Unformatted text preview: = (p 1)(q 1). These numbers appear in some publickey algorithms; this is why. According to Euler’s generalization of Fermat’s little theorem, if gcd(a,n) = 1, then aÆ(n) mod n = 1 Now it is easy to compute a1 mod n: x = aÆ(n)1 mod n For example, what is the inverse of 5, modulo 7? Since 7 is prime, Æ(7) = 7 1 = 6. So, the inverse of 5, modulo 7, is 561 mod 7 = 55 mod 7 = 3 Both methods for calculating inverses can be extended to solve for x in the general problem (if gcd(a,n) = 1): (a*x) mod n = b Using Euler’s generalization, solve x = (b*a Æ(n)1) mod n Using Euclid’s algorithm, solve x = (b*(a1 mod n)) mod n In general, Euclid’s algorithm is faster than Euler’s generalization for calculating inverses, especially for numbers in the 500bit range. If gcd(a,n) ` 1, all is not lost. In this general case, (a*x) mod n = b, can have multiple solutions or no solution. Chinese Remainder Theorem
If you know the prime factorization of n, then you can use something called the Chinese remainder theorem to solve a whole system of equations. The basic version of this theorem was discovered by the firstcentury Chinese mathematician, Sun Tse. 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 © 19962000 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: Protocols, Algorthms, and Source Code in C (cloth)
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Brief Full Advanced Search Search Tips (Publisher: John Wiley & Sons, Inc.) Author(s): Bruce Schneier ISBN: 0471128457 Publication Date: 01/01/96 Search this book:
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 In general, if the prime factorization of n is p1*p2*...*pt, then the system of equations (x mod pi) = ai, where i = 1, 2,..., t has a unique solution, x, where x is less than n. (Note that s...
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 Cryptography, Bruce Schneier, Applied Cryptography, EarthWeb, Search Search Tips

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