Generating Prime Order Elliptic Curves: Difficulties and Efficiency Considerations

2005 ◽  
pp. 261-278 ◽  
Author(s):  
Elisavet Konstantinou ◽  
Aristides Kontogeorgis ◽  
Yannis C. Stamatiou ◽  
Christos Zaroliagis
Keyword(s):  
Elliptic Curves ◽  
Prime Order

Elliptic Curves of Nearly Prime Order

2021 ◽  
pp. 923-932
Author(s):  
Daniele Di Tullio ◽  
Manoj Gyawali
Keyword(s):  
Elliptic Curves ◽  
Prime Order

Solving the Multi–discrete Logarithm Problems over a Group of Elliptic Curves with Prime Order

2005 ◽  
Vol 21 (6) ◽  
pp. 1443-1450 ◽  
Author(s):  
Jun Quan Li ◽  
Mu Lan Liu ◽  
Liang Liang Xiao
Keyword(s):  
Elliptic Curves ◽  
Prime Order ◽  
Discrete Logarithm

On the Efficient Generation of Prime-Order Elliptic Curves

2009 ◽  
Vol 23 (3) ◽  
pp. 477-503 ◽  
Author(s):  
Elisavet Konstantinou ◽  
Aristides Kontogeorgis ◽  
Yannis C Stamatiou ◽  
Christos Zaroliagis
Keyword(s):  
Elliptic Curves ◽  
Prime Order ◽  
Efficient Generation

scholarly journals Pairing-Friendly Elliptic Curves of Prime Order

2006 ◽  
pp. 319-331 ◽  
Author(s):  
Paulo S. L. M. Barreto ◽  
Michael Naehrig
Keyword(s):  
Elliptic Curves ◽  
Prime Order

Isolated elliptic curves and the MOV attack

2017 ◽  
Vol 11 (3) ◽  
Author(s):  
Travis Scholl
Keyword(s):  
Elliptic Curves ◽  
Prime Order ◽  
Prime Fields ◽  
Embedding Degree

AbstractWe present a variation on the CM method that produces elliptic curves over prime fields with nearly prime order that do not admit many efficiently computable isogenies. Assuming the Bateman–Horn conjecture, we prove that elliptic curves produced this way almost always have a large embedding degree, and thus are resistant to the MOV attack on the ECDLP.

scholarly journals On the Construction of Prime Order Elliptic Curves

2003 ◽  
pp. 309-322 ◽  
Author(s):  
Elisavet Konstantinou ◽  
Yannis C. Stamatiou ◽  
Christos Zaroliagis
Keyword(s):  
Elliptic Curves ◽  
Prime Order

RATIONAL TORSION ON OPTIMAL CURVES

2005 ◽  
Vol 01 (04) ◽  
pp. 513-531 ◽  
Author(s):  
NEIL DUMMIGAN
Keyword(s):  
Elliptic Curves ◽  
Prime Order ◽  
Rational Points ◽  
Root Number ◽  
Hecke Eigenforms ◽  
Modular Parametrization ◽  
Local Root

Vatsal has proved recently a result which has consequences for the existence of rational points of odd prime order ℓ on optimal elliptic curves over ℚ. When the conductor N is squarefree, ℓ ∤ N and the local root number wp= -1 for at least one prime p | N, we offer a somewhat different proof, starting from an explicit cuspidal divisor on X0(N). We also prove some results linking the vanishing of L(E,1) with the divisibility by ℓ of the modular parametrization degree, fitting well with the Bloch–Kato conjecture for L( Sym2E,2), and with an earlier construction of elements in Shafarevich–Tate groups. Finally (following Faltings and Jordan) we prove an analogue of the result on ℓ-torsion for cuspidal Hecke eigenforms of level one (and higher weight), thereby strengthening some existing evidence for another case of the Bloch–Kato conjecture.

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