Counting points on hyperelliptic curves of type $y^2=x^{2g+1}+a{x}^{g+1}+bx$

Abstract

In this work, we investigate hyperelliptic curves of type $C:y^2=x^{2g+1}+a{x}^{g+1}+bx$ over the finite field ${\mathbb{F}}_q,q=p^n,p>2$. For the case of $g=3$ and $4$ we propose algorithms to compute the number of points on the Jacobian of the curve with complexity $\tilde{O}({\log }^{4}p)$ and $\tilde{O}({\log }^{8}p)$. For curves of genus $2-7$ we give a complete list of the characteristic polynomials of Frobenius endomorphism modulo $p$.