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

2 set j 0 and set z am mod p 3 if z 1 or if z

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Unformatted text preview: ch of length 100 digits through 500 digits in steps of 10 digits (plus one additional number, 129 digits long). At the time of writing, RSA-100, RSA-110, RSA-120, and RSA-129 have been factored, all using the QS. RSA-130 might be next (using the NFS), or the factoring champions might skip directly to RSA-140. This is a fast-moving field. It is difficult to extrapolate factoring technology because no one can predict advances in mathematical theory. Before the NFS was discovered, many people conjectured that the QS was asymptotically as fast as any factoring method could be. They were wrong. Near-term advances in the NFS are likely to come in the form of bringing down the constant: 1.923. Some numbers of a special form, like Fermat numbers, have a constant more along the lines of 1.5 [955, 954]. If the hard numbers used in public-key cryptography had that kind of constant, 1024-bit numbers could be factored today. One way to lower the constant is to find better ways of representing numbers as polynomials with small coefficients. The problem hasn’t been studied very extensively yet, but it is probable that advances are coming [949]. For the most current results from the RSA Factoring Challenge, send e-mail to [email protected] Square Roots Modulo n If n is the product of two primes, then the ability to calculate square roots mod n is computationally equivalent to the ability to factor n [1283, 35, 36, 193]. In other words, someone who knows the prime factors of n can easily compute the square roots of a number mod n, but for everyone else the computation has been proven to be as hard as computing the prime factors of n. 11.5 Prime Number Generation Public-key algorithms need prime numbers. Any reasonably sized network needs lots of them. Before discussing the mathematics of prime number generation, I will answer a few obvious questions. 1. If everyone needs a different prime number, won’t we run out? No. In fact, there are approximately 10151 primes 512 bits in length or less. For numbers near n, the probability that a random number is prime is...
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