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28
Factoring Integers and Computing Discrete Logarithms via Diophantine Approximation
, 1993
"... Let N be an integer with at least two distinct prime factors. We reduce the problem of factoring N to the task of finding t + 2 integer solutions (e1 ; : : : ; e t ) 2 ZZ t of the inequalities fi fi fi fi fi t X i=1 e i log p i \Gamma log N fi fi fi fi fi N \Gammac p o(1) t t X i=1 j ..."
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Cited by 30 (5 self)
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Let N be an integer with at least two distinct prime factors. We reduce the problem of factoring N to the task of finding t + 2 integer solutions (e1 ; : : : ; e t ) 2 ZZ t of the inequalities fi fi fi fi fi t X i=1 e i log p i \Gamma log N fi fi fi fi fi N \Gammac p o(1) t t X i=1 je i log p i j (2c \Gamma 1) log N + 2 log p t ; where c ? 1 is fixed and p1 ; : : : ; p t are the first t primes. We show, under a reasonable hypothesis, that there are N "+o(1) many solutions (e1 ; : : : ; e t ) where " = c\Gamma1\Gamma(2c\Gamma1)=ff with p t = (log N) ff . Here we have " ? 0 if and only if ff ? (2c \Gamma 1)=(c \Gamma 1). We associate with the primes p1 ; : : : ; p t a lattice L ae IR t+1 of rank t and we associate with N a point N 2 IR t+1 . The above problem of diophantine approximation amounts to finding lattice vectors z that are sufficiently close to N in the 1norm. We also reduce the problem of computing, for a prime N , discrete logarithms of the units...
Asymptotic semismoothness probabilities
 Mathematics of computation
, 1996
"... Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with res ..."
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Cited by 22 (1 self)
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Abstract. We call an integer semismooth with respect to y and z if each of its prime factors is ≤ y, and all but one are ≤ z. Such numbers are useful in various factoring algorithms, including the quadratic sieve. Let G(α, β)bethe asymptotic probability that a random integer n is semismooth with respect to n β and n α. We present new recurrence relations for G and related functions. We then give numerical methods for computing G,tablesofG, and estimates for the error incurred by this asymptotic approximation. 1.
Algorithms for Finding Almost Irreducible and Almost Primitive Trinomials
 in Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams, Fields Institute
, 2003
"... Consider polynomials over GF(2). We describe ecient algorithms for nding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree r for all Mersenne exponents r = 3 mod 8 in the range 5 < r < 10 , although t ..."
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Cited by 17 (6 self)
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Consider polynomials over GF(2). We describe ecient algorithms for nding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree r for all Mersenne exponents r = 3 mod 8 in the range 5 < r < 10 , although there is no irreducible trinomial of degree r.
On the asymptotic distribution of large prime factors
 J. London Math. Soc
, 1993
"... A random integer N, drawn uniformly from the set {1,2,..., n), has a prime factorization of the form N = a1a2...aM where ax ^ a2>... ^ aM. We establish the asymptotic distribution, as «» • oo, of the vector A(«) = (loga,/logiV: i:> 1) in a transparent manner. By randomly reordering the comp ..."
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Cited by 16 (0 self)
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A random integer N, drawn uniformly from the set {1,2,..., n), has a prime factorization of the form N = a1a2...aM where ax ^ a2>... ^ aM. We establish the asymptotic distribution, as «» • oo, of the vector A(«) = (loga,/logiV: i:> 1) in a transparent manner. By randomly reordering the components of A(«), in a sizebiased manner, we obtain a new vector B(n) whose asymptotic distribution is the GEM distribution with parameter 1; this is a distribution on the infinitedimensional simplex of vectors (xv x2,...) having nonnegative components with unit sum. Using a standard continuity argument, this entails the weak convergence of A(/i) to the corresponding PoissonDirichlet distribution on this simplex; this result was obtained by Billingsley [3]. 1.
Integers, without large prime factors, in arithmetic progressions, II
"... : We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q 1+" and log x=log q ! 1 as y ! 1: this improves on previous estimates. y An Alfred P. Sloan R ..."
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Cited by 8 (1 self)
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: We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q 1+" and log x=log q ! 1 as y ! 1: this improves on previous estimates. y An Alfred P. Sloan Research Fellow. Supported, in part, by the National Science Foundation Integers, without large prime factors, in arithmetic progressions, II Andrew Granville 1. Introduction. The study of the distribution of integers with only small prime factors arises naturally in many areas of number theory; for example, in the study of large gaps between prime numbers, of values of character sums, of Fermat's Last Theorem, of the multiplicative group of integers modulo m, of Sunit equations, of Waring's problem, and of primality testing and factoring algorithms. For over sixty years this subject has received quite a lot of attention from analytic number theorists and we have recently begun to attain a very pre...
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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Cited by 3 (2 self)
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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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Cited by 3 (0 self)
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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.
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.
The Number of Relations in the Quadratic Sieve Algorithm
, 1996
"... The subject of our study is the single large prime variation of the quadratic sieve algorithm. We derive a formula for the average numbers of complete and incomplete relations per polynomial, directly generated by the algorithm. The number of additional complete relations from the incomplete relatio ..."
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Cited by 2 (0 self)
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The subject of our study is the single large prime variation of the quadratic sieve algorithm. We derive a formula for the average numbers of complete and incomplete relations per polynomial, directly generated by the algorithm. The number of additional complete relations from the incomplete relations is then computed by a known formula. Hence practical hints for the optimal choice of the parameter values can be derived. We further compare theoretical estimates for the total number of smooth integers in an interval with countings in practice. AMS Subject Classification (1991): 11A51, 11Y05 CR Subject Classification (1991): F.2.1 Keywords & Phrases: Factorization, Multiple Polynomial Quadratic Sieve, Vector supercomputer, Cluster of work stations 1. Introduction We assume that the reader is familiar with the multiple polynomial quadratic sieve algorithm [Bre89, Pom85, PST88, Sil87, RLW89]. We consider the single large prime variation of the algorithm and write MPQS for short. If we ...