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Smooth Orders and Cryptographic Applications
 Lecture Notes in Comptuer Science
, 2002
"... We obtain rigorous upper bounds on the number of primes x for which p1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker re ..."
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Cited by 5 (1 self)
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We obtain rigorous upper bounds on the number of primes x for which p1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker result for almost all odd numbers n. We also discuss some cryptographic applications.
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 3 (1 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.
On the distribution of the Euler function of shifted smooth numbers’, Preprint, 2008, (available from http://arxiv.org/abs/0810.1093
"... We give asymptotic formulas for some average values of the Euler function on shifted smooth numbers. The result is based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to A. Granville and É. Fouvry & G. Tenenbaum. 1 ..."
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Cited by 1 (1 self)
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We give asymptotic formulas for some average values of the Euler function on shifted smooth numbers. The result is based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to A. Granville and É. Fouvry & G. Tenenbaum. 1
Smoothing "Smooth" Numbers
"... : An integer is called ysmooth if all of its prime factors are y. An important problem is to show that the ysmooth integers up to x are equidistributed amongst short intervals. In particular, for many applications we would like to know that if y is an arbitrarily small, fixed power of x then al ..."
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: An integer is called ysmooth if all of its prime factors are y. An important problem is to show that the ysmooth integers up to x are equidistributed amongst short intervals. In particular, for many applications we would like to know that if y is an arbitrarily small, fixed power of x then all intervals of length p x, up to x, contain, asymptotically, the same number of ysmooth integers. We come close to this objective by proving that such ysmooth integers are so equidistributed in intervals of length p xy 2+" , for any fixed " ? 0. * Partially supported by NSERC grant A5123 y An Alfred P. Sloan Research Fellow. Partially supported by the NSF. Smoothing "Smooth" Numbers John B. Friedlander and Andrew Granville Introduction. In this paper we will investigate the distribution, in short intervals, of integers without large prime factors . Good estimates, in such questions, have turned out to be surprisingly difficult. For instance, one might suppose that proving the ...
Divisibility, Smoothness and Cryptographic Applications
, 2008
"... This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in var ..."
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This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth numbers play a pivotal role. 1 1
THE DISTRIBUTION OF SMOOTH NUMBERS IN ARITHMETIC PROGRESSIONS
, 707
"... We say that a number n is ysmooth if all the prime factors of n lie below y. Let S(y) denote the set of all ysmooth numbers, and let S(x, y) denote the set of ysmooth numbers below x. Let Ψ(x, y) denote the number of smooth integers below x; thus Ψ(x, y) is the cardinality of S(x, y). In this not ..."
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We say that a number n is ysmooth if all the prime factors of n lie below y. Let S(y) denote the set of all ysmooth numbers, and let S(x, y) denote the set of ysmooth numbers below x. Let Ψ(x, y) denote the number of smooth integers below x; thus Ψ(x, y) is the cardinality of S(x, y). In this note we consider the distribution of smooth numbers among