ece190_lect9 - Fermat's little theorem Periklis A...

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Fermat’s little theorem Periklis A. Papakonstantinou ? University of Toronto We prove “Fermat’s little theorem” . That is, for every a ( Z /p Z ) * it holds that a p - 1 1(mod p ), where p is a prime integer. On the way we go through equivalence classes, modular arithmetic and elements of finite group theory. All these are very important issues in Discrete Mathematics and tight nicely together. Fermat’s little theorem seems to be just a mathematical statement. Surprisingly its consequences are very important for the Applied Cryptography; e.g. electronic transactions. In this handout we introduce very elementary concepts from Finite Group Theory and on the way we prove an important result, namely the Fermat’s little theorem. In general, Group Theory aims to abstract the notion of some restricted type of algebraic systems and prove results in this abstract setting. Among the dozens of reasons for doing so two are that when we prove any theorem we might be interested in two questions: (i) what is the minimum “set of properties” that we need in order to prove the result and (ii) to which more abstract settings this result may be also true. For example, when we prove a theorem about integers it may be the case that we do not use the full power of the structure of the set of integers. So how much of the properties of integers are we using? The other question is that if instead of integers we had something else how does this something else look like. Central objects in Group Theory are groups . These are sets with some additional algebraic structure. The familiar systems of integers, and rational numbers and real numbers are groups in some sense. In discrete mathematics we are especially interested in finite “analogs” of these systems. Groups Definition and examples A group (often denoted as ( G, · )) is a set G together with a binary operation · which is closed for the group elements. That is, if a, b G then ( a · b ) G . In addition the following axioms should hold for the group. There is an element 1 G such that for every g G we have 1 · g = g · 1 = g . The element 1 is called the identity of the group. Let a, b, c G . Then ( a · b ) · c = a · ( b · c ). That is, the binary operation is associative. For every g G there is a h G such that gh = hg = 1. We denote h by g - 1 = h and we call g - 1 the inverse of g . In other words this axiom says that every element of G has an inverse in G . ? This document possibly contains typos. Please submit any typos, remarks, suggestions to [email protected] .
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2 Periklis Papakonstantinou (ECE 190, Fall 2006) Note that we didn’t mention anything about commutativity. That is in an arbitrary group G it is possible that a · b 6 = b · a . If for a group G and for every two elements a, b G we have that a · b = b · a then the group is called commutative or abelian .
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