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On the discrete logarithm problem in elliptic curves

Part of: Arithmetic problems. Diophantine geometry Computational number theory

Published online by Cambridge University Press:  15 October 2010

Claus Diem
Claus Diem*
Affiliation:
University of Leipzig, Mathematical Institute, Johannisgasse 26, 04103 Leipzig, Germany (email: [email protected])
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Abstract

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We study the elliptic curve discrete logarithm problem over finite extension fields. We show that for any sequences of prime powers (qi)i∈ℕ and natural numbers (ni)i∈ℕ with ni⟶∞ and ni/log (qi)⟶0 for i⟶∞, the elliptic curve discrete logarithm problem restricted to curves over the fields 𝔽qnii can be solved in subexponential expected time (qnii)o(1). We also show that there exists a sequence of prime powers (qi)i∈ℕ such that the problem restricted to curves over 𝔽qi can be solved in an expected time of e𝒪(log (qi)2/3).

Keywords

elliptic curvesdiscrete logarithm problem

MSC classification

Secondary: 11Y16: Algorithms; complexity 14G15: Finite ground fields 14G50: Applications to coding theory and cryptography
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 1 , January 2011 , pp. 75 - 104
Copyright
Copyright © Foundation Compositio Mathematica 2010

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