## Fast generation of prime numbers and secure public-key cryptographic parameters (1995)

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Venue: | Journal of Cryptology |

Citations: | 21 - 0 self |

### BibTeX

@ARTICLE{Maurer95fastgeneration,

author = {Ueli M. Maurer},

title = {Fast generation of prime numbers and secure public-key cryptographic parameters},

journal = {Journal of Cryptology},

year = {1995},

volume = {8},

pages = {123--155}

}

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### Abstract

Abstract. Avery e cient 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 pseudo-prime of the same size that passes the Miller-Rabin test for only one base. Therefore our algorithm is even faster than presently-used algorithms for generating only pseudo-primes because several Miller-Rabin tests with independent bases must be applied for achieving a su cient con dence level. Heuristic arguments suggest that the generated primes are close to uniformly distributed over the set of primes in the speci ed 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 public-key cryptosystem is presented. The prime generation algorithm can easily be modi ed to generate nearly random primes or RSA-moduli that satisfy these security constraints. Further results described in this paper include an analysis of the optimal upper bound for trial division in the Miller-Rabin test as well as an analysis of the distribution of the number of bits of the smaller prime factor of a random k-bit RSA-modulus, givenasecurity bound on the size of the two primes.