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classof-4-2-10pdf - CIPHERS USING SOME ELEMENTARY NUMBER...

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CIPHERS USING SOME ELEMENTARY NUMBER THEORY next
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Modular Arithmetic A = B q + r quotient remainder y = x (mod p ) y = p q + x y = “a -1 y a = 1 (mod p) 1. For small p inverses are obtained from the multiplication modulo p table 2. For large p we use the euclidean algorithm (0 r < B ) (0 x<p ) next
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The primes less than 1000 next
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The Addition Table Mod 26 5 next
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The multiplication table modulo 26 Since 2 and 13 are the prime factors of 26 then 13 and all even numbers have no inverse mod 26 The Inverse of x mod p exists only if x and p have no factor in common next
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The multiplication table modulo 29 18 21 If p is a prime all x>0 have an inverse mod p next
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Invertibles Mod 26 7 - 1 15 ( mod 26 ) 15 - 1 7 ( mod 26 ) 3 - 1 9 ( mod 26 ) 9 - 1 3 ( mod 26 ) 5 - 1 21 ( mod 26 ) 21 - 1 5 ( mod 26 ) 11 - 1 19 ( mod 26 ) 19 - 1 11 ( mod 26 ) 17 - 1 23 ( mod 26 ) 23 - 1 17 ( mod 26 ) 25 - 1 25 ( mod 26 ) 1 - 1 1 ( mod 26 ) next
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Solving Linear Equations modulo 26 7 x + 5 y (mod 26) 7 x y - 5 (mod 26) x 7 ( y - 5 ) (mod 26) -1 x 15 y + 3 (mod 26) 7 7 1 (mod 26) x -1 7 15 (mod 26) -1 next
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This note was uploaded on 09/22/2010 for the course MATH MATH187 taught by Professor Math187 during the Spring '10 term at UCSD.

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classof-4-2-10pdf - CIPHERS USING SOME ELEMENTARY NUMBER...

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