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In real-world implementations, prime generation goes quickly. (1) Generate a random n- bit number, p. (2) Set the high-order and low-order bit to 1. (The high-order bit ensures that the prime is of the required length and the low-order bit ensures that it is odd.) (3) Check to make sure p is not divisible by any small primes: 3, 5, 7, 11, and so on. Many implementations test p for divisibility by all primes less than 256. The most efficient is to test for divisibility by all primes less than 2000 . You can do this efficiently using a wheel . (4) Perform the Rabin-Miller test for some random a. If p passes, generate another random a and go through the test again. Choose a small value of a to make the calculations go quicker. Do five tests . (One might seem like enough, but do five.) If p fails one of the tests, generate another p and try again. Another option is not to generate a random p each time, but to incrementally search through numbers starting at a random point until you find a prime. Step (3) is optional, but it is a good idea. Testing a random odd p to make sure it is not divisible by 3, 5, and 7 eliminates 54 percent of the odd numbers before you get to step (4). Testing against all primes less than 100 eliminates 76 percent of the odd numbers; testing against all primes less than 256 eliminates 80 percent. In general, the fraction of odd candidates that is not a multiple of any prime less than n is 1.12/ln n. The larger the n you test up to, the more precomputation is required before you get to the Rabin-Miller test. Previous Table of Contents Next Products | Contact Us | About Us | Privacy | Ad Info | Home Use of this site is subject to certain Terms & Conditions, Copyright © 1996-2000 EarthWeb Inc. All rights reserved. Reproduction whole or in part in any form or medium without express written permission of EarthWeb is prohibited. Read EarthWeb's privacy statement. To access the contents, click the chapter and section titles. Applied Cryptography, Second Edition: Pro...
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This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.
- Fall '10