By Lynn Margaret Batten
This booklet covers public-key cryptography, describing extensive all significant public-key cryptosystems in present use, together with ElGamal, RSA, Elliptic Curve, and electronic signature schemes. It explains the underlying arithmetic had to construct those schemes, and examines the commonest suggestions utilized in attacking them. Illustrated with many examples, the ebook presents a fantastic origin for execs in govt, cloud provider prone, and big organizations utilizing public-key platforms to safe their information. it's also precious for these taking the CISSP examination (Certified info structures protection Professional).
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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.
Public Key Cryptography: Applications and Attacks by Lynn Margaret Batten