Efficient Algorithms for the Jacobian Variety of Hyperelliptic Curves y2=xp-x+1 Over a Finite Field of Odd Characteristic p

We develop efficient algorithms for the Jacobian of the hyperelliptic curve defined by the equation y2=xp-x+1 over a finite field F _p n of odd characteristic p. We first determine the zeta function of the curve which yields the order of the Jacobian. We also investigate the Frobenius operator and use it to show that, for field extensionsequation y2=xp-x+1 over a finite field F _p n, of degree n prime to p, the Jacobian has a cyclic group structure. We furthermore propose a method for faster scalar multiplication in the Jacobian by using efficient operators other than the Frobenius that have smaller eigenvalues.