Construction of hyperelliptic curves with CM and its application to cryptosystems

Abstract

Construction of secure hyperelliptic curves is of most important yet most difficult problem in design of cryptosystems based on the discrete logarithm problems on hyperelliptic curves. Presently the only accessible approach is to use CM curves. However, to find models of the CM curves is nontrivial. The popular approach uses theta functions to derive a projective embedding of the Jacobian varieties, which needs to calculate the theta functions to very high precision. As we show in this paper, it costs computation time of an exponential function in the discriminant of the CM field. This paper presents new algorithms to find explicit models of hyperelliptic curves with CM. Algorithms for CM test of Jacobian varieties of algebraic curves and to lift from small finite fields both the models and the invariants of CM curves are presented. We also show that the proposed algorithm for invariants lifting has complexity of a polynomial time in the discriminant of the CM field.