Counting points on hyperelliptic curves of type y2 = x2g+1 + axg+1 + bx

Abstract

In this work, we investigate hyperelliptic curves of type C:y2=x2g+1+axg+1+bx over the finite field Fq,q=pn,p>2. For the case of g=3 we propose an algorithm to compute the number of points on the Jacobian of the curve with complexity O˜(log4⁡p) over Fp. In case of g=4 we present a point counting algorithm with complexity O˜(log8⁡q) over Fq. The Jacobian JC splits over an extension of the field Fq on the Jacobians of the curves defined by the Dickson polynomials Dg(x,1) of degree g. For these curves of genus 2,3,5 with equation y2=Dg(x,1)+a and curves of genus 2,4 with equation y2=(x+2)(Dg(x,1)+a), we give the lists of possible characteristic polynomials of the Frobenius endomorphism modulo p.