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w-cnt - Chapter 9 Computational Number Theory 9.1 The basic...

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Chapter 9 Computational Number Theory 9.1 The basic groups We let Z = { . . . , 2 , 1 , 0 , 1 , 2 , . . . } denote the set of integers. We let Z + = { 1 , 2 , . . . } denote the set of positive integers and N = { 0 , 1 , 2 , . . . } the set of non-negative 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 non-negative 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 = { 0 , 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 ϕ : Z + N by ϕ ( N ) = | Z * N | for all N Z + . That is, ϕ ( N ) is the size of the set Z * N . 9.1.2 Groups Let G be a non-empty 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 non-empty 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 .
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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 . In Z N , the identity element is 0 and the inverse of a is a mod N = N a . In Z * N , the identity element is 1 and the inverse of a is a b Z * N such that ab 1 (mod N ). In may not be obvious why such a b even exists, but it does. We do not prove the above fact here.
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