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399
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 202 (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 147 (8 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 92 (3 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 49 (16 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
Fourier Coefficients of Half Integral Weight Modular Forms Modulo l
 Annals of Mathematics
, 1998
"... Dedicated to the memory of S. Chowla. ..."
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 46 (19 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
Defect Zero pBlocks for Finite SIMPLE GROUPS
 Trans. Amer. Math. Soc
, 1996
"... . We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a pblock with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p\Gammablocks remained unclassified were the alternating groups An . Here we show that these ..."
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Cited by 46 (5 self)
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. We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a pblock with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p\Gammablocks remained unclassified were the alternating groups An . Here we show that these all have a pblock with defect 0 for every prime p 5. This follows from proving the same result for every symmetric group Sn , which in turn follows as a consequence of the tcore partition conjecture, that every nonnegative integer possesses at least one tcore partition, for any t 4. For t 17, we reduce this problem to Lagrange's Theorem that every nonnegative integer can be written as the sum of four squares. The only case with t ! 17, that was not covered in previous work, was the case t = 13. This we prove with a very different argument, by interpreting the generating function for tcore partitions in terms of modular forms, and then controlling the size of the coefficients usin...
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 43 (8 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.
Periodic cohomology theories defined by elliptic curves
 Contemp. Math
, 1995
"... Abstract. We use bordism theory to construct periodic cohomology theories, which we call elliptic cohomology, for which the cohomology of a point is a ring of modular functions. These are complexoriented multiplicative cohomology theories, with formal groups associated to the universal elliptic g ..."
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Cited by 43 (2 self)
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Abstract. We use bordism theory to construct periodic cohomology theories, which we call elliptic cohomology, for which the cohomology of a point is a ring of modular functions. These are complexoriented multiplicative cohomology theories, with formal groups associated to the universal elliptic genus studied by a number of authors ([CC, LS, O, W1, Z]). We are unable to find a geometric description for these theories. 1.
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 41 (8 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