7 - Lecture #6 SS G513 Network Security Agenda Essential...

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Lecture #6 SS G513 Network Security
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Agenda Essential Number Theory Fundamental Number Theorem GCD, Euclid’s algorithm Linear combinations Modular Arithmetic Euler’s Totient Function
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Fundamental Theorem of Arithmetic All numbers are expressible as a unique product of primes 10 = 2 * 5, 60 = 2 * 2 * 3 * 5 Proof in two parts 1. All numbers are expressible as products of primes 2. There is only one such product sequence per number
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GCD (Greatest Common Divisor) gcd(a,b) = the greatest of the divisors of a,b Many ways to compute gcd Extract common prime factors Express a, b as products of primes Extract common prime factors gcd(18, 66) = gcd( 2*3 *3, 2*3 *11) = 2*3 = 6 Factoring is hard. Not practical Euclid’s algorithm
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Euclid's GCD Algorithm an efficient way to find the GCD(a,b) uses theorem that: GCD(a,b) = GCD(b, a mod b) Euclid's Algorithm to compute GCD(a,b): A=a, B=b while B>0 R = A mod B A = B, B = R return A
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r r 1 r r = a % b Euclid’s algorithm a b b r % r 1 = 0. gcd (a,b) = r 1 r 1 = b % r 1 2 3
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Example GCD(1970,1066) 1970 = 1 x 1066 + 904 gcd(1066, 904) 1066 = 1 x 904 + 162 gcd(904, 162) 904 = 5 x 162 + 94 gcd(162, 94) 162 = 1 x 94 + 68 gcd(94, 68) 94 = 1 x 68 + 26 gcd(68, 26) 68 = 2 x 26 + 16 gcd(26, 16) 26 = 1 x 16 + 10 gcd(16, 10) 16 = 1 x 10 + 6 gcd(10, 6) 10 = 1 x 6 + 4 gcd(6, 4) 6 = 1 x 4 + 2 gcd(4, 2) 4 = 2 x 2 + 0 gcd(2, 0)
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Linear Combination a x + b y = “linear combination” of a and b 12x + 20y = {…, -12,-8,-4,0,4,8,12, … } The minimum positive linear combination of a & b = gcd(a,b)
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All numbers are expressible as unique products of prime numbers GCD calculated using Euclid’s algorithm gcd(a,b) = 1 gcd(a,b) equals the minimum positive ax+by linear combination
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This note was uploaded on 03/14/2010 for the course CSE SS ZG513 taught by Professor Sundarb during the Summer '10 term at Birla Institute of Technology & Science.

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7 - Lecture #6 SS G513 Network Security Agenda Essential...

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