H. J. J. te Riele, W. M. Lioen and D. T. Winter, Factoring with the quadratic sieve on large vector computers, Belgian J. Comp. Appl. Math. 27 (1989), 267278.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algorithm (mpqs, the fastest practical general purpose factoring algorithm) have been polished both theoretically and practically. Although these efforts have pushed our factorization capabilities from the eighty digit range, through the nineties, to integers having more than one hundred digits [1, 4, 10, 14], cryptographers still feel confident basing the security of some of their cryptosystems on the supposed intractability of the factoring problem. It is unlikely that the method presented here will have a major impact on the security of cryptosystems. However, it does make the integer factoring ....

H.J.J. te Riele, W.M. Lioen, D.T. Winter, "Factoring with the quadratic sieve on large vector computers," report NM-R8805, 1988, Centrum voor Wiskunde en Informatica, Amsterdam.

....polynomials so that many n s could be factored simultaneously by sieving. Pomerance s quadratic sieve is a simpli cation of Schroeppel s linear sieve. Each method seems to always succeed in time y 2 o(1) with y as above. See [140] 79] 168] 62] 141] 65] 63] 165] 46] 150] 64] [155], 106] 13] 144] 156] 166] 133] 66] 9] 11] 27] and [52] The algorithm in this paper can be used to indirectly speed up sieving, as described above. Furthermore, a reduction in the sieve array size allows a reduction in the size of n; see, e.g. 52] Pollard s number eld ....

Herman te Riele, Walter Lioen, Dik Winter, Factoring with the quadratic sieve on large vector computers, Journal of Computational and Applied Mathematics 27 (1989), 267-278. MR 90h:11111.

....be found. If not, we need to obtain a di#erent linear dependency and try again. In quadratic sieve algorithms the numbers w i are the values of one (or more) quadratic polynomials with integer coe#cients. This makes it easy to factor the w i by sieving. For details of the process, we refer to [11,32,35,46,49,52,59]. The best quadratic sieve algorithm (MPQS) can, under plausible assumptions, factor a number N in time #(exp(c(ln N ln ln N) 1 2 ) where c # 1. The constants involved are such that MPQS is usually faster than ECM if N is the product of two primes which both exceed N 1 3 . This is because ....

....for Integer Factorisation Algorithms 9 could volunteer to contribute 2 . The final stage of MPQS Gaussian elimination to combine the relations is not so easily distributed. We discuss this in 7 below. 4. 2 MPQS Examples MPQS has been used to obtain many impressive factorisations [10,32,52,59]. At the time of writing (April 2000) the largest number factored by MPQS is the 129 digit RSA Challenge [54] number RSA129. It was factored in 1994 by Atkins et al. [1] The relations formed a sparse matrix with 569466 columns, which was reduced to a dense matrix with 188614 columns; a ....

H. J. J. te Riele, W. Lioen and D. Winter, Factoring with the quadratic sieve on large vector computers, Belgian J. Comp. Appl. Math. 27 (1989), 267--278.

....6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 p D Figure 2: p D versus year Y for ECM In quadratic sieve algorithms the numbers w i are the values of one (or more) quadratic polynomials with integer coecients. This makes it easy to factor the w i by sieving. For details of the process, we refer to [11, 32, 35, 46, 49, 52, 59]. The best quadratic sieve algorithm (MPQS) can, under plausible assumptions, factor a number N in time (exp(c(ln N ln ln N) 1=2 ) where c 1. The constants involved are such that MPQS is usually faster than ECM if N is the product of two primes which both exceed N 1=3 . This is because ....

.... anyone with access to electronic mail and a C compiler could volunteer to contribute 3 . The nal stage of MPQS Gaussian elimination to combine the relations is not so easily distributed. We discuss this in x7. 4. 2 MPQS examples MPQS has been used to obtain many impressive factorisations [10, 32, 52, 59]. At the time of writing (April 2000) the largest number factored by MPQS is the 129 digit RSA Challenge [55] 3 This idea of using machines on the Internet as a free supercomputer has been adopted by several other computation intensive projects. 5 number RSA129. It was factored in 1994 by ....

H. J. J. te Riele, W. Lioen and D. Winter, Factoring with the quadratic sieve on large vector computers, Belgian J. Comp. Appl. Math. 27 (1989), 267-278.

....(1991) 11A51, 11Y05 CR Subject Classification (1991) F.2.1 Keywords Phrases: Factorization, Multiple Polynomial Quadratic Sieve, Vector supercomputer, Cluster of work stations 1. Introduction We assume that the reader is familiar with the multiple polynomial quadratic sieve algorithm [Bre89, Pom85, PST88, Sil87, RLW89]. We consider the single large prime variation of the algorithm and write MPQS for short. If we can predict the rate by which the complete relations in MPQS are generated as a function of the various parameters in the algorithm, then we can determine a good choice of the parameter values. Here we ....

H. J. J. te Riele, W. M. Lioen, and D. T. Winter. Factoring with the quadratic sieve on large vector computers. Journal of Computational and Applied Mathematics, 27:267--278, 1989.

....a computationally expensive task with the best known algorithms. The development of new algorithms and faster machines has made the factorization of general integers with 100 120 digits feasible. Several authors have considered vector and parallel implementations of the MPQS and NFS algorithms [5, 8, 10, 14, 15]. These algorithms have the property that the run time depends mainly on the size of the number N to be factored. For another class of algorithms the run time depends mainly on the size of the factor found. This class includes Lenstra s elliptic curve method (ECM) 11] and Pollard s rho method ....

H. J. J. te Riele, W. Lioen and D. Winter, "Factoring with the quadratic sieve on large vector computers", Belgian J. Comp. Appl. Math. 27(1989), 267--278.

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H. J. J. te Riele, W. M. Lioen and D. T. Winter, Factoring with the quadratic sieve on large vector computers, Belgian J. Comp. Appl. Math. 27 (1989), 267278.

No context found.

H. J. J. te Riele, W. M. Lioen and D. T. Winter, Factoring with the quadratic sieve on large vector computers, Belgian J. Comp. Appl. Math. 27 (1989), 267278.

No context found.

H. J. J. te Riele, W. M. Lioen and D. T. Winter, Factoring with the quadratic sieve on large vector computers, Belgian J. Comp. Appl. Math. 27 (1989), 267278.

No context found.

Herman te Riele, Walter Lioen, and Dik Winter. Factoring with the quadratic sieve on large vector computers. J. Comp. Appl. Math., 27:267--278, 1989.

....of n if X 6j SigmaY (mod n) If X and Y are randomly chosen subject to (1.1) then this yields a proper factor of n in at least 50 of the tries. This principle is the basis for the best known 1. Introduction 2 general factorization methods, namely, the multi polynomial quadratic sieve (MPQS [Bre89, Pom85, PST88, Sil87, RLW89]) and the number field sieve (NFS [LL93] In this paper we discuss and compare the single large prime variation (PMPQS) and the double large prime variation (PPMPQS) of MPQS, and we factor many numbers in the 66 88 decimal digits range, mainly with PPMPQS, both on SGI workstations, and on a ....

....slightly varying. We conclude that, in order also to minimize the amount of memory for storage of the relations, the optimal choice of B 2 =B 1 is about 400. 7. Implementation and experiments 13 7. Implementation and experiments For our PMPQS experiments we used the implementation described in [RLW89]. Almost all our subroutines are written in Fortran. We have originally implemented the PPMPQS algorithm on a supercomputer like the Cray C90 vectorcomputer. We used the same implementation on Silicon Graphics workstations (although we now have written a program especially designed for ....

H.J.J. te Riele, W.M. Lioen, and D.T. Winter. Factoring with the quadratic sieve on large vector computers. J. Comp. Appl. Math., 27:267--278, 1989.

No context found.

Herman te Riele, Walter Lioen, and Dik Winter. Factoring with the quadratic sieve on large vector computers. J. Comp. Appl. Math., 27:267--278, 1989.

....[6] very useful, but needed to extend them to higher bases. For example, we needed many factorizations of a n Gamma 1 for a = 13; 19; 31; 127. The majority were computed using Lenstra s Elliptic Curve Method (ECM) 13] and in some difficult cases the Multiple Polynomial Quadratic Sieve (MPQS) [17, 18, 20]. The factors were kept in a machine readable file which has been distributed together with a simple factorization program factor for IBM PC and compatible computers [3] The program factor should be considered primarily as a means of accessing a file of known factors, rather than as a ....

....program factor for IBM PC and compatible computers [3] The program factor should be considered primarily as a means of accessing a file of known factors, rather than as a general purpose factorization program. For surveys of factorization algorithms and programs, we refer the reader to [1, 2, 7, 8, 10, 13, 14, 16, 17, 18, 19, 20]. Over the past few years we have systematically extended our list of factors, concentrating on numbers a n Sigma 1 for 13 a 100, n 100, but also considering some larger values of the exponent n for the smaller bases a. The tables are now complete for n 46 and include no composites with ....

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H. J. J. te Riele, W. M. Lioen and D. T. Winter, Factoring with the quadratic sieve on large vector computers, Belgian J. Comp. Appl. Math. 27 (1989), 267278.

....methods are given in [1, 3] ECM is useful for finding factors up to about 30 decimal digits, or up to about 40 digits if we use a lot of computer time and are lucky. If the number remaining on division by known factors is composite, but not too large, the factorization can be completed by MPQS [9, 10]. The old Pollard p Sigma 1 methods are still useful: 38 new factors were found by the p Gamma 1 method, and 16 by the p 1 method. The Special Number Field Sieve (SNFS) 6, 7] was not used in the computation of the original tables [2] but SNFS was used to complete 37 difficult factorizations ....

H. J. J. te Riele, W. M. Lioen and D. T. Winter, Factoring with the quadratic sieve on large vector computers, Belgian J. Comp. Appl. Math. 27 (1989), 267278.

No context found.

H. J. J. te Riele, W. Lioen and D. Winter, Factoring with the quadratic sieve on large vector computers, Belgian J. Comp. Appl. Math. 27 (1989), 267--278.

No context found.

H. J. J. te Riele, W. Lioen and D. Winter, Factoring with the quadratic sieve on large vector computers, Belgian J. Comp. Appl. Math. 27 (1989), 267--278.

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