Counting hyperelliptic curves that admit a Koblitz model

Let k = Fq 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 nonpointed 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)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)q + O(q) and (3641/2880)q + O(q).