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32
Constructing Elliptic Curves with Prescribed Embedding Degrees
, 2002
"... Pairingbased cryptosystems depend on the existence of groups where the Decision DiffieHellman problem is easy to solve, but the Computational DiffieHellman problem is hard. Such is the case of elliptic curve groups whose embedding degree is large enough to maintain a good security level, but smal ..."
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Cited by 51 (16 self)
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Pairingbased cryptosystems depend on the existence of groups where the Decision DiffieHellman problem is easy to solve, but the Computational DiffieHellman problem is hard. Such is the case of elliptic curve groups whose embedding degree is large enough to maintain a good security level, but small enough for arithmetic operations to be feasible. However, the embedding degree is usually enormous, and the scarce previously known suitable elliptic groups had embedding degree k <= 6. In this note, we examine criteria for curves with larger k that generalize prior work by Miyaji et al. based on the properties of cyclotomic polynomials, and propose efficient representations for the underlying algebraic structures.
Explicit 4descents on an elliptic curve
 Acta Arith
, 1996
"... Abstract. It is shown that the obvious method of descending from an element of the 2Selmer group of an elliptic curve, E, will indeed give elements of order 1, 2 or 4 in the WeilChatelet group of E. Explicit algorithms for such a method are given. 1. ..."
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Cited by 20 (3 self)
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Abstract. It is shown that the obvious method of descending from an element of the 2Selmer group of an elliptic curve, E, will indeed give elements of order 1, 2 or 4 in the WeilChatelet group of E. Explicit algorithms for such a method are given. 1.
Primitive divisors of elliptic divisibility sequences
 J. Number Theory
"... Abstract. Silverman proved the analogue of Zsigmondy’s Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the ..."
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Cited by 15 (7 self)
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Abstract. Silverman proved the analogue of Zsigmondy’s Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. 1.
PRIME POWERS IN ELLIPTIC DIVISIBILITY SEQUENCES
, 2005
"... Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In some cases the methods prove that only finitely many terms are divisible by a bounded number of distinct primes. ..."
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Cited by 10 (7 self)
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Certain elliptic divisibility sequences are shown to contain only finitely many prime power terms. In some cases the methods prove that only finitely many terms are divisible by a bounded number of distinct primes.
On Sums of Consecutive Squares
 J. Number Th
, 1997
"... In this paper we consider the problem of characterizing those perfect squares that can be expressed as the sum of consecutive squares where the initial term in this sum is k 2 . This problem is intimately related to that of finding all integral points on elliptic curves belonging to a certain fami ..."
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Cited by 8 (4 self)
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In this paper we consider the problem of characterizing those perfect squares that can be expressed as the sum of consecutive squares where the initial term in this sum is k 2 . This problem is intimately related to that of finding all integral points on elliptic curves belonging to a certain family which can be represented by a Weierstraß equation with parameter k. All curves in this family have positive rank, and for those of rank 1 a most likely candidate generator of infinite order can be explicitly given in terms of k. We conjecture that this point indeed generates the free part of the MordellWeil group, and give some heuristics to back this up. We also show that a point which is modulo torsion equal to a nontrivial multiple of this conjectured generator cannot be integral. For k in the range 1 k 100 the corresponding curves are closely examined, all integral points are determined and all solutions to the original problem are listed. It is worth mentioning that all curves of ...
Solving Elliptic Diophantine Equations: The General Cubic Case
 Acta Arith
, 1999
"... In this paper we consider binary cubic diophantine equations of every form and shape, solely subjected to the requirement that they represent elliptic curves defined over Q . How to deal with standard Weierstraß equations is well understood, but comparatively little is known about elliptic equations ..."
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Cited by 8 (5 self)
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In this paper we consider binary cubic diophantine equations of every form and shape, solely subjected to the requirement that they represent elliptic curves defined over Q . How to deal with standard Weierstraß equations is well understood, but comparatively little is known about elliptic equations of different type. We give a detailed analysis of the general situation and subsequently apply the elliptic logarithm method to a variety of unusual elliptic equations, notably to some equations directly related to Krawtchouk polynomials. 1991 Mathematics subject classification: 11D25, 11G05, 11Y50, 12D10 Key words and phrases: cubic diophantine equation, elliptic curve, elliptic logarithm, LLLreduction, binary Krawtchouk polynomial Econometric Institute, Erasmus University, P.O.Box 1738, 3000 DR Rotterdam, The Netherlands, email: stroeker@few.eur.nl, internet URL of my homepage: http://www.few.eur.nl/few/people/stroeker/ y Sportsingel 30, 2924 XN Krimpen aan den IJssel, The Netherla...
Descents on Curves of Genus 1
, 1995
"... This thesis is available for Library use on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. I certify that all the material in this thesis which is not my own work has been clearly identified and that no material ..."
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Cited by 7 (4 self)
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This thesis is available for Library use on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. I certify that all the material in this thesis which is not my own work has been clearly identified and that no material is included for which a degree has previously been conferred upon me.
ELLIPTIC BINOMIAL DIOPHANTINE EQUATIONS
, 1999
"... The complete sets of solutions of the equation ( n) ( m) = are k ℓ ..."
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Cited by 6 (4 self)
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The complete sets of solutions of the equation ( n) ( m) = are k ℓ
Multiples of integral points on elliptic curves
 J. Number Theory
"... Abstract. We show that if Lang’s conjectured lower bound on heights of points on elliptic curves exists, then there is an absolute constant C such that for any integral point P on a minimal elliptic curve with integral coefficients, nP is integral for at most one value of n> C. As a corollary, we sh ..."
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Cited by 4 (1 self)
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Abstract. We show that if Lang’s conjectured lower bound on heights of points on elliptic curves exists, then there is an absolute constant C such that for any integral point P on a minimal elliptic curve with integral coefficients, nP is integral for at most one value of n> C. As a corollary, we show that if E/Q is a fixed elliptic curve, then for all twists E ′ of E of sufficient height, and all torsionfree, rankone subgroups Γ ⊆ E ′ (Q), Γ contains at most 6 integral points. Explicit computations for congruent number curves are included.
Effective solution of two simultaneous Pell equations by the Elliptic Logarithm Method
"... this paper is to present a hopefully practical and uniform general method 1 for solving explicitly any specic system of two Pell equations ..."
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Cited by 3 (0 self)
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this paper is to present a hopefully practical and uniform general method 1 for solving explicitly any specic system of two Pell equations