CSCI6268L10 - Foundations of Network and Computer Security...

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Foundations of Network and Foundations of Network and Computer Security Computer Security J J ohn Black Lecture #10 Sep 18 th 2009 CSCI 6268/TLEN 5550, Fall 2009
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But first, a little math… A group is a nonempty set G along with an operation # : G x G → G such that for all a, b, c G (a # b) # c = a # (b # c) (associativity) e G such that e # a = a # e = a (identity) a -1 G such that a # a -1 = e (inverses) If a,b G, a # b = b # a we say the group is “commutative” or “abelian” All groups in this course will be abelian
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Examples of Groups Z (the integers) under + ? Q, R, C, under + ? N under + ? Q under x ? Z under x ? 2 x 2 matrices with real entries under x ? Invertible 2 x 2 matrices with real entries under x ? Note all these groups are infinite Meaning there are an infinite number of elements in them Can we have finite groups?
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Finite Groups Simplest example is G = {0} under + Called the “trivial group” Almost as simple is G = {0, 1} under addition mod 2 Let’s generalize – Z m is the group of integers modulo m – Z m = {0, 1, …, m-1} Operation is addition modulo m Identity is 0 Inverse of any a Z m is m-a Also abelian
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