Generators of Elliptic Curves over Finite Fields
by Jose Felipe Voloch   Igor E. Shparlinski  

Vol. 9 No. 4 (2014) P.657~P.670

ABSTRACT

We prove estimates on character sums on the subset of points of an elliptic curve over $F_{q^n}$ with $x$-coordinate of the form $\alpha + t$ where $t \in F_q$ varies and fixed $\alpha$ is such that $F_{q^n} = F_q(\alpha)$. We deduce that, for a suitable choice of $\alpha$, this subset has a point of maximal order in $E(F_{q^n})$. This provides a deterministic algorithm for finding a point of maximal order which for a very wide class of finite fields is faster than other available algorithms.

KEYWORDS
Elliptic curves, generators, finite fields.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 11G20,11Y16, 11T23.

MILESTONES

Received: 2013-11-25
Revised : 2014-06-06
Accepted: 2014-06-06

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