By Susan Landau, Whitfield Diffie

ISBN-10: 0262256010

ISBN-13: 9780262256018

A penetrating and insightful research of privateness and protection in telecommunications for a post-9/11, post-Patriot Act international.

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**Sample text**

Public Key Cryptography: Applications and Attacks, First Edition. Lynn Margaret Batten. © 2013 by The Institute of Electrical and Electronics Engineers, Inc. Published 2013 by John Wiley & Sons, Inc. 45 46 THE ELGAMAL SCHEME For instance, φ(15) = 8; equivalently, eight of the numbers in the range 1–14 have inverses modulo 15. For n a prime, it should be clear that φ(n) = n − 1. The Euler φ function has some properties that are worth noting including those in the following proposition that we give without proof.

Thus, it looks like 9999991 is a prime. The Maple command isprime(9999991); confirms with true. 2. 3 modulo n where n is composite, then we say that n is a pseudoprime base a. The site in question 1 also finds these for certain possible bases. Use the site to determine all pseudoprimes less than 1000 to base 2. Solution. The output is Pseudoprimes base 2 less than 1000: 341 561 645. 3. 3, it is given a special name. It is called a Carmichael number. On the same site, find as many Carmichael numbers as possible.

Solution. This is a CRT problem. We can reformulate it as: x ≡ 17 (mod 101) and x ≡ 18 (mod 201) and x = 19 (mod 301). Using the Maple command: > chrem([17, 18, 19], [101, 201, 301]); produces 61122. You can check the answer by reducing 61122 by each of the moduli in turn. Obviously, this notation generalizes to as many equations as you want. 3. Find a common solution to the system 3x ≡ 2 (mod 3127) and 4x ≡ 5 (mod 2563). Solution. First we check gcd(3127, 2563); which yields 1. It is easy to see that gcd(3, 3127) and gcd(4, 2563) are also 1.

### Privacy on the Line: The Politics of Wiretapping and Encryption by Susan Landau, Whitfield Diffie

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