Some Cryptographic Applications Of Hyperelliptic Curves

In recent years elliptic and hyperelliptic cryptosystem have been extensively studied and deployed in the real world. In EUROCRYPT 2009, Galbraith, Lin and Scott constructed an efficiently computable endomorphism for a large family of elliptic curves defned over finite fields of large characteristic. They demonstrated that the endomorphism can be used to accelerate scalar multiplication in the elliptic curve cryptosystem based on these curves. Secret sharing is an important cryptographic primitive. One of the main open problems in secret sharing is the characterization of the access structures of ideal secret sharing schemes. Chen, Ling and Xing completely characterized the access structures for the elliptic secret sharing schemes from algebraic-geometric codes associated with elliptic curves.In this thesis we extend the method of Galbraith et al. to any genus 2 hyperelliptic curve de ned over a finite field of even characteristic. We propose an efficient algorithm to generate a random genus 2 hyperelliptic curve and its quadratic twist equipped with a fast endomor-phism on the Jacobian; Secondly, we generalize the method of Chen et al. to the jacobians of hyperelliptic curves of any genus. Our result includes the access structures of elliptic secret sharing schemes as a special case.