Results 1  10
of
12
The Number Field Sieve
, 1990
"... The number field sieve is an algorithm to factor integers of the form r e s for small positive r and s . This note is intended as a `report on work in progress' on this algorithm. We informally describe the algorithm, discuss several implementation related aspects, and present some of the fact ..."
Abstract

Cited by 82 (4 self)
 Add to MetaCart
The number field sieve is an algorithm to factor integers of the form r e s for small positive r and s . This note is intended as a `report on work in progress' on this algorithm. We informally describe the algorithm, discuss several implementation related aspects, and present some of the factorizations obtained so far. We also
Factoring by electronic mail
, 1990
"... In this paper we describe our distributed implementation of two factoring algorithms. the elliptic curve method (ecm) and the multiple polynomial quadratic sieve algorithm (mpqs). Since the summer of 1987. our ermimplementation on a network of MicroVAX processors at DEC’s Systems Research Center h ..."
Abstract

Cited by 60 (10 self)
 Add to MetaCart
In this paper we describe our distributed implementation of two factoring algorithms. the elliptic curve method (ecm) and the multiple polynomial quadratic sieve algorithm (mpqs). Since the summer of 1987. our ermimplementation on a network of MicroVAX processors at DEC’s Systems Research Center has factored several most and more wanted numbers from the Cunningham project. In the summer of 1988. we implemented the multiple polynomial quadratic sieve algorithm on rhe same network On this network alone. we are now able to factor any!@I digit integer, or to find 35 digit factors of numbers up to 150 digits long within one month. To allow an even wider distribution of our programs we made use of electronic mail networks For the distribution of the programs and for interprocessor communicatton. Even during the mitial stage of this experiment machines all over the United States and at various places in Europe and Ausnalia conhibuted 15 percent of the total factorization effort. At all the sites where our program is running we only use cycles that would otherwise have been idle. This shows that the enormous computational task of factoring 100 digit integers with the current algoritluns can be completed almost for free. Since we use a negligible fraction of the idle cycles of alI the machines on the worldwide elecnonic mail networks. we could factor 100 digit integers within a few days with a little more help.
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 ..."
Abstract

Cited by 44 (17 self)
 Add to MetaCart
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.
Recent progress and prospects for integer factorisation algorithms
 In Proc. of COCOON 2000
, 2000
"... Abstract. The integer factorisation and discrete logarithm problems are of practical importance because of the widespread use of public key cryptosystems whose security depends on the presumed difficulty of solving these problems. This paper considers primarily the integer factorisation problem. In ..."
Abstract

Cited by 24 (1 self)
 Add to MetaCart
(Show Context)
Abstract. The integer factorisation and discrete logarithm problems are of practical importance because of the widespread use of public key cryptosystems whose security depends on the presumed difficulty of solving these problems. This paper considers primarily the integer factorisation problem. In recent years the limits of the best integer factorisation algorithms have been extended greatly, due in part to Moore’s law and in part to algorithmic improvements. It is now routine to factor 100decimal digit numbers, and feasible to factor numbers of 155 decimal digits (512 bits). We outline several integer factorisation algorithms, consider their suitability for implementation on parallel machines, and give examples of their current capabilities. In particular, we consider the problem of parallel solution of the large, sparse linear systems which arise with the MPQS and NFS methods. 1
The Number of Relations in the Quadratic Sieve Algorithm
, 1996
"... The subject of our study is the single large prime variation of the quadratic sieve algorithm. We derive a formula for the average numbers of complete and incomplete relations per polynomial, directly generated by the algorithm. The number of additional complete relations from the incomplete relatio ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
The subject of our study is the single large prime variation of the quadratic sieve algorithm. We derive a formula for the average numbers of complete and incomplete relations per polynomial, directly generated by the algorithm. The number of additional complete relations from the incomplete relations is then computed by a known formula. Hence practical hints for the optimal choice of the parameter values can be derived. We further compare theoretical estimates for the total number of smooth integers in an interval with countings in practice. AMS Subject Classification (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 ...
Factorization beyond the googol with MPQS on a single computer
 CWI Quarterly
, 1991
"... For the first time a number of more than 100 decimal digits has been factorized on a single computer by means of the Multiple Polynomial Quadratic Sieve method of Kraïtchik and Pomerance (with improvements by Montgomery and Silverman). This method (MPQS) is the best one known to handle numbers which ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
For the first time a number of more than 100 decimal digits has been factorized on a single computer by means of the Multiple Polynomial Quadratic Sieve method of Kraïtchik and Pomerance (with improvements by Montgomery and Silverman). This method (MPQS) is the best one known to handle numbers which are the product of two large, approximately equal prime factors. These numbers are being used in cryptography as keys in publickey cryptosystems. The safety of such cryptosystems depends on our ability to factorize these keys. The computer used is the fourprocessor Cray YMP4/464 which was installed
Abstract Primality Testing and 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 ..."
Abstract
 Add to MetaCart
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 recent algorithms for primality testing and factorisation, give examples of their use and outline some applications. 1.
Factorisation of Large Integers on some Vector and Parallel Computers
 The Australian National University TRCS9501
, 1995
"... ..."
Integer Factorisation on the AP1000
, 1995
"... We compare implementations of two integer factorisation algorithms, the elliptic curve method (ECM) and a variant of the Pollard "rho " method, on three machines (the Fujitsu AP1000, VP2200 and VPP500) with parallel and/or vector architectures. ECM is scalable and well suited for b ..."
Abstract
 Add to MetaCart
We compare implementations of two integer factorisation algorithms, the elliptic curve method (ECM) and a variant of the Pollard &quot;rho &quot; method, on three machines (the Fujitsu AP1000, VP2200 and VPP500) with parallel and/or vector architectures. ECM is scalable and well suited for both vector and parallel architectures.