Counting hyperelliptic curves that admit a Koblitz model

Abstract

Let 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 q 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) for cryptographic applications is asymptotically (1 – e–1)2q2g–1, and not 2q2g–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)q3 + O(q2) and (3641/2880)q5 + O(q4).