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Parallel Algorithms for Integer Factorisation
"... The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends o ..."
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Cited by 43 (17 self)
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The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the RivestShamirAdelman (RSA) system, depends on the difficulty of factoring the public keys. In recent years the best known integer factorisation algorithms have improved greatly, to the point where it is now easy to factor a 60decimal digit number, and possible to factor numbers larger than 120 decimal digits, given the availability of enough computing power. We describe several algorithms, including the elliptic curve method (ECM), and the multiplepolynomial quadratic sieve (MPQS) algorithm, and discuss their parallel implementation. It turns out that some of the algorithms are very well suited to parallel implementation. Doubling the degree of parallelism (i.e. the amount of hardware devoted to the problem) roughly increases the size of a number which can be factored in a fixed time by 3 decimal digits. Some recent computational results are mentioned – for example, the complete factorisation of the 617decimal digit Fermat number F11 = 2211 + 1 which was accomplished using ECM.
A Noninteractive PublicKey Distribution System
"... An identitybased noninteractive public key distribution system is presented that is based on a novel trapdoor oneway function allowing a trusted authority to compute the discrete logarithms modulo a publicly known composite number m while this is infeasible for an adversary not knowing the fac ..."
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Cited by 32 (0 self)
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An identitybased noninteractive public key distribution system is presented that is based on a novel trapdoor oneway function allowing a trusted authority to compute the discrete logarithms modulo a publicly known composite number m while this is infeasible for an adversary not knowing the factorization of m. Without interaction with a key distribution center or with the recipient of a given message, a user can generate a mutual secure cipher key based solely on the recipient's identity and his own secret key, and subsequently send the message, encrypted with the generated cipher used in a conventional cipher, over an insecure channel to the recipient. In contrast to previously proposed identitybased systems, no public keys, certificates for public keys or other information need to be exchanged and thus the system is suitable for certain applications that do not allow for interaction. The paper solves an open problem proposed by Shamir in 1984.
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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Cited by 27 (0 self)
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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...
Factorization Of The Tenth Fermat Number
 MATH. COMP
, 1999
"... We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factor ..."
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Cited by 24 (10 self)
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We describe the complete factorization of the tenth Fermat number F 10 by the elliptic curve method (ECM). F 10 is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40digit factor was found after about 140 Mflopyears of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of F 5 ; : : : ; F 11 .
Factorization of the tenth and eleventh Fermat numbers
, 1996
"... . We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a ..."
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Cited by 17 (8 self)
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. We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27decimal digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40digit factor of the tenth Fermat number was found after about 140 Mflopyears of computation. We discuss aspects of the practical implementation of ECM, including the use of specialpurpose hardware, and note several other large factors found recently by ECM. 1. Introduction For a nonnegative integer n, the nth Fermat number is F n = 2 2 n + 1. It is known that F n is prime for 0 n 4, and composite for 5 n 23. Also, for n 2, the factors of F n are of th...
Implementing the Elliptic Curve Method of Factoring in Reconfigurable Hardware
"... A novel portable hardware architecture of the Elliptic Curve Method of factoring, designed and optimized for application in the relation collection step of the Number Field Sieve, is described and analyzed. A comparison with an earlier proofofconcept design by Pelzl, Simka, et al. has been perform ..."
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Cited by 14 (3 self)
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A novel portable hardware architecture of the Elliptic Curve Method of factoring, designed and optimized for application in the relation collection step of the Number Field Sieve, is described and analyzed. A comparison with an earlier proofofconcept design by Pelzl, Simka, et al. has been performed, and a substantial improvement has been demonstrated in terms of both the execution time and the areatime product. The ECM architecture has been ported across five different families of FPGA devices in order to select the family with the best performance to cost ratio. A timing comparison with the highly optimized software implementation, GMPECM, has been performed. Our results indicate that lowcost families of FPGAs, such as Spartan3 and Spartan3E, offer at least an order of magnitude improvement over the same generation of microprocessors in terms of the performance to cost ratio. 1.
Implementation of Parallel Graph Algorithms on a Massively Parallel SIMD Computer with Virtual Processing
, 1995
"... We describe our implementation of several PRAM graph algorithms on the massively parallel computer MasPar MP1 with 16,384 processors. Our implementation incorporated virtual processing and we present extensive test data. In a previous project [13], we reported the implementation of a set of paralle ..."
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Cited by 13 (3 self)
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We describe our implementation of several PRAM graph algorithms on the massively parallel computer MasPar MP1 with 16,384 processors. Our implementation incorporated virtual processing and we present extensive test data. In a previous project [13], we reported the implementation of a set of parallel graph algorithms with the constraint that the maximum input size was restricted to be no more than the physical number of processors on the MasPar. The MasPar language MPL that we used for our code does not support virtual processing. In this paper, we describe a method of simulating virtual processors on the MasPar. We recoded and finetuned our earlier parallel graph algorithms to incorporate the usage of virtual processors. Under the current implementation scheme, there is no limit on the number of virtual processors that one can use in the program as long as there is enough main memory to store all the data required during the computation. We also give two general optimization techniq...
On The Oracle Complexity Of Factoring Integers
 COMPUTATIONAL COMPLEXITY
, 1996
"... The problem of factoring integers in polynomial time with the help of an (infinitely powerful) oracle who answers arbitrary questions with yes or no is considered. The goal is to minimize the number of oracle questions. Let N be a given composite nbit integer to be factored, where n = dlog 2 ..."
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Cited by 6 (0 self)
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The problem of factoring integers in polynomial time with the help of an (infinitely powerful) oracle who answers arbitrary questions with yes or no is considered. The goal is to minimize the number of oracle questions. Let N be a given composite nbit integer to be factored, where n = dlog 2 Ne. The trivial method of asking for the bits of the smallest prime factor of N requires n/2 questions in the worst case. A nontrivial algorithm of Rivest and Shamir requires only n/3 questions for the special case where N is the product of two n/2bit primes. In this paper, a polynomialtime oracle factoring algorithm for general integers is presented which, for any ffl ? 0, asks at most ffln oracle questions for sufficiently large N , thus solving an open problem posed by Rivest and Shamir. Based on a plausible conjecture related to Lenstra's conjecture on the running time of the elliptic curve factoring algorithm it is shown that the algorithm fails with probability at most N ...