Vol. 7, No. 6, 2013

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On the discrete logarithm problem in elliptic curves II

Claus Diem

Vol. 7 (2013), No. 6, 1281–1323
Abstract

We continue our study on the elliptic curve discrete logarithm problem over finite extension fields. We show, among others, the following results:

For sequences of prime powers (qi)i and natural numbers (ni)i with ni and nilog(qi)2 0 for i , the discrete logarithm problem in the groups of rational points of elliptic curves over the fields Fqini can be solved in subexponential expected time (qini)o(1).

Let a, b > 0 be fixed. Then the problem over fields Fqn, where q is a prime power and n a natural number with a log(q)13 n b log(q), can be solved in an expected time of eO(log(qn)34) .

Keywords
elliptic curves, discrete logarithm problem
Mathematical Subject Classification 2010
Primary: 11Y16
Secondary: 14H52, 11G20
Milestones
Received: 28 July 2011
Revised: 12 June 2012
Accepted: 15 July 2012
Published: 19 September 2013
Authors
Claus Diem
Mathematical Institute
University of Leipzig
Augustusplatz 10
D-04109 Leipzig
Germany