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Lecture Data security and encryption - Lecture 12: Number Theory and Finite Fields

This chapter presents the following content: Number theory, divisibility & GCD, modular arithmetic with integers, Euclid’s algorithm for GCD & inverse, Group, Ring, Field, finite fields GF(p), polynomial arithmetic in general and in GF(2n). | Data Security and Encryption (CSE348) Lecture slides by Lawrie Brown for “Cryptography and Network Security”, 5/e, by William Stallings, briefly reviewing the text outline from Ch 0, and then presenting the content from Chapter 1 – “Introduction”. Lecture # 12 Review Number Theory divisibility & GCD modular arithmetic with integers Euclid’s algorithm for GCD & Inverse Chapter 4 summary. Group Groups, rings, and fields are the fundamental elements of a branch of mathematics known as abstract algebra, or modern algebra In abstract algebra, we are concerned with sets on whose elements we can operate algebraically That is, we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set Groups, rings, and fields are the fundamental elements of a branch of mathematics known as abstract algebra, or modern algebra. In abstract algebra, we are concerned with sets on whose elements we can operate algebraically; that is, we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set. These operations are subject to specific rules, which define the nature of the set. By convention, the notation for the two principal classes of operations on set elements is usually the same as the notation for addition and multiplication on ordinary numbers. However, it is important to note that, in abstract algebra, we are not limited to ordinary arithmetical operations. A group G, sometimes denoted by {G, • }, is a set of elements with a binary operation, denoted by •, that associates to each ordered pair (a, b) of elements in G an element (a • b) in G, such that the following axioms are obeyed: Closure, Associative, Identity element, Inverse element. Note - we have used . as operator: could be addition +, multiplication x or any other mathematical operator. A group can have a finite (fixed) number of elements, or it may be infinite. Note that integers (+ve, -ve and 0) using addition | Data Security and Encryption (CSE348) Lecture slides by Lawrie Brown for “Cryptography and Network Security”, 5/e, by William Stallings, briefly reviewing the text outline from Ch 0, and then presenting the content from Chapter 1 – “Introduction”. Lecture # 12 Review Number Theory divisibility & GCD modular arithmetic with integers Euclid’s algorithm for GCD & Inverse Chapter 4 summary. Group Groups, rings, and fields are the fundamental elements of a branch of mathematics known as abstract algebra, or modern algebra In abstract algebra, we are concerned with sets on whose elements we can operate algebraically That is, we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set Groups, rings, and fields are the fundamental elements of a branch of mathematics known as abstract algebra, or modern algebra. In abstract algebra, we are concerned with sets on whose elements we can operate algebraically; that is, we .