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Unformatted text preview: 1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 2.1, pp. 4354 Intro to Crypto and Mod Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen 2 Godfrey Harold Hardy b. 1877. d. 1947 was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. from http://en.wikipedia.org/wiki/G. H. Hardy In his 1940 autobiography A Mathematician’s Apology , Hardy wrote ”The great modern achievments of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present, almost as ‘useless’ as the theory of numbers.” ”... then the great bulk of higher mathematics is useless. Modern Geometry and algebra, the theory of numbers , the theory of aggregates and functions, relativity, quantum mechanics – no one of them stands the test much better than another, . . .” and 3 If he could see the world now, G.H. Hardy would be spinning in his grave. Number theory , introduced in this lecture, is the basis of modern coding theory. Computer security and ecommerce would be impossible without it. relativity and quantum theory turned out to be pretty useful as well . . .. At one point, not long ago, the largest employer of math ematicians in the United States, and therefore probably the world, was the National Security Agency (NSA). The NSA is the largest spy agency in the US – bigger than the CIA – and has the responsibility for code design and breaking. 4 (Euclid’s Division Theorem) Let n be a positive integer. Then for every integer m , there exist unique integers q and r such that m = nq + r and ≤ r < n . This will be proven in next lecture. It says that m mod n is uniquely defined. 5 2.1 Cryptography and Modular Arithmetic • Introduction to Cryptography • PrivateKey Cryptography • PublicKey Cryptography • Arithmetic Modulo n • Caesar Ciphers: Cryptography Using Addition mod n • Cryptography Using Multiplication mod n 6 A quick review of the laws of arithmetic over the real numbers • The commutative laws for addition and multiplication a + b = b + a ; ab = ba Ex: 3 + 7 . 2 = 7 . 2 + 3 ; 3 · 5 = 5 · 3 . • The associative laws for addition and multiplication a + ( b + c ) = ( a + b ) + c ; a ( bc ) = ( ab ) c Ex: 5 + (3 + 7) = (5 + 3) + 7 ; 5 · (3 · 7) = (5 · 3) · 7 • The distributive law ( a + b ) c = ac + bc (5 + 3) · 7 = 5 · 7 + 3 · 7 • Every number a 6 = 0 has a multiplicative inverse a 1 s.t. aa 1 = 1 . Ex: 5 · 1 5 = 1 . • Every number a has an additive inverse a such that a + ( a ) = 0 . Ex: 5 + ( 5) = 0 . 7 2.1 Cryptography and Modular Arithmetic • Introduction to Cryptography • PrivateKey Cryptography • PublicKey Cryptography • Arithmetic Modulo n • Caesar Ciphers: Cryptography Using Addition mod n • Cryptography Using Multiplication mod n 8 Modular Arithmetic Definition (1st version) ; Let m, n...
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This note was uploaded on 08/25/2010 for the course COMP COMP170 taught by Professor M.j.golin during the Spring '10 term at HKUST.
 Spring '10
 M.J.Golin
 Computer Science

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