Notes on Number Theory and Cryptography

Many interesting and useful properties of the set of integers Z (the whole numbers . . . ,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . . ) can be studied by thinking in terms of divisibility by a particular base or modulus, a positive integer m. For example, it is frequently important to know whether a given number k is even or odd, that is to say, whether or not k is (evenly) divisible by 2. (An even number of young swimmers can be paired off in a buddy system, but if there are an odd number, a problem arises.) The entire set Z of integers breaks into two disjoint subsets, the evens and the odds. Note that two numbers x and y are in the same class if and only if their difference is even, that is, 2 divides x − y. Further, the sum of any two even numbers is even, the sum of any two odd numbers is even, and the sum of an even number and an odd number is odd. This allows us to do arithmetic with the two classes of even and odd numbers. Similar statements hold for divisibility by any positive integer modulus m, and we now give those statements formally.