Algebra & Number Theory

On the discrete logarithm problem in elliptic curves II

Claus Diem

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 $(qi)i∈ℕ$ and natural numbers $(ni)i∈ℕ$ with $ni→∞$ and $ni∕log(qi)2→0$ 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 $a⋅ log(q)1∕3≤n≤b⋅ log(q)$, can be solved in an expected time of $eO(log(qn)3∕4)$.

Article information

Source
Algebra Number Theory, Volume 7, Number 6 (2013), 1281-1323.

Dates
Revised: 12 June 2012
Accepted: 15 July 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730029

Digital Object Identifier
doi:10.2140/ant.2013.7.1281

Mathematical Reviews number (MathSciNet)
MR3107564

Zentralblatt MATH identifier
1300.11132

Citation

Diem, Claus. On the discrete logarithm problem in elliptic curves II. Algebra Number Theory 7 (2013), no. 6, 1281--1323. doi:10.2140/ant.2013.7.1281. https://projecteuclid.org/euclid.ant/1513730029

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