(A) study of some problems of hyperelliptic curves

Abstract

Public key cryptosystems related to hyperelliptic curves over finite fields are represented by the ideals in hyperelliptic function fields. Let k be a field of odd characteristic. Then a hyperelliptic function field K over k of genus g can be generated over the rational function field by the square root of a squarefree polynomial of degree 2g+1 or 2g+2 which is called an imaginary or a real quadratic function field respectively. In each case, different discrete logarithm problems were defined. In this paper, we consider the correspondence and the equivalence between an imaginary quadratic representation and a real quadratic representation of a function field K. As an application, we show that the discrete logarithm problems defined on them are equivalent in case of small genera and we consider computation of regulators of real quadratic function fields. In elliptic curve or hyperelliptic curve cryptographic schemes, the dominantly costing operation is a divisor multiplication by an integer. Recently, a fast method which is applicable to a family of hyperelliptic curves having efficiently computable endomorphisms was presented. We analyze the proposed hyperelliptic curves of genus two having efficiently computable endomorphisms. In some cases, they are supersingular and in another cases they are reducible over the defining field. So, their Jacobian varieties can`t have any large prime order subgroup and we propose an efficient method by which one can classify some of such curves. This will help one to find a good curve whose Jacobian variety has a large prime order subgroup.