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11
An analysis of the generalized binary GCD algorithm
 HIGH PRIMES AND MISDEMEANORS, LECTURES IN HONOUR OF HUGH COWIE
, 2007
"... In this paper we analyze a slight modification of Jebelean’s version of the kary GCD algorithm. Jebelean had shown that on nbit inputs, the algorithm runs in O(n²) time. In this paper, we show that the average running time of our modified algorithm is O(n²/ log n). This analysis involves explori ..."
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In this paper we analyze a slight modification of Jebelean’s version of the kary GCD algorithm. Jebelean had shown that on nbit inputs, the algorithm runs in O(n²) time. In this paper, we show that the average running time of our modified algorithm is O(n²/ log n). This analysis involves exploring the behavior of spurious factors introduced during the main loop of the algorithm. We also introduce a Jebeleanstyle leftshift kary GCD algorithm with a similar complexity that performs well in practice.
Distribution of the Number of Factors in Random Ordered Factorizations of Integers
 J. Number Theory
, 1998
"... We study in detail the asymptotic behavior of the number of ordered factorizations with a given number of factors. Asymptotic formulae are derived for almost all possible values of interest. In particular, the distribution of the number of factors is asymptotically normal. Also we improve the error ..."
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We study in detail the asymptotic behavior of the number of ordered factorizations with a given number of factors. Asymptotic formulae are derived for almost all possible values of interest. In particular, the distribution of the number of factors is asymptotically normal. Also we improve the error term in Kalmar's problem of "factorisatio numerorum" and investigate the average number of distinct factors in a random ordered factorization.
Arbitrarily Tight Bounds On The Distribution Of Smooth Integers
 Proceedings of the Millennial Conference on Number Theory
, 2002
"... This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve ..."
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This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.
Fast Bounds on the Distribution of Smooth Numbers
, 2006
"... Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our fi ..."
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Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our first, easy improvement runs in time roughly y 2/3. Then, assuming the Riemann Hypothesis, we show how to drastically improve this. In particular, if log y is a fractional power of log x, which is true in applications to factoring and cryptography, then our new algorithm has a running time that is polynomial in log y, and gives bounds as tight as, and often tighter than, Bernstein’s algorithm.
Approximating the number of integers without large prime factors
 Mathematics of Computation
, 2004
"... Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ <y ≤ x,whereɛ is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, ..."
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Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ <y ≤ x,whereɛ is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate Ψ(x, y). The computational complexity of this algorithm is O ( � (log x)(log y)). We give numerical results which show that this algorithm provides accurate estimates for Ψ(x, y) andisfaster than conventional methods such as algorithms exploiting Dickman’s function. 1.
Dickman polylogarithms and their constants, preprint arXiv:1004.0519 v1. [4.0
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Uniform asymptotics of some Abel sums arising in coding theory
"... We derive uniform asymptotic expressions of some Abel sums appearing in some problems in coding theory and indicate the usefulness of these sums in other fields, like empirical processes, machine maintenance, analysis of algorithms, probabilistic number theory, queuing models, etc. Key words: Abel s ..."
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We derive uniform asymptotic expressions of some Abel sums appearing in some problems in coding theory and indicate the usefulness of these sums in other fields, like empirical processes, machine maintenance, analysis of algorithms, probabilistic number theory, queuing models, etc. Key words: Abel sums, coding theory, Mellin transforms, Wfunction, uniform asymptotics. 1
A Sublinear Time Parallel GCD Algorithm for the EREW PRAM
, 2009
"... We present a parallel algorithm that computes the greatest common divisor of two integers of n bits in length that takes O(n log log n / logn) expected time using n 6+ǫ processors on the EREW PRAM parallel model of computation. We believe this to be the first sublinear time algorithm on the EREW PRA ..."
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We present a parallel algorithm that computes the greatest common divisor of two integers of n bits in length that takes O(n log log n / logn) expected time using n 6+ǫ processors on the EREW PRAM parallel model of computation. We believe this to be the first sublinear time algorithm on the EREW PRAM for this problem.
A Randomized Sublinear Time Parallel GCD Algorithm for the EREW PRAM
, 2010
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