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applied cryptography - protocols, algorithms, and source code in c

# Development of the standard in the early 1970s

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Unformatted text preview: tocols, Algorthms, and Source Code in C (cloth) Go! Keyword 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: Go! Previous Table of Contents Next ----------- One implementation of this method on a Sparc II was able to find 256-bit primes in an average of 2.8 seconds, 512-bit primes in an average of 24.0 seconds, 768-bit primes in an average of 2.0 minutes, and 1024-bit primes in an average of 5.1 minutes [918]. Strong Primes If n is the product of two primes, p and q, it may be desirable to use strong primes for p and q. These are prime numbers with certain properties that make the product n difficult to factor by specific factoring methods. Among the properties suggested have been [1328,651]: The greatest common divisor of p - 1 and q - 1 should be small. Both p - 1 and q - 1 should have large prime factors, respectively p’ and q’. Both p’ - 1 and q’ - 1 should have large prime factors. Both p + 1 and q + 1 should have large prime factors. Both (p - 1)/2 and (q - 1)/2 should be prime [182]. (Note that if this condition is true, then so are the first two.) Whether strong primes are necessary is a subject of debate. These properties were designed to thwart some older factoring algorithms. However, the fastest factoring algorithms have as good a chance of factoring numbers that meet these criteria as they do of factoring numbers that do not [831]. I recommend against specifically generating strong primes. The length of the primes is much more important than the structure. Moreover, structure may be damaging because it is less random. This may change. New factoring techniques may be developed that work better on numbers with certain properties than on numbers without them. If so, strong primes may be required once again. Check current theoretical mathematics journals for updates. 11.6 Discrete Logarithms in a Finite Field Modular exponentiation is another one-way function used frequently in cr...
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