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49
Solving the Pell Equation
, 2008
"... We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly a ..."
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We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly accelerated. The most promising method depends on the use of smooth numbers. As with many algorithms depending on smooth numbers, its run time can presently only conjecturally be established; giving a rigorous analysis is one of the many open problems surrounding the Pell equation.
Computing the endomorphism ring of an ordinary elliptic curve over a finite field
 Journal of Number Theory
"... Abstract. We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the se ..."
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Cited by 26 (12 self)
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Abstract. We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the second algorithm depends primarily on log DE, where DE is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed. 1.
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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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
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
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EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS
, 2013
"... Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments invol ..."
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Cited by 12 (1 self)
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Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant. Contents
Smooth orders and cryptographic applications
 Lect. Notes in Comp. Sci
"... Abstract. We obtain rigorous upper bounds on the number of primes p ≤ x for which p−1 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 ..."
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Cited by 9 (2 self)
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Abstract. We obtain rigorous upper bounds on the number of primes p ≤ x for which p−1 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. 1
On values taken by the largest prime factor of shifted primes
 J. Aust. Math. Soc
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Another generalization of Wiener’s attack on RSA
 Africacrypt 2008. LNCS
, 2008
"... Abstract. A wellknown attack on RSA with low secretexponent d was given by Wiener in 1990. Wiener showed that using the equation ed − (p − 1)(q − 1)k = 1 and continued fractions, one can efficiently recover the secretexponent d and factor N = pq from the public key (N, e) as long as d < 1 3 N ..."
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Cited by 5 (5 self)
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Abstract. A wellknown attack on RSA with low secretexponent d was given by Wiener in 1990. Wiener showed that using the equation ed − (p − 1)(q − 1)k = 1 and continued fractions, one can efficiently recover the secretexponent d and factor N = pq from the public key (N, e) as long as d < 1 3 N 1 4. In this paper, we present a generalization of Wiener’s attack. We show that every public exponent e that satisfies eX − (p − u)(q − v)Y = 1 with 1 ≤ Y < X < 2 − 1 4 N 1 4, u  < N 1 [ 4, v = − qu p − u and all prime factors of p − u or q − v are less than 10 50 yields the factorization of N = pq. We show that the number of these exponents is at least N 1 2 −ε.
Nicolaas Govert de Bruijn, the enchanter of friable integers
 Indagationes Math
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