MATH 245 CH12 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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Unformatted text preview: DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1997, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 12 PUBLIC KEY CRYPTOGRAPHY 12.1. Basic Number Theory A hugely successful public key cryptosystem is based on two simple results in number theory and the currently very low computer speed. In this section, we shall discuss the two results in elementary number theory. PROPOSITION 12A. Suppose that a,m N and ( a,m ) = 1 . Then there exists d Z such that ad 1 (mod m ) . Proof. Since ( a,m ) = 1, it follows from Proposition 4J that there exist d,v Z such that ad + mv = 1. Hence ad 1 (mod m ). Definition. The Euler function : N N is defined for every n N by letting ( n ) denote the number of elements of the set S n = { x { 1 , 2 ,...,n } : ( x,n ) = 1 } ; in other words, ( n ) denotes the number of integers among 1 , 2 ,...,n that are coprime to n . Example 12.1.1. We have (4) = 2, (5) = 4 and (6) = 2. Example 12.1.2. We have ( p ) = p- 1 for every prime p . Example 12.1.3. Suppose that p and q are distinct primes. Consider the number pq . To calculate ( pq ), note that we start with the numbers 1 , 2 ,...,pq and eliminate all the multiples of p and q . Now among these pq numbers, there are clearly q multiples of p and p multiples of q , and the only common multiple of both p and q is pq . Hence ( pq ) = pq- p- q + 1 = ( p- 1)( q- 1). Chapter 12 : Public Key Cryptography page 1 of 5 Discrete Mathematics c W W L Chen, 1997, 2008 PROPOSITION 12B. Suppose that a,n N and ( a,n ) = 1 . Then a ( n ) 1 (mod n ) ....
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This note was uploaded on 12/20/2009 for the course MATH 245 taught by Professor ramaswamy during the Spring '08 term at San Diego State.

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MATH 245 CH12 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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