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11
Discrete logarithms in gf(p) using the number field sieve
 SIAM J. Discrete Math
, 1993
"... Recently, several algorithms using number field sieves have been given to factor a number n in heuristic expected time Ln[1/3; c], where Ln[v; c] = exp{(c + o(1))(log n) v (log log n) 1−v}, for n → ∞. In this paper we present an algorithm to solve the discrete logarithm problem for GF (p) with heur ..."
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Cited by 78 (1 self)
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Recently, several algorithms using number field sieves have been given to factor a number n in heuristic expected time Ln[1/3; c], where Ln[v; c] = exp{(c + o(1))(log n) v (log log n) 1−v}, for n → ∞. In this paper we present an algorithm to solve the discrete logarithm problem for GF (p) with heuristic expected running time Lp[1/3; 3 2/3]. For numbers of a special form, there is an asymptotically slower but more practical version of the algorithm.
Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients
, 1996
"... This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a nontrivial problem to actually do so. In particular so that the constants involv ..."
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Cited by 23 (1 self)
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This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a nontrivial problem to actually do so. In particular so that the constants involved do not render the explicit estimates useless in practical applications. We have used the practical bounds that are needed to prove Theorem 1 as motivation for our results here, though we hope that this work will be applicable to a variety of other problems which routinely apply these or related exponential sum estimates. In particular our results here can be used to say something about the questions of estimating the number of integers free of large prime factors in short intervals (see [FL]), and of the largest prime factor of an integer in an interval (see [J]). Our key result is
Smooth Numbers in Short Intervals
, 2006
"... We show that for any ɛ> 0, there exists c> 0, such that for all x sufficiently large, there are x 1/2 (log x) − log 4−o(1) integers n ∈ [x, x + c √ x], all of whose prime factors are ≤ x 47/(190 √ e)+ɛ. AMS Suject Classification. 11N25. Keywords. Smooth Numbers, Multiplicative Number Theory. ..."
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Cited by 7 (0 self)
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We show that for any ɛ> 0, there exists c> 0, such that for all x sufficiently large, there are x 1/2 (log x) − log 4−o(1) integers n ∈ [x, x + c √ x], all of whose prime factors are ≤ x 47/(190 √ e)+ɛ. AMS Suject Classification. 11N25. Keywords. Smooth Numbers, Multiplicative Number Theory.
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 4 (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.
Lower bounds for the number of smooth values of a polynomial
, 1998
"... We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct ord ..."
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Cited by 3 (1 self)
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We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct order of magnitude has hitherto been established only in a few special cases. The purpose of this paper is to provide such a lower bound for an arbitrary polynomial. Various generalizations to subsets of the set of values taken by a polynomial are also obtained.
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 all ..."
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Cited by 1 (0 self)
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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.
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
Grimm’s Conjecture and Smooth Numbers
 MICHIGAN MATH. J. 61 (2012), 151–160
, 2012
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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