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28
Random Mapping Statistics
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of ..."
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Cited by 78 (6 self)
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Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.
Boltzmann Samplers For The Random Generation Of Combinatorial Structures
 Combinatorics, Probability and Computing
, 2004
"... This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combina ..."
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Cited by 67 (2 self)
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This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class  an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on realarithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.
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.
On Random Walks For Pollard's Rho Method
 Mathematics of Computation
, 2000
"... . We consider Pollard's rho method for discrete logarithm computation. Usually, in the analysis of its running time the assumption is made that a random walk in the underlying group is simulated. We show that this assumption does not hold for the walk originally suggested by Pollard: its performa ..."
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Cited by 31 (5 self)
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. We consider Pollard's rho method for discrete logarithm computation. Usually, in the analysis of its running time the assumption is made that a random walk in the underlying group is simulated. We show that this assumption does not hold for the walk originally suggested by Pollard: its performance is worse than in the random case. We study alternative walks that can be efficiently applied to compute discrete logarithms. We introduce a class of walks that lead to the same performance as expected in the random case. We show that this holds for arbitrarily large prime group orders, thus making Pollard's rho method for prime group orders about 20% faster than before. 1. Introduction Let G be a finite cyclic group, written multiplicatively, and generated by the group element g. We define the discrete logarithm problem (DLP) as follows: given a group element h, find the least nonnegative integer x such that h = g x . We write x = log g h and call it the discrete logarithm of h...
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 .
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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Cited by 17 (8 self)
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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...
20 years of ECM
 In Proceedings of the 7th Algorithmic Number Theory Symposium (ANTS VII
, 2006
"... Abstract. The Elliptic Curve Method for integer factorization (ECM) was invented by H. W. Lenstra, Jr., in 1985 [14]. In the past 20 years, many improvements of ECM were proposed on the mathematical, algorithmic, and implementation sides. This paper summarizes the current stateoftheart, as implem ..."
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Cited by 14 (1 self)
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Abstract. The Elliptic Curve Method for integer factorization (ECM) was invented by H. W. Lenstra, Jr., in 1985 [14]. In the past 20 years, many improvements of ECM were proposed on the mathematical, algorithmic, and implementation sides. This paper summarizes the current stateoftheart, as implemented in the GMPECM software.
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.
Primality proving using elliptic curves: An update
 In Proceedings of ANTS III
, 1998
"... Abstract. In 1986, following the work of Schoof on counting points on elliptic curves over finite fields, new algorithms for primality proving emerged, due to Goldwasser and Kilian on the one hand, and Atkin on the other. The latter algorithm uses the theory of complex multiplication. The algorithm, ..."
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Cited by 8 (1 self)
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Abstract. In 1986, following the work of Schoof on counting points on elliptic curves over finite fields, new algorithms for primality proving emerged, due to Goldwasser and Kilian on the one hand, and Atkin on the other. The latter algorithm uses the theory of complex multiplication. The algorithm, now called ECPP, has been used for nearly ten years. The purpose of this paper is to give an account of the recent theoretical and practical improvements of ECPP, as well as new benchmarks for integers of various sizes and a new primality record. 1
Integer Factoring
, 2000
"... Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization. ..."
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Cited by 2 (0 self)
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Using simple examples and informal discussions this article surveys the key ideas and major advances of the last quarter century in integer factorization.