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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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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
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
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A crossprotocol attack on the TLS protocol
 In ACM CCS
, 2012
"... This paper describes a crossprotocol attack on all versions of TLS; it can be seen as an extension of the Wagner and Schneier attack on SSL 3.0. The attack presents valid explicit elliptic curve DiffieHellman parameters signed by a server to a client that incorrectly interprets these parameters as ..."
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This paper describes a crossprotocol attack on all versions of TLS; it can be seen as an extension of the Wagner and Schneier attack on SSL 3.0. The attack presents valid explicit elliptic curve DiffieHellman parameters signed by a server to a client that incorrectly interprets these parameters as valid plain DiffieHellman parameters. Our attack enables an adversary to successfully impersonate a server to a random client after obtaining 2 40 signed elliptic curve keys from the original server. While attacking a specific client is improbable due to the high number of signed keys required during the lifetime of one TLS handshake, it is not completely unrealistic for a setting where the server has high computational power and the attacker contents itself with recovering one out of many session keys. We remark that popular opensource server implementations are not susceptible to this attack, since they typically do not support the explicit curve option. Finally we propose a fix that renders the protocol immune to this family of crossprotocol attacks. Categories andSubjectDescriptors
Average running time of the fast Fourier transform
 J. Algorithms
, 1980
"... We compare several algorithms for computing the discrete Fourier transform of n numbers. The number of “operations ” of the original CooleyTukey algorithm is approximately 2n A(n), where A(n) is the sum of the prime divisors of n. We show that the average number of operations satisfies (l/x)Z,,,2n ..."
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Cited by 6 (1 self)
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We compare several algorithms for computing the discrete Fourier transform of n numbers. The number of “operations ” of the original CooleyTukey algorithm is approximately 2n A(n), where A(n) is the sum of the prime divisors of n. We show that the average number of operations satisfies (l/x)Z,,,2n A(n)(n2/9)(x2/log x). The average is not a good indication of the number of operations. For example, it is shown that for about half of the integers n less than x, the number of “operations ” is less than n i 61. A similar analysis is given for Good’s algorithm and for two algorithms that compute the discrete Fourier transform in O(n log n) operations: the chirpz transform and the mixedradix algorithm that computes the transform of a series of prime length p in O(p log p) operations. 1.
The twoparameter PoissonDirichlet point process
, 2007
"... The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtai ..."
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The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Relying on this, we apply the theory of point processes to reveal mathematical structure of the twoparameter PoissonDirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we will be able to extend several results previously known for the oneparameter case, and the MarkovKrein identity for the generalized Dirichlet process is discussed from a point of view of functional analysis based on the twoparameter PoissonDirichlet distribution. 1
On Quadratic Polynomials for the Number Field Sieve
 Australian Computer Science Communications
, 1997
"... . The newest, and asymptotically the fastest known integer factorisation algorithm is the number field sieve. The area in which the number field sieve has the greatest capacity for improvement is polynomial selection. The best known polynomial selection method finds quadratic polynomials. In this pa ..."
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. The newest, and asymptotically the fastest known integer factorisation algorithm is the number field sieve. The area in which the number field sieve has the greatest capacity for improvement is polynomial selection. The best known polynomial selection method finds quadratic polynomials. In this paper we examine the smoothness properties of integer values taken by these polynomials. Given a quadratic NFS polynomial f , let \Delta be its discriminant. We show that a prime p can divide values taken by f only if (\Delta=p) = 1. We measure the effect of this residuosity property on the smoothness of fvalues by adapting a parameter ff, developed for analysis of MPQS, to quadratic NFS polynomials. We estimate the yield of smooth values for these polynomials as a function of ff, and conclude that practical changes in ff might bring significant changes in the yield of smooth and almost smooth polynomial values. Keywords: integer factorisation, number field sieve 1
Improved CRT algorithm for class polynomials in genus 2.” In: Algorithmic Number Theory — ANTSX. Edited by Everett Howe and Kiran Kedlaya
 Mathematical Science Publishers
"... Abstract. We present a generalization to genus 2 of the probabilistic algorithm in Sutherland [28] for computing Hilbert class polynomials. The improvement over the algorithm presented in [5] for the genus 2 case, is that we do not need to find a curve in the isogeny class with endomorphism ring whi ..."
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Abstract. We present a generalization to genus 2 of the probabilistic algorithm in Sutherland [28] for computing Hilbert class polynomials. The improvement over the algorithm presented in [5] for the genus 2 case, is that we do not need to find a curve in the isogeny class with endomorphism ring which is the maximal order: rather we present a probabilistic algorithm for “going up ” to a maximal curve (a curve with maximal endomorphism ring), once we find any curve in the right isogeny class. Then we use the structure of the Shimura class group and the computation of (ℓ, ℓ)isogenies to compute all isogenous maximal curves from an initial one. This article is an extended version of the version published at ANTS X. 1.
On the amount of dependence in the prime factorization of a uniform random integer
 In Contemporary Combinatorics
, 2002
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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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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.
On the discrete logarithm problem in finite fields of fixed characteristic
"... Abstract. For q a prime power, the discrete logarithm problem (DLP) in F×q consists in finding, for any g ∈ F×q and h ∈ 〈g〉, an integer x such that g x = h. For each prime p we exhibit infinitely many extension fields Fpn for which the DLP in F pn can be solved in expected quasipolynomial time. 1 ..."
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Abstract. For q a prime power, the discrete logarithm problem (DLP) in F×q consists in finding, for any g ∈ F×q and h ∈ 〈g〉, an integer x such that g x = h. For each prime p we exhibit infinitely many extension fields Fpn for which the DLP in F pn can be solved in expected quasipolynomial time. 1