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Unformatted text preview: Chapter 9 Computational Number Theory 9.1 The basic groups We let Z = { ... , 2 , 1 , , 1 , 2 ,... } denote the set of integers. We let Z + = { 1 , 2 ,... } denote the set of positive integers and N = { , 1 , 2 ,... } the set of nonnegative integers. 9.1.1 Integers mod N If a,b are integers, not both zero, then their greatest common divisor, denoted gcd( a,b ), is the largest integer d such that d divides a and d divides b . If gcd( a,b ) = 1 then we say that a and b are relatively prime. If a,N are integers with N > 0 then there are unique integers r,q such that a = Nq + r and 0 r < N . We call r the remainder upon division of a by N , and denote it by a mod N . We note that the operation a mod N is defined for both negative and nonnegative values of a , but only for positive values of N . (When a is negative, the quotient q will also be negative, but the remainder r must always be in the indicated range 0 r < N .) If a,b are any integers and N is a positive integer, we write a b (mod N ) if a mod N = b mod N . We associate to any positive integer N the following two sets: Z N = { , 1 ,... ,N 1 } Z * N = { i Z : 1 i N 1 and gcd( i,N ) = 1 } The first set is called the set of integers mod N . Its size is N , and it contains exactly the integers that are possible values of a mod N as a ranges over Z . We define the Euler Phi (or totient) function : Z + N by ( N ) =  Z * N  for all N Z + . That is, ( N ) is the size of the set Z * N . 9.1.2 Groups Let G be a nonempty set, and let be a binary operation on G . This means that for every two points a,b G , a value a b is defined. Definition 9.1.1 Let G be a nonempty set and let denote a binary operation on G . We say that G is a group if it has the following properties: 1. Closure: For every a,b G it is the case that a b is also in G . 2 COMPUTATIONAL NUMBER THEORY 2. Associativity: For every a,b,c G it is the case that ( a b ) c = a ( b c ). 3. Identity: There exists an element 1 G such that a 1 = 1 a = a for all a G . 4. Invertibility: For every a G there exists a unique b G such that a b = b a = 1 . The element b in the invertibility condition is referred to as the inverse of the element a , and is denoted a 1 . We now return to the sets we defined above and remark on their group structure. Let N be a positive integer. The operation of addition modulo N takes input any two integers a,b and returns ( a + b ) mod N . The operation of multiplication modulo N takes input any two integers a,b and returns ab mod N . Fact 9.1.2 Let N be a positive integer. Then Z N is a group under addition modulo N , and Z * N is a group under multiplication modulo N ....
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This note was uploaded on 08/31/2011 for the course CSE 207 taught by Professor Daniele during the Winter '08 term at UCSD.
 Winter '08
 daniele

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