identity 35 Revisiting Euclid’s Algorithm for the Calculation of GCD 37 What Conclusions Can We Draw From the Remainders? 39 Rewriting GCD Recursion in the Form of Derivations for the Remainders 40 Two Examples That Illustrate the Extended Euclid’s Algorithm 42 The Extended Euclid’s Algorithm in Perl and Python 43 Homework Problems 50 Computer and Network Security by Avi Kak Lecture 5 : MODULAR ARITHMETIC NOTATION • Given any integer a and a positive integer n, and given a division of a by n that leaves the remainder between
