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Multiplicative structure of values of the Euler function
 High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams , Fields Institute Communications
, 2004
"... Dedicated to Hugh Williams on the occasion of his sixtieth birthday. Abstract. We establish upper bounds for the number of smooth values of the Euler function. In particular, although the Euler function has a certain “smoothing ” effect on its integer arguments, our results show that, in fact, most ..."
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Cited by 16 (11 self)
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Dedicated to Hugh Williams on the occasion of his sixtieth birthday. Abstract. We establish upper bounds for the number of smooth values of the Euler function. In particular, although the Euler function has a certain “smoothing ” effect on its integer arguments, our results show that, in fact, most values produced by the Euler function are not smooth. We apply our results to study the distribution of “strong primes”, which are commonly encountered in cryptography. We also consider the problem of obtaining upper and lower bounds for the number of positive integers n ≤ x for which the value of the Euler function ϕ(n) is a perfect square and also for the number of n ≤ x such that ϕ(n) is squarefull. We give similar bounds for the Carmichael function λ(n). 1
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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Cited by 7 (2 self)
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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.
BombieriVinogradov and BarbanDavenportHalberstam type theorems for smooth numbers. prépublication
, 2012
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Additive decompositions of sets with restricted prime factors. Arxiv preprint arXiv:1309.0593
, 2013
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Smooth Values of Shifted Primes in Arithmetic Progressions
"... We study the problem of bounding the number of primes p ≤ x in an arithmetic progression for which the largest prime factor of p − h does not exceed y. 1 ..."
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We study the problem of bounding the number of primes p ≤ x in an arithmetic progression for which the largest prime factor of p − h does not exceed y. 1
Anatomy of Integers and Cryptography
, 2008
"... It is wellknown that heuristic and rigorous analysis of many integer factorisation and discrete logarithm algorithms depends on our various results about the distribution of smooth numbers. Here we give a survey of some other important cryptographic algorithms which rely on our knowledge and under ..."
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It is wellknown that heuristic and rigorous analysis of many integer factorisation and discrete logarithm algorithms depends on our various results about the distribution of smooth numbers. Here we give a survey of some other important cryptographic algorithms which rely on our knowledge and understanding of the multiplicative structure of “typical ” integers and also “typical ” terms of various sequences such as shifted primes, polynomials, totients and so on. Part I