Constructing Hyperelliptic Covers For Elliptic Curves Over Quadratic Extension Fields

Elliptic curves and hyperelliptic curves over finite fields are of great interest in public key cryptography. Using much smaller field for same security makes the genus 2 curves more competitive than elliptic curves. However, point counting algorithms for the Jacobians of genus 2 curves are not as efficient as what we have for elliptic curves. We give a method to generate genus 2 curves for which the point counting problems can be easily solved with efficient algorithms for elliptic curves. As an application, an example of a hyperelliptic curve whose order is a 256-bit prime is given. The method relies on the construction of a cover map from a hyperelliptic curve to an elliptic curve. Another important application of the construction is to generate the cover for the cover-decomposition attack on the discrete logarithm problems in elliptic curves.