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A Montgomerylike Square Root for the Number Field Sieve
, 1998
"... The Number Field Sieve (NFS) is the asymptotically fastest factoring algorithm known. It had spectacular successes in factoring numbers of a special form. Then the method was adapted for general numbers, and recently applied to the RSA130 number [6], setting a new world record in factorization. Th ..."
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The Number Field Sieve (NFS) is the asymptotically fastest factoring algorithm known. It had spectacular successes in factoring numbers of a special form. Then the method was adapted for general numbers, and recently applied to the RSA130 number [6], setting a new world record in factorization. The NFS has undergone several modifications since its appearance. One of these modifications concerns the last stage: the computation of the square root of a huge algebraic number given as a product of hundreds of thousands of small ones. This problem was not satisfactorily solved until the appearance of an algorithm by Peter Montgomery. Unfortunately, Montgomery only published a preliminary version of his algorithm [15], while a description of his own implementation can be found in [7]. In this paper, we present a variant of the algorithm, compare it with the original algorithm, and discuss its complexity.
An Implementation of the Number Field Sieve
 EXPERIMENTAL MATHEMATICS
, 1996
"... This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical sieving and a special form of lattice sieving (line sieving), the block Lanczos method and a new square root algorithm. Finally some data on factorizations obtained with this implem ..."
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This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical sieving and a special form of lattice sieving (line sieving), the block Lanczos method and a new square root algorithm. Finally some data on factorizations obtained with this implementation are listed, including the record factorization of 12^151 1.
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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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.
The ThreeLargePrimes Variant of the Number Field Sieve
"... The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This method was proposed by John Pollard [20] in 1988. Since then several variants have been implemented with the objective of improving the siever which is the most time consuming part of this ..."
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The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This method was proposed by John Pollard [20] in 1988. Since then several variants have been implemented with the objective of improving the siever which is the most time consuming part of this method (but fortunately, also the easiest to parallelise). Pollard's original method allowed one large prime. After that the twolargeprimes variant led to substantial improvements [11]. In this paper we investigate whether the threelargeprimes variant may lead to any further improvement. We present theoretical expectations and experimental results. We assume the reader to be familiar with the NFS.
An Implementation of the Number Field Sieve
 EXPERIMENTAL MATHEMATICS
, 1995
"... The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical and lattice sieving, the block Lanczos method and a new square root algorith ..."
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The Number Field Sieve (NFS) is the asymptotically fastest known factoring algorithm for large integers. This article describes an implementation of the NFS, including the choice of two quadratic polynomials, both classical and lattice sieving, the block Lanczos method and a new square root algorithm. Finally some data on factorizations obtained with this implementation are listed, including the record factorization of 12^151  1.
FPGA and ASIC implementation of rho and p1 methods of factoring
, 2007
"... FPGA and ASIC Implementation of rho and p1 methods of factoring ..."
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FPGA and ASIC Implementation of rho and p1 methods of factoring
Evaluation Report on the Factoring Problem
 Society International Conference, September N., (1981C).  23rd IEEE
, 2001
"... This document is an evaluation of the factoring problem, as a basis for designing cryptographic schemes. It relies on the analysis of numerous research papers on the subject. The present report is organized as follows: firstly, we review the factoring problem ..."
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This document is an evaluation of the factoring problem, as a basis for designing cryptographic schemes. It relies on the analysis of numerous research papers on the subject. The present report is organized as follows: firstly, we review the factoring problem
Reconfigurable Computing: A Design and Implementation Study of Elliptic Curve Method
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THE NUMBER FIELD SIEVE FACTORING ALGORITHM
"... In my last paper, I described the Quadratic Sieve (QS) and it’s variants, including a very abbreviated history of the factoring problem. In this paper I will discuss the Number Field Sieve (NFS), that is, the algorithm itself, along with several variants, and how it applies to ..."
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In my last paper, I described the Quadratic Sieve (QS) and it’s variants, including a very abbreviated history of the factoring problem. In this paper I will discuss the Number Field Sieve (NFS), that is, the algorithm itself, along with several variants, and how it applies to