L05_InversesGCDs_print

L05_InversesGCDs_print - 1 COMP170 Discrete Mathematical...

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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 2.2, pp. 56-69 Inverses and GCDs Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen 2 2.2 Inverses and Greatest Common Divisors • Solutions to Equations and Inverses mod n • Converting Modular Equations to Normal Equations • Greatest Common Divisors • Euclid’s Division Theorem • Euclid’s GCD Algorithm • Extended GCD Agorithm • Computing Inverses 3 Greatest Common Divisors: Definitions Euclid’s Division Theorem Euclid’s GCD Algorithm Solutions to Eqns & Inverses mod n Converting mod n Eqns to Normal Ones Extended GCD Algorithm Computing Inverses mod n Proving tool from prev lecture This lecture develops lots of tools for later use. Here’s a chart describing the relationship between the various parts Introduction to proof by smallest counterexample Introduction to proof by contradiction 4 Definition: • Positive integer m is a divisor of integer n if n = mq for some integer q • if m is a divisor of n we write m | n . (say) “ m divides n ” • if m is a not a divisor of n we write m 6 | n . (say) “ m does not divide n ” Examples: • 1 | 30 , 5 | 30 , 5 | 35 , 5 6 | 31 5 • If p is a divisor of both m and n then p is a common divisor of m and n • gcd ( m,n ) denotes the greatest common divisor of m and n . 1 is aways a common divisor of m and n Definition: Examples: • { 1 , 2 , 3 , 6 } are all of the common divisors of 24 and 30 . • gcd (24 , 30) = 6 6 Definition: • Positive integer p > 1 is prime if its only divisors are 1 and itself . If p is not prime, it is composite . • m and n are relatively prime if they have no common divisor other than 1 , i.e., gcd ( m,n ) = 1 . Examples: • 2 , 3 , 5 , 7 , 11 are prime. 33 = 3 · 11 is composite • gcd (77 , 34) = 1 , so 77 and 34 are relatively prime gcd (77 , 33) = 11 , so 77 and 33 are not relatively prime 7 Theorem 2.15 : Two positive integers j,k are relatively prime , i.e., gcd ( j,k ) = 1 , if and only if there are integers x and y such that jx + ky = 1 . The main goal of this lecture is to prove the Theorem and Corollary below and also to show how to calculate the corre- sponding x and y and multiplicative inverses . Corollary 2.16 : For any positive integer n , an element a ∈ Z n has a multiplicative inverse if and only if gcd ( a,n ) = 1 . In order to get to that point we will have to develop a lot of auxillary machinery. We will see in the next lecture that this auxillary machinery will be useful for implementing RSA public-key cryptography. 8 2.2 Inverses and Greatest Common Divisors • Solutions to Equations and Inverses mod n • Converting Modular Equations to Normal Equations • Greatest Common Divisors • Euclid’s Division Theorem • Euclid’s GCD Algorithm • Extended GCD Agorithm • Computing Inverses 9 Recall that in the last section we learnt about Euclid’s division theorem...
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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.

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L05_InversesGCDs_print - 1 COMP170 Discrete Mathematical...

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