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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 42 (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.
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 22 (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 .
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 21 (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
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 ..."
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Cited by 2 (0 self)
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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 ...
• Some Statistics for NFS Factorizations
, 2005
"... This talk outlines how very large, sparse linear systems arise in the solution of problems of interest in computational number theory and publickey cryptography, such as the integer factorization and discrete logarithm problems. The linear systems are over finite fields, often the field GF(2) of tw ..."
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This talk outlines how very large, sparse linear systems arise in the solution of problems of interest in computational number theory and publickey cryptography, such as the integer factorization and discrete logarithm problems. The linear systems are over finite fields, often the field GF(2) of two elements. We describe some algorithms for solving large sparse linear systems over GF(2), and compare them with algorithms for the real field. In particular, some ”iterative ” algorithms which are wellknown to numerical analysts, such as the conjugate gradient and Lanczos algorithms, can be adapted to work over GF(2), but there are significant differences between algorithms for the real field and for GF(2). Finite Fields 2 In computational number theory and cryptographic applications, we often have to work over finite fields. A finite field F is a finite set with operations “+ ” and “× ” which satisfy the usual associative, commutative and distributive laws: (x + y) + z = x + (y + z), (xy)z = x(yz),
Lecture 1 Fast Algorithms for Elementary Functions,
"... 12 Highprecision arithmetic We are concerned with highprecision computations on integers or floatingpoint approximations to real numbers. If N is a (large) integer we can represent N using base (or radix) fi, say ..."
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12 Highprecision arithmetic We are concerned with highprecision computations on integers or floatingpoint approximations to real numbers. If N is a (large) integer we can represent N using base (or radix) fi, say