Counting hyperelliptic curves that admit a Koblitz model

Let k be a finite field of odd characteristic. We find a closed formula for the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic curves of genus g over k, admitting a Koblitz model. These numbers are expressed as a polynomial in the cardinality q of k, with integer coefficients (for pointed curves) and rational coefficients (for non-pointed curves). The coefficients depend on g and the set of divisors of q-1 and q+1. These formulas show that the number of hyperelliptic curves of genus g suitable (in principle) of cryptographic applications is asymptotically (1-e^{-1})2q^{2g-1}, and not 2q^{2g-1} as it was believed. The curves of genus g=2 and g=3 are more resistant to the attacks to the DLP; for these values of g the number of curves is respectively (91/72)q^3+O(q^2) and (3641/2880)q^5+O(q^4)

Enumerating Curves of Genus 2 Over Finite Fields

In this thesis, we investigate curves over finite fields. More precisely, fixing a base field F_q and a genus g, we aim to enumerate a representative from each isogeny and isomorphism class of curves defined over that field and of that genus. As a step towards this goal, in this work we provide code that, given a finite field of any characteristic, generates a list of models of hyperelliptic curves of genus 2 which we can guarantee contains one representative from each isomorphism class of curves defined over that field. Furthermore, our code allows us to sort these models into isogeny classes. Finally, if the field is of odd characteristic, we can further sort the models into isomorphism classes. As an application of our software, we obtain representatives for every isogeny class of hyperelliptic curves of genus 2 defined over the finite field F_2. We also give a model for each isogeny and isomorphism class of hyperelliptic curves of genus 2 defined over F_3. In these investigations, we discovered that Theorem 5 of Isomorphism Classes of Genus-2 Hyperelliptic Curves Over Finite Fields by Encinas, Menezes, and Masqué may be more accurately stated as giving the number of isomorphism classes of pointed hyperelliptic curves rather than isomorphism classes of hyperelliptic curves

Isomorphism classes of genus-2 hyperelliptic curves over finite fields

We propose a reduced equation for hyperelliptic curves of genus $2$ over finite fields \fq of $q$ elements with characteristic different from $2$ and $5$. We determine the number of isomorphism classes of genus-2 hyperelliptic curves having an \fq-rational Weierstrass point. These results have applications to hyperelliptic curve cryptography.Peer reviewe

Isomorphism classes of genus-2 hyperelliptic curves over finite fields F5m\mathbb{F}_{5^m}

The reduced equations for the hyperelliptic curves of genus 2 over $\mathbb{F}_{5^m}$ is given and the number of isomorphism classes of these curves is computed. These results are relevant for the genus-2 hyperelliptic curve classification and have applications to cryptographyPeer reviewe