33 - Modular Discrete Mathematics Andrei Bulatov Discrete...

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Introduction Modular Arithmetic Discrete Mathematics Andrei Bulatov

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Discrete Mathematics – Modular Arithmetic II 33-2 Previous Lecture Relatively prime, Euler’s totient function Congruences Residues Residues and arithmetic operations
Discrete Mathematics – Modular Arithmetic II 33-3 Residues (cntd) The residue of an integer a modulo m is such a number b that a b (mod m) and 0 b < m In other words the residue of a modulo m is the remainder of a when divided by m Let denote the set {0,1,2,…,n – 1}. This is the set of all possible remainders of integers when divided by n It is called the set of residues , and its members are called residues n Z

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Discrete Mathematics – Modular Arithmetic II 33-4 Modular Arithmetic We define addition, subtraction, and multiplication of residues: Let a,b . Then a + b (mod n) is the element c such that c a + b (mod n) a – b (mod n) is the element c such that c a – b (mod n) a b (mod n) is the element c such that c a b (mod n) Example. Construct operation tables for n Z n Z n Z n Z 5 Z + 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 1 2 3 4 0 2 3 4 0 1 3 4 0 1 2 4 0 1 2 3 0 0 0 0 0 0 1 2 3 4 0 2 4 1 3 0 3 1 4 2 0 4 3 2 1
Discrete Mathematics – Modular Arithmetic II 33-5 Applications: Criptography One of the oldest cryptosystems is the Caesar cipher. He made messages secret by shifting each letter three letters forward. Thus B becomes E, and X is sent to A To express this process mathematically we first replace letters by integers from 0 to 25. For example, A is replaced by 0, K by 10. Next, to encrypt a message we add 3 modulo 25 to every letter.

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33 - Modular Discrete Mathematics Andrei Bulatov Discrete...

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