Counting hyperelliptic curves that admit a Koblitz model

TL;DR

This paper derives explicit formulas for counting hyperelliptic curves with a Koblitz model over finite fields, revealing their distribution and cryptographic relevance, especially for genus 2 and 3 curves.

Contribution

It provides the first closed-form polynomial formulas for the number of hyperelliptic curves with a Koblitz model over finite fields, including pointed and non-pointed cases.

Findings

Number of such curves is asymptotically (1-e^{-1})2q^{2g-1}.

Formulas depend on divisors of q-1 and q+1.

Genus 2 and 3 curves are more resistant to DLP attacks.

Abstract

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).