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223
Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 162 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
Dbranes and short distances in string theory
"... We study the behavior of Dbranes at distances far shorter than the string length scale ls. We argue that shortdistance phenomena are described by the IR behavior of the Dbrane worldvolume quantum theory. This description is valid until the brane motion becomes relativistic. At weak string coupli ..."
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Cited by 122 (7 self)
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We study the behavior of Dbranes at distances far shorter than the string length scale ls. We argue that shortdistance phenomena are described by the IR behavior of the Dbrane worldvolume quantum theory. This description is valid until the brane motion becomes relativistic. At weak string coupling gs this corresponds to momenta and energies far above string scale. We use 0brane quantum mechanics to study 0brane collisions and find structure at length scales corresponding to the elevendimensional Planck length (l 11 P ∼ g1/3 s ls) and to the radius of the eleventh dimension in Mtheory (R11 ∼ gsls). We use 0branes to probe nontrivial geometries and topologies at substringy scales. We study the 0brane 4brane system, calculating the 0brane moduli space metric, and find the bound state at threshold, which has characteristic size l11 P. We examine the blowup of an orbifold and are able to resolve the resulting S2 down to size l11 P. A 0brane with momentum approaching 1/R11 is able to explore a larger configuration space in which the blowup is embedded. Analogous phenomena occur for small instantons. We finally turn to 1branes and calculate the size of a bound state to be ∼ g 1/2 s ls, the 1brane tension scale. August
Pairingbased Cryptography at High Security Levels
 Proceedings of Cryptography and Coding 2005, volume 3796 of LNCS
, 2005
"... Abstract. In recent years cryptographic protocols based on the Weil and Tate pairings on elliptic curves have attracted much attention. A notable success in this area was the elegant solution by Boneh and Franklin [7] of the problem of efficient identitybased encryption. At the same time, the secur ..."
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Cited by 78 (2 self)
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Abstract. In recent years cryptographic protocols based on the Weil and Tate pairings on elliptic curves have attracted much attention. A notable success in this area was the elegant solution by Boneh and Franklin [7] of the problem of efficient identitybased encryption. At the same time, the security standards for public key cryptosystems are expected to increase, so that in the future they will be capable of providing security equivalent to 128, 192, or 256bit AES keys. In this paper we examine the implications of heightened security needs for pairingbased cryptosystems. We first describe three different reasons why highsecurity users might have concerns about the longterm viability of these systems. However, in our view none of the risks inherent in pairingbased systems are sufficiently serious to warrant pulling them from the shelves. We next discuss two families of elliptic curves E for use in pairingbased cryptosystems. The first has the property that the pairing takes values in the prime field Fp over which the curve is defined; the second family consists of supersingular curves with embedding degree k = 2. Finally, we examine the efficiency of the Weil pairing as opposed to the Tate pairing and compare a range of choices of embedding degree k, including k = 1 and k = 24. Let E be the elliptic curve 1.
On the frequency of vanishing of quadratic twists of modular Lfunctions
 in Number theory for the millennium, I (Urbana, IL, 2000), 301–315, A K Peters
, 2002
"... Abstract. We present theoretical and numerical evidence for a random matrix theoretical approach to a conjecture about vanishings of quadratic twists of certain Lfunctions. In this paper we 1 present some evidence that methods from random matrix theory can give insight into the frequency of vanishi ..."
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Cited by 38 (15 self)
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Abstract. We present theoretical and numerical evidence for a random matrix theoretical approach to a conjecture about vanishings of quadratic twists of certain Lfunctions. In this paper we 1 present some evidence that methods from random matrix theory can give insight into the frequency of vanishing for quadratic twists of modular Lfunctions. The central question is the following: given a holomorphic newform f with integral coefficients and associated Lfunction Lf(s), for how many fundamental discriminants d with d  ≤ x, does Lf(s, χd), the Lfunction twisted by the real, primitive, Dirichlet character associated with the discriminant d, vanish at the center of the critical strip to order at least 2? This question is of particular interest in the case that the Lfunction is associated with an elliptic curve, in light of the conjecture of Birch and SwinnertonDyer. This case corresponds to weight k = 2. We will focus on this case for most of the paper, though we do make some remarks about higher weights (see (26) and below). Suppose that E/Q is an elliptic curve with associated Lfunction (1) LE(s) = for ℜs> 1. Then, as a consequence of the TaniyamaShimura conjecture, recently solved by Wiles, Taylor, ([W], [TW]), and Breuil, Conrad, and Diamond, LE is entire and satisfies a functional equation n=1 a ∗ n n s
Reach Set Computations Using Real Quantifier Elimination
, 2000
"... Reach set computations are of fundamental importance in control theory. We consider the reach set problem for openloop systems described by parametric inhomogeneous linear dierential systems and use real quanti er elimination methods to get exact and approximate solutions. For simple elementar ..."
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Cited by 33 (1 self)
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Reach set computations are of fundamental importance in control theory. We consider the reach set problem for openloop systems described by parametric inhomogeneous linear dierential systems and use real quanti er elimination methods to get exact and approximate solutions. For simple elementary functions we give an exact calculation of the cases where exact semialgebraic transcendental implicitization is possible. For the negative cases we provide approximate alternating using discrete point checking or safe estimations of reach sets and control parameter sets. The method employs a reduction of forward and backward reach set and control parameter set problem to the transcendental implicitization problem for the components of special solutions of simpler nonparametric systems. Numerous examples are computed using the redlog and qepcad packages.
Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary
 Sém. Lotharingien de Combinatoire 42 (1999), 63 pp.; in The Andrews Festschrift
, 2001
"... When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s taufunction τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising mat ..."
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Cited by 26 (13 self)
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When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s taufunction τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising material for insertion in the
Nonvanishing of quadratic twists of modular Lfunctions
 Invent. Math
, 1998
"... If F (z) = ∑ ∞ n=1 a(n)qn ∈ S2k(Γ0(M), χ0) (note: q: = e 2πiz throughout) is a newform of even integer weight 2k with trivial character χ0, then let L(F, s) be its Lfunction ..."
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Cited by 25 (5 self)
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If F (z) = ∑ ∞ n=1 a(n)qn ∈ S2k(Γ0(M), χ0) (note: q: = e 2πiz throughout) is a newform of even integer weight 2k with trivial character χ0, then let L(F, s) be its Lfunction
Relations between the ranks and the cranks of partitions
 RAMANUJAN J
"... New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions. ..."
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Cited by 22 (6 self)
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New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.
Distribution Of The Partition Function Modulo Composite Integers M
 M, MATH. ANNALEN
, 2000
"... ..."
Crossing probabilities and modular forms
 J. Statist. Phys
"... Abstract. We examine crossing probabilities and free energies for conformally invariant critical 2D systems in rectangular geometries, derived via conformal field theory and Stochastic Löwner Evolution methods. These quantities are shown to exhibit interesting modular behavior, although the physica ..."
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Cited by 20 (1 self)
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Abstract. We examine crossing probabilities and free energies for conformally invariant critical 2D systems in rectangular geometries, derived via conformal field theory and Stochastic Löwner Evolution methods. These quantities are shown to exhibit interesting modular behavior, although the physical meaning of modular transformations in this context is not clear. We show that in many cases these functions are completely characterized by very simple transformation properties. In particular, Cardy’s function (including the conformal dimension 1/3), follows from a simple modular argument. A new type of “higherorder modular form ” arises and its properties are discussed briefly. 1. Introduction. There