Skew-Frobenius Maps on Hyperelliptic Curves

The hyperelliptic curve cryptosystems take most of the time for computing a scalar multiplication kD of an element D in the Jacobian JC of a hyperelliptic curve C for an integer k. Therefore its efficiency depends on the scalar multiplications. Among the fast scalar multiplication methods, there is a method using a Frobenius map. It uses a Jacobian defined over an extension field of the definition field of C, so that the Jacobian cannot be a 160 bit prime order. Therefore there is a loss of efficiency in that method. Iijima et al. proposed a method using a Frobeinus map on the quadratic twist of an elliptic curve, which is called a skew-Frobenius map in this paper. This paper shows constructions of the skew-Frobenius maps on hyperelliptic curves of genus 2 and 3.