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# 31 - Chapter 31 Euclidean Algorithm Euclidean Algorithm...

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June 1, 2004 Computer Security: Art and Science ©2002-2004 Matt Bishop Slide #31-1 Chapter 31: Euclidean Algorithm Euclidean Algorithm Extended Euclidean Algorithm Solving ax mod n = 1 Solving ax mod n = b

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June 1, 2004 Computer Security: Art and Science ©2002-2004 Matt Bishop Slide #31-2 Overview Solving modular equations arises in cryptography Euclidean Algorithm From Euclid to solving ax mod n = 1 From ax mod n = 1 to solving ax mod n = b
June 1, 2004 Computer Security: Art and Science ©2002-2004 Matt Bishop Slide #31-3 Euclidean Algorithm Given positive integers a and b , find their greatest common divisor Idea if x is the greatest common divisor of a and b , then x divides r = a b reduces problem to finding largest x that divides r and b iterate

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June 1, 2004 Computer Security: Art and Science ©2002-2004 Matt Bishop Slide #31-4 Example 1 Take a = 15, b = 12 a b q r 15 12 1 3 q = 15/12 = 1 r = 15 – 1 × 12 12 3 4 0 q = 12/3 = 4 r = 12 – 4 × 3 so gcd (15, 12) = 3 The b for which r is 0
June 1, 2004 Computer Security: Art and Science ©2002-2004 Matt Bishop

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