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Average twin prime conjecture for elliptic curves
, 2007
"... Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy and Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s co ..."
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Cited by 7 (3 self)
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Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy and Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s conjecture is still widely open. In this paper we prove that Koblitz’s conjecture is true on average over a twoparameter family of elliptic curves. One of the key ingredients in the proof is a short average distribution result in the style of BarbanDavenportHalberstam,
Reductions of an elliptic curve with almost prime orders
"... 1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exis ..."
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Cited by 3 (0 self)
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1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exist infinitely many primes p for which #Ep(Fp) has at most 5 prime factors. We also obtain upper bounds for the number of primes p ≤ x for which #Ep(Fp) is a prime. 1
Almost prime values of the order of elliptic curves over finite fields
, 2008
"... Abstract. Let E be an elliptic curve over Q without complex multiplication, and which is not isogenous to a curve with nontrivial rational torsion. For each prime p of good reduction, let E(Fp)  be the order of the group of points of the reduced curve over Fp. We prove in this paper that, under t ..."
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Cited by 2 (1 self)
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Abstract. Let E be an elliptic curve over Q without complex multiplication, and which is not isogenous to a curve with nontrivial rational torsion. For each prime p of good reduction, let E(Fp)  be the order of the group of points of the reduced curve over Fp. We prove in this paper that, under the GRH, there are at least 2.778Ctwin E x/(log x)2 primes p such that E(Fp)  has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [18] and Murty & Miri [13]. This is also the first result where the dependence on the conjectural constant Ctwin E appearing in the twin prime conjecture for elliptic curves (also known as Koblitz’s conjecture) is made explicit. This is achieved by sieving a slightly different sequence than the one of [18] and [13]. By sieving the same sequence and using Selberg’s linear sieve, we can also improve the constant of Zywina [22] appearing in the upper bound for the number of primes p such that E(Fp)  is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH. 1.
Divisibility, Smoothness and Cryptographic Applications
, 2008
"... This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in var ..."
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This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth numbers play a pivotal role. 1 1