congruences - Math 470.502&200 Communication and...

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Math 470.502&200 Communication and Cryptography Spring 2012 Congruences Congruence is a fundamental concept of number theory and of central importance in this course. Here we supplement the discussion in T&W, pp. 70-76. Basics Definition. Reference T&W, pp. 70-72. Let a,b,n be integers with n negationslash =0 . We say that a b (mod n ) if a b is a multiple (positive or negative) of n . We say that a is congruent to b mod n . Remark. It is important to note that a b (mod n ) if and only if a = b + nk for some integer k , positive or negative. For example, all numbers that are congruent to x mod 3 are of the form x,x ± 3 ,x ± 6 ,... . It follows that we every integer is congruent mod n to exactly one of the integers 0 , 1 , 2 ,...,n 1 . This leads to a new definition. Definition. Let Z n denote the integers mod n . In view of the preceding remark, we may regard Z n as the set { 0 , 1 ,...,n 1 } . Remark. Z n , the set of integers mod n , is a very important set in cryptography. There are several different ways to look at Z n , all useful: Integers mod n . All possible remainders when n arbitrary integers are divided by n . Specifically, if m Z , then m = qn + r for some r Z n , so m r (mod n ) . Representatives of all congruence classes mod n . The congruence class of m mod n is the set of all integers r such that m r (mod n ) . Proposition. Let a,b,c,d,n be integers with n negationslash =0 . (1) a 0 (mod n ) if and only if n divides a . (2) a a (mod n ) . (3) a b (mod n ) if and only if b a (mod n ) . (4) If a b (mod n ) and b c (mod n ) , then a c (mod n ) . Congurences Copyright c circlecopyrt 2012by Jon Pitts Page 1 of 4
Math 470.502&200 Communication and Cryptography Spring 2012 (5)
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