Counting hyperelliptic curves that admit a Koblitz model

Abstract

Abstract. 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 )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 ).

Orbits of rational n-sets of projective spaces under the action of the linear group

“…For instance, this technique has been used in the papers [3] and [11] to count, for any given value of the genus g > 1, the number of k-isomorphism classes of hyperelliptic curves, of pointed hyperelliptic curves, and of hyperelliptic curves having a rational Weierstrass point. N = 1, 2 The description of the subtypes of conjugacy classes of PGL N +1 (k) and the computation of the coefficients c α can be deduced from [7, Proposition 2.3, Lemma 2.4] for N = 1 and from [9, for N = 2.…”