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CS284GCD - GCD CSCI 284/162 Spring 2009 GW Zm Definition a...

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CSCI 284/162 Spring 2009 GW GCD
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05/13/09 CS284-162/Spring09/GW/Vora/GCD 2 Z m Definition: a b ( mod m) m divides a-b Z m is the “ring” of integers modulo m : 0, 1, 2, …m-1 with normal addition and multiplication, performed modulo m We define a mod m to be the unique remainder of a when divided by m , i.e. a mod m Z m
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05/13/09 CS284-162/Spring09/GW/Vora/GCD 3 Examples: multiplicative inverses Inverse of -1 mod m (for any m ) Or m -1 mod m
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05/13/09 CS284-162/Spring09/GW/Vora/GCD 4 Affine Cipher P = C = R K R × R e K (x) = ax + b d K (x) = a -1 (x – b) Key may be written as: (a, b) or a=; b= Example How many keys when R = Z 4
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05/13/09 CS284-162/Spring09/GW/Vora/GCD 5 To know if a is invertible, need definition of GCD The gcd (Greatest Common Divisor) of two integers m and n denoted gcd(m , n ) is the largest non-negative integer that divides both m and n .
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05/13/09 CS284-162/Spring09/GW/Vora/GCD 6 Multiplicative inverse of a in Z m Theorem: The multiplicative inverse of a mod m Z m , denoted a -1 , exists if and only if gcd(m, a) = 1 Need show: i. a -1 exists gcd(m, a) = 1 ii. gcd(m, a) = 1 a -1 exists
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05/13/09 CS284-162/Spring09/GW/Vora/GCD 7 Proof: (i) a -1 exists gcd(m, a) = 1 Suppose a -1 exists, call it t at 1 (mod m) at + ms = 1 for some integer s gcd(m, a) = 1 (because the gcd divides both sides of above equation, and only 1 can divide the rhs)
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05/13/09 CS284-162/Spring09/GW/Vora/GCD 8 Proof: ii. gcd(m, a) = 1 a -1 exists This involves a bit more work. We show the following, A. 5 integers s , t , such that ms + at = gcd(m, a) Hence, gcd(m, a) = 1 5 integers s , t , such that ms + at = 1 B. 5 integers s , t , such that ms + at = 1 a -1 exists
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05/13/09 CS284-162/Spring09/GW/Vora/GCD 9 Proof of ii A: 5 s , t , such that ms + at = gcd(m, a) Let x
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