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

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 Rivest-Shamir-Adelman (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 60-decimal 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 multiple-polynomial 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 617-decimal digit Fermat number F11 = 2211 + 1 which was accomplished using ECM.

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 100-decimal 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

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 lO65-digit (2^3539+ l)/3, the first ordinary Titanic prime.

This article discusses the approach of developing MuPAD as an open and parallel problem solving environment (PSE) for mathematical and technical problems, including so-called 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 so-called real world applications, meaning large physical systems and applications from industry, in a reasonable amount of time. Our approach to solve this dis...

These notes informally review the most common methods from computational number theory that have applications in public key cryptology.

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 Rivest-Shamir-Adelman (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 60-decimal 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

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.

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Introduction L'apparition des syst`emes de chiffrement `a clefs publiques de fa¸con g'en'erale [DH76], et du syst`eme de chiffrement RSA en particulier [ARS78], a caus'e un regain d'int'eret pour la th'eorie des nombres et en particulier l'arithm'etique dans ses aspects calculatoires. Pour r'epondre `a des questions aussi simples que celles concernant la d'ecomposition des nombres en facteurs premiers, il a fallu donner des r'eponses algorithmiques prenant en compte la faisabilit'e des calculs ainsi que le temps imparti pour donner une r'eponse satisfaisante. Cela a provoqu'e l'essor de la th'eorie algorithmique des nombres. Cet expos'e est destin'e `a mettre en lumi`ere les progr`es accomplis depuis une dizaine d'ann'ees dans les domaines de la primalit'e des entiers (comment peut-on prouver qu'un entier de quelques centaines de chiffres d'ecimaux est premier) ; factorisation des entiers (quels sont les facteurs d'un nombre qui n'est pas premier) ; logarithme