Mathematics for Cryptography

1.2 Definition (modular reduction). Suppose we divide a and b by m, obtaining integer quotients and remainders, where the remainders are between 0 and m − 1. That is, a = q1m + r1 and b = q2m+ r2, where 0 ≤ r1 ≤ m− 1 and 0 ≤ r2 ≤ m− 1. Then it is not difficult to see that a ≡ b (mod m) if and only if r1 = r2. We will use the notation a mod m (without parentheses) to denote the remainder when a is divided by m, i.e., the value r1 above. Thus a ≡ b (mod m) if and only if a mod m = b mod m. If we replace a by a mod m, we say that a is reduced modulo m.