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Unformatted text preview: IS 5371 Cryptography CIS 5371 Cryptography 5. Algebraic foundations 1 Groups Group ( G , ) A set G with a binary operation “ “ for which we have Closure Associativity An identity Each element has an inverse 2 Groups xamples Examples ( Z ,+), (Z p *, ), ( Z n *, ) are all groups Here: {0 1 2 } Z n ={0,1,2, … , n1}, Z p * = {1,2, … , p1}, for prime p , Z n * = {all integers k, 0 < k < n, with gcd ( k,n )=1}. So : Z 15 * = {1,2,,4,7,8,11,13,14}. 3 We have, ) (   * n Z n Lagrange’s theorem Lagrange’s theorem If H is a subgroup of G then: H is a factor of G If G is a finite group and a G then ord ( a ) is a factor of G. xamples Examples ({1,2,4}, ) is a subgroup of Z 7 * 4 The order of 2 in Z 7 * is 3: ) 7 (mod 1 8 2 3 Cyclic groups group is it has an element whose order is A group is cyclic if it has an element whose order is the same as the cardinality of the group....
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This note was uploaded on 12/11/2011 for the course CIS 5371 taught by Professor Mascagni during the Fall '11 term at FSU.
 Fall '11
 Mascagni

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