On maximal curves over finite fields of small order

On maximal curves over finite fields of small order

On the order of points on curves over finite fields

We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We prove that for points on a plane curve, one of the coordinates has to have high order. We also discuss a conjecture of Poonen for subvarieties of semiabelian varieties for which our result is a weak special case. Finally, we look at some special cases where we...

On Curves over Finite Fields with Jacobians of Small Exponent

We show that finite fields over which there is a curve of a given genus g ≥ 1 with its Jacobian having a small exponent, are very rare. This extends a recent result of W. Duke in the case g = 1. We also show when g = 1 or g = 2 that our bounds are best possible.

On counting and generating curves over small finite fields

We consider curves defined over small finite fields with points of large prime order over an extension field. Such curves are often referred to as Koblitz curves and are of considerable cryptographic interest. An interesting question is whether such curves are easy to construct as the target point order grows asymptotically. We show that under certain number theoretic conjecture, if q is a prim...

Curves of Genus 3 over Small Finite Fields