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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 44 (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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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
Factorization of the tenth and eleventh Fermat numbers
, 1996
"... . We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a ..."
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. We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27decimal digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40digit factor of the tenth Fermat number was found after about 140 Mflopyears of computation. We discuss aspects of the practical implementation of ECM, including the use of specialpurpose hardware, and note several other large factors found recently by ECM. 1. Introduction For a nonnegative integer n, the nth Fermat number is F n = 2 2 n + 1. It is known that F n is prime for 0 n 4, and composite for 5 n 23. Also, for n 2, the factors of F n are of th...
Optimal Scheduling for Disconnected Cooperation
, 2001
"... We consider a distributed environment consisting of n processors that need to perform t tasks. We assume that communication is initially unavailable and that processors begin work in isolation. At some unknown point of time an unknown collection of processors may establish communication. Before proc ..."
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Cited by 9 (3 self)
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We consider a distributed environment consisting of n processors that need to perform t tasks. We assume that communication is initially unavailable and that processors begin work in isolation. At some unknown point of time an unknown collection of processors may establish communication. Before processors begin communication they execute tasks in the order given by their schedules. Our goal is to schedule work of isolated processors so that when communication is established for the rst time, the number of redundantly executed tasks is controlled. We quantify worst case redundancy as a function of processor advancements through their schedules. In this work we rene and simplify an extant deterministic construction for schedules with n t, and we develop a new analysis of its waste. The new analysis shows that for any pair of schedules, the number of redundant tasks can be controlled for the entire range of t tasks. Our new result is asymptotically optimal: the tails of these schedules are within a 1 +O(n 1 4 ) factor of the lower bound. We also present two new deterministic constructions one for t n, and the other for t n 3=2 , which substantially improve pairwise waste for all prexes of length t= p n, and oer near optimal waste for the tails of the schedules. Finally, we present bounds for waste of any collection of k 2 processors for both deterministic and randomized constructions. 1
Running time predictions for factoring algorithms
 Algorithmic Number Theory, ANTS VIII, Banff, Springer LNCS 5011
, 2008
"... Partiellement soutenu par une bourse de la Conseil de recherches en sciences naturelles et en génie du Canada. 3 Supported in part by NSF Grant DMS0103635. In 1994, Carl Pomerance proposed the following problem: Select integers a1, a2,..., aJ at random from the interval [1, x], stopping when some ..."
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Partiellement soutenu par une bourse de la Conseil de recherches en sciences naturelles et en génie du Canada. 3 Supported in part by NSF Grant DMS0103635. In 1994, Carl Pomerance proposed the following problem: Select integers a1, a2,..., aJ at random from the interval [1, x], stopping when some (nonempty) subsequence, {ai: i ∈ I} where I ⊆ {1, 2,..., J}, has a square product (that is ∏ i∈I ai ∈ Z2). What can we say about the possible stopping times, J? A 1985 algorithm of Schroeppel can be used to show that this process stops after selecting (1 + ɛ)J0(x) integers aj with probability 1 − o(1) (where the function J0(x) is given explicitly in (1) below). Schroeppel’s algorithm actually finds the square product, and this has subsequently been adopted, with relatively minor modifications, by all factorers. In 1994 Pomerance showed that, with probability 1−o(1), the
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.
Quantum and Post Quantum Cryptography
"... Public key cryptography is widely used for signing contracts, electronic voting, encryption, and to secure transactions over the Internet. The discovery by Peter Shor, in 1994, of an efficient algorithm based on quantum mechanics for factoring large integers and computing discrete logarithms undermi ..."
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Public key cryptography is widely used for signing contracts, electronic voting, encryption, and to secure transactions over the Internet. The discovery by Peter Shor, in 1994, of an efficient algorithm based on quantum mechanics for factoring large integers and computing discrete logarithms undermined the security assumptions upon which currently used public key cryptographic algorithms are based, like RSA, El Gamal and ECC. However, some cryptosystems, called post quantum cryptosystems, while not currently in widespread use are believed to be resistant to quantum computing based attacks. In this paper, we provide a survey of quantum and post quantum cryptography. We review the principle of a quatum computer as well as Shor’s algorithm and quantum key distribution. Then, we review some cryptosystems undermined by Shor’s algorithm as well as some post quantum cryptosystems, that are believed to resist classical and quantum computers. 1
Modern Factoring Algorithms
"... ... both Gauss and lesser mathematicians may be justified in rejoicing that there is one science [number theory] at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. G.H. Hardy, A Mathematician’s Apology, 1940 Unfortunately for Hardy ..."
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... both Gauss and lesser mathematicians may be justified in rejoicing that there is one science [number theory] at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. G.H. Hardy, A Mathematician’s Apology, 1940 Unfortunately for Hardy, nowadays, number theoretic results form the basis of several applications in the field of cryptography. In this survey, we will present the most recent approaches on solving the factoring problem. In particular, we will study Pollard’s ρ algorithm, the Quadratic and Number Field Sieve and finally, we will give a brief overview on factoring in quantum computers. 1.