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Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
Fast Generation of Prime Numbers and Secure PublicKey Cryptographic Parameters
, 1995
"... A very efficient recursive algorithm for generating nearly random provable primes is presented. The expected time for generating a prime is only slightly greater than the expected time required for generating a pseudoprime of the same size that passes the MillerRabin test for only one base. The ..."
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A very efficient recursive algorithm for generating nearly random provable primes is presented. The expected time for generating a prime is only slightly greater than the expected time required for generating a pseudoprime of the same size that passes the MillerRabin test for only one base. Therefore our algorithm is even faster than presentlyused algorithms for generating only pseudoprimes because several MillerRabin tests with independent bases must be applied for achieving a sufficient confidence level. Heuristic arguments suggest that the generated primes are close to uniformly distributed over the set of primes in the specified interval. Security constraints on the prime parameters of certain cryptographic systems are discussed, and in particular a detailed analysis of the iterated encryption attack on the RSA publickey cryptosystem is presented. The prime generation algorithm can easily be modified to generate nearly random primes or RSAmoduli that satisfy t...
Finding prime pairs with particular gaps
 Math. Comp
, 2002
"... Abstract. By a prime gap of size g, we mean that there are primes p and p + g such that the g − 1 numbers between p and p + g are all composite. It is widely believed that infinitely many prime gaps of size g exist for all even integers g. However, it had not previously been known whether a prime ga ..."
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Abstract. By a prime gap of size g, we mean that there are primes p and p + g such that the g − 1 numbers between p and p + g are all composite. It is widely believed that infinitely many prime gaps of size g exist for all even integers g. However, it had not previously been known whether a prime gap of size 1000 existed. The objective of this article was to be the first to find a prime gap of size 1000, by using a systematic method that would also apply to finding prime gaps of any size. By this method, we find prime gaps for all even integers from 746 to 1000, and some beyond. What we find are not necessarily the first occurrences of these gaps, but, being examples, they give an upper bound on the first such occurrences. The prime gaps of size 1000 listed in this article were first announced on the Number Theory Listing to the World Wide Web on Tuesday, April 8, 1997. Since then, others, including Sol Weintraub and A.O.L. Atkin, have found prime gaps of size 1000 with smaller integers, using more ad hoc methods. At the end of the article, related computations to find prime triples of the form 6m +1, 12m − 1, 12m + 1 and their application
On the primality of F 4723 and F 5387
, 1999
"... Introduction We follow the notations of both [2] and [3]. Let F n (resp. L n ) be the nth Fibonacci number (resp. Lucas number). The aim of this informal note is to describe a short proof of primality for both F 4723 and F 5387 . See the paper [5] for more on this topic. 2 F 4723 From [3, (4.1)], ..."
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Introduction We follow the notations of both [2] and [3]. Let F n (resp. L n ) be the nth Fibonacci number (resp. Lucas number). The aim of this informal note is to describe a short proof of primality for both F 4723 and F 5387 . See the paper [5] for more on this topic. 2 F 4723 From [3, (4.1)], one has F 4k+3 \Gamma 1 = F k+1 L k+1 L 2k+1 : (1) Here k = 1180, k + 1 = 1181 and 2k + 1 = 2361 = 3 \Theta 787. From [3] and with the help of factors found by Montgomery and Silverman [7, 8, 9], we get F 1181 = 5453857 \Theta C 240 ; L 1181 = 59051 \Theta<F27.43