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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 factoriza ..."
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Cited by 69 (2 self)
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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
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 41 (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.
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 ..."
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Cited by 20 (1 self)
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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
Integer Factorization
, 2006
"... Factorization problems are the “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic,” Gauss wrote in his Disquisitiones Arithmeticae in 1801. “The dignity of the sc ..."
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Cited by 10 (1 self)
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Factorization problems are the “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic,” Gauss wrote in his Disquisitiones Arithmeticae in 1801. “The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.” But what exactly is the problem? It turns out that there are many different factorization problems, as we will discuss in this paper.
DISTRIBUTED PRIMALITY PROVING AND THE PRIMALITY OF (2^3539+ 1)/3
, 1991
"... We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certification of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the lO65digit (2^3539+ l)/3, the fir ..."
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Cited by 2 (1 self)
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We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certification of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the lO65digit (2^3539+ l)/3, the first ordinary Titanic prime.
Towards HighPerformance Symbolic Computing in MuPAD: MultiPolynomial Quadratic Sieve using Macro Parallelism and Dynamic Modules
, 1998
"... This article discusses the approach of developing MuPAD as an open and parallel problem solving environment (PSE) for mathematical and technical problems, including socalled real world applications. It describes the new implementation of macro parallelism, based on dynamic modules, which now cove ..."
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Cited by 1 (1 self)
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This article discusses the approach of developing MuPAD as an open and parallel problem solving environment (PSE) for mathematical and technical problems, including socalled real world applications. It describes the new implementation of macro parallelism, based on dynamic modules, which now covers three fields of parallel programming: message passing, network variables and work groups. Parallel algorithms including benchmarks and examples of using MuPAD as a PSE are demonstrated. 1 Introduction Interactive general purpose computer algebra systems (CAS) are very good to define mathematical descriptions of technical problems, i.e. algorithms, systems of equations etc. They can transform them, do several kinds of symbolic computations and also display solutions graphically. Alas, they are mostly not efficient enough to solve socalled real world applications, meaning large physical systems and applications from industry, in a reasonable amount of time. Our approach to solve this dis...
Computational Methods in Public Key Cryptology
, 2002
"... These notes informally review the most common methods from computational number theory that have applications in public key cryptology. ..."
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These notes informally review the most common methods from computational number theory that have applications in public key cryptology.
Integer Factorization
, 1994
"... 6.19> public key cryptosystems (also known as asymmetric cryptosystems and open encryption key cryptosystems) [12, 13]. The security of such systems depends on the (assumed) difficulty of factoring the product of two large primes. This is a practical motivation for the current interest in integer f ..."
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Cited by 1 (0 self)
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6.19> public key cryptosystems (also known as asymmetric cryptosystems and open encryption key cryptosystems) [12, 13]. The security of such systems depends on the (assumed) difficulty of factoring the product of two large primes. This is a practical motivation for the current interest in integer factorisation algorithms. Parallel algorithms We would hope that an algorithm which required time T 1 on a computer with one processor could be implemented to run in time T P ¸ T 1 =P on a computer with P independent processors. This is not always the case, since it may be impossible to use all P processors effectively. However, it is true for many integer factorisation algorithms, provided that P is not too large. Integer factorization algorithms There are many algorithms for finding a nontrivial fac
Integer Factorization Summary
, 1994
"... 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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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 factorization 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. However, the problem of integer factorization still appears difficult, both in a practical sense (for numbers of more than about 80 decimal digits), and in a theoretical sense (because none of the algorithms run in polynomial time). We outline several recent integer factorization algorithms, including the elliptic curve algorithm (ECM), the multiple polynomial quadratic sieve (MPQS), and the special/general number field sieve (NFS), give examples of their use, and mention some applications. Public key cryptography Large primes have at least one practical application – they can be used to construct public key
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 both vect ..."
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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 both vector and parallel architectures.