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374
Identity Based Authenticated Key Agreement Protocols from Pairings
 In: Proc. 16th IEEE Security Foundations Workshop
, 2002
"... We investigate a number of issues related to identity based authenticated key agreement protocols in the DiffieHellman family enabled by the Weil or Tate pairings. These issues include how to make protocols efficient; to avoid key escrow by a Trust Authority (TA) who issues identity based private k ..."
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Cited by 65 (2 self)
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We investigate a number of issues related to identity based authenticated key agreement protocols in the DiffieHellman family enabled by the Weil or Tate pairings. These issues include how to make protocols efficient; to avoid key escrow by a Trust Authority (TA) who issues identity based private keys for users, and to allow users to use different TAs. We describe a few authenticated key agreement (AK) protocols and AK with key confirmation (AKC) protocols by modifying Smart's AK protocol [Sm02]. We discuss the security of these protocols heuristically and give formal proofs of security for our AK and AKC protocols (using a security model based on the model defined in [BJM97]). We also prove that our AK protocol has the key compromise impersonation property. We also show that our second protocol has the TA forward secrecy property (which we define to mean that the compromise of the TA's private key will not compromise previously established session keys), and we note that this also implies that it has the perfect forward secrecy property.
Lowlying zeros of families of elliptic curves
, 2006
"... There is a growing body of evidence giving strong evidence that zeros of families of Lfunctions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the KatzSarnak philosophy. The random matrix model makes predictions for the average di ..."
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Cited by 59 (2 self)
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There is a growing body of evidence giving strong evidence that zeros of families of Lfunctions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the KatzSarnak philosophy. The random matrix model makes predictions for the average distribution of zeros near the central point for families of Lfunctions. We study the lowlying zeros for families of elliptic curve Lfunctions. For these Lfunctions there is special arithmetic interest in any zeros at the central point (by the conjecture of Birch and SwinnertonDyer and the impressive partial results towards resolving the conjecture). We calculate the density of the lowlying zeros for various families of elliptic curves. Our main foci are the family of all elliptic curves and a large family with positive rank. A main challenge has been to obtain results with test functions that are concentrated close to the origin since the central point is a location of great interest. An application is an improvement on the upper bound of the average rank of the family of all elliptic curves. We show that there is an extra contribution to the density of the lowlying zeros from the family with positive rank (presumably from the “extra ” zero at the central point). 1
Equidistribution of small points, rational dynamics, and potential theory
 Ann. Inst. Fourier (Grenoble
, 2006
"... Abstract. Given a dynamical system associated to a rational function ϕ(T) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µϕ,v on the Berkovich space P 1 Berk,v /Cv such that if {zn} is a sequence of points ..."
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Cited by 47 (7 self)
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Abstract. Given a dynamical system associated to a rational function ϕ(T) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µϕ,v on the Berkovich space P 1 Berk,v /Cv such that if {zn} is a sequence of points in P 1 (k) whose ϕcanonical heights tend to zero, then the zn’s and their Galois conjugates are equidistributed with respect to µϕ,v. In the archimedean case, µϕ,v coincides with the wellknown canonical measure associated to ϕ. This theorem generalizes a result of BakerHsia [BH] when ϕ(z) is a polynomial. The proof uses a polynomial lift F (x, y) = (F1(x, y), F2(x, y)) of ϕ to construct a twovariable ArakelovGreen’s function gϕ,v(x, y) for each v. The measure µϕ,v is obtained by taking the Berkovich space Laplacian of gϕ,v(x, y), using a theory developed in [RB]. The other ingredients in the proof are (i) a potentialtheoretic energy minimization principle which says that � � gϕ,v(x, y) dν(x)dν(y) is uniquely minimized over all probability measures ν on P 1 Berk,v when ν = µϕ,v, and (ii) a formula for homogeneous transfinite diameter of the vadic filled Julia set KF,v ⊂ C 2 v in terms of the resultant Res(F) of F1 and F2. The resultant formula, which generalizes a formula of DeMarco [DeM], is proved using results
Primes In Elliptic Divisibility Sequences
"... Morgan Ward pursued the study of elliptic divisibility sequences initiated by Lucas, and Chudnovsky and Chudnovsky suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here from a theoretical and a practical vie ..."
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Cited by 46 (13 self)
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Morgan Ward pursued the study of elliptic divisibility sequences initiated by Lucas, and Chudnovsky and Chudnovsky suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here from a theoretical and a practical viewpoint. We exhibit calculations, together with a heuristic argument, to suggest that these sequences contain only finitely many primes.
Bounding the Number of Rational Points on Certain Curves of High Rank
, 1997
"... Let K be a number eld and let C be a curve of genus g > 1 dened over K. In this dissertation we describe techniques for bounding the number of Krational points on C. In Chapter I we discuss Chabauty techniques. This is a review and synthesis of previously known material, both published and unpu ..."
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Cited by 38 (2 self)
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Let K be a number eld and let C be a curve of genus g > 1 dened over K. In this dissertation we describe techniques for bounding the number of Krational points on C. In Chapter I we discuss Chabauty techniques. This is a review and synthesis of previously known material, both published and unpublished. We have tried to eliminate unnecessary restrictions, such as assumptions of good reduction or the existence of a known rational point on the curve. We have also attempted to clearly state the circumstances under which Chabauty techniques can be applied. Our primary goal is to provide a exible and powerful tool for computing on specic curves. In Chapter II we develop a technique which, given a Krational isogeny to the Jacobian of C, produces a positive integer n and a collection of covers of C with the property that the set of Krational points in the collection is in nto1 correspondence with the set of Krational points on C. If Chabauty is applicable to every curve in the collection, then we can use the covers to bound the number of Krational points on C. The examples in Chapters I and II are taken from problem VI.17 in the Arabic text of the Arithmetica. Chapter III is devoted to the background calculations for this problem. When we assemble the pieces, we discover that the solution given by Diophantus is the only positive rational solution to this problem. Contents 1. Preface 4 Chapter 1. Chabauty bounds 5 1.
Ordinary abelian varieties having small embedding degree
 IN PROC. WORKSHOP ON MATHEMATICAL PROBLEMS AND TECHNIQUES IN CRYPTOLOGY
, 2004
"... Miyaji, Nakabayashi and Takano (MNT) gave families of group orders of ordinary elliptic curves with embedding degree suitable for pairing applications. In this paper we generalise their results by giving families corresponding to nonprime group orders. We also consider the case of ordinary abelia ..."
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Cited by 38 (1 self)
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Miyaji, Nakabayashi and Takano (MNT) gave families of group orders of ordinary elliptic curves with embedding degree suitable for pairing applications. In this paper we generalise their results by giving families corresponding to nonprime group orders. We also consider the case of ordinary abelian varieties of dimension 2. We give families of group orders with embedding degrees 5, 10 and 12.
Ranks of twists of elliptic curves and Hilbert’s tenth problem, arxiv:0904.3709v2 [math.NT
"... Abstract. In this paper we investigate the 2Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2Selmer rank, and we give lower bounds for the number of twists (with bound ..."
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Cited by 35 (4 self)
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Abstract. In this paper we investigate the 2Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial MordellWeil group, and (assuming the ShafarevichTate conjecture) many others with infinite cyclic MordellWeil group. Using work of Poonen and Shlapentokh, it follows from our results that if the ShafarevichTate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field. 1. Introduction and
Independence of rational points on twists of a given curve, to appear
 in Compositio Math. arXiv: math.NT/0603557 School of Engineering and Science, International University Bremen, P.O.Box 750561, 28725
"... Abstract. In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of Krational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly g ..."
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Cited by 30 (12 self)
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Abstract. In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of Krational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J ′ (K) and c is a constant depending on C. For the proof, we use a refinement of the method of ChabautyColeman; the main new ingredient is to use it for an extension field of Kv, where v is a place of bad reduction for C ′. 1.