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Lecture notes on Computer and network security: Lecture 4 - Avinash Kak

Lecture 4: Finite fields (Part 1: Groups, rings, and fields theoretical underpinnings of modern cryptography). This chapter includes contents: Why study finite fields? What does it take for a set of objects to? infinite groups and abelian groups, rings, integral domain, fields. | Lecture 4: Finite Fields (PART 1) PART 1: Groups, Rings, and Fields Theoretical Underpinnings of Modern Cryptography Lecture Notes on “Computer and Network Security” by Avi Kak (kak@purdue.edu) January 21, 2016 1:21am c 2016 Avinash Kak, Purdue University Goals: • To answer the question: Why study finite fields? • To review the concepts of groups, rings, integral domains, and fields CONTENTS Section Title Page 4.1 Why Study Finite Fields? 3 4.2 What Does It Take for a Set of Objects to? Form a Group 6 4.2.1 Infinite Groups vs. Finite Groups (Permutation Groups) 8 4.2.2 An Example That Illustrates the Binary Operation of Composition of Two Permutations 11 4.2.3 What About the Other Three Conditions that Sn Must Satisfy if it is a Group? 13 4.3 Infinite Groups and Abelian Groups 4.3.1 4.4 If the Group Operator is Referred to as Addition, Then The Group Also Allows for Subtraction Rings 15 17 19 4.4.1 Rings: Properties of the Elements with Respect to the Ring Operator 20 4.4.2 Examples of Rings 21 4.4.3 Commutative Rings 22 4.5 Integral Domain 23 4.6 Fields 24 4.6.1 4.7 Positive and Negative Examples of Fields Homework Problems 25 26 2 Computer and Network Security by Avi Kak Lecture 4 4.1: WHY STUDY FINITE FIELDS? • It is almost impossible to fully understand practically any facet of modern cryptography and several important aspects of general computer security if you do not know what is meant by a finite field. • For example, without understanding the notion of a finite field, you will not be able to understand AES (Advanced Encryption Standard) that we will take up in Lecture 8. As you will recall from Lecture 3, AES is supposed to be a modern replacement for DES. The substitution step in AES is based on the concept of a multiplicative inverse in a finite field. • For another example, without understanding finite fields, you will NOT be able to understand the derivation of the RSA algorithm for public-key cryptography that we .