superelliptic curves, jacobians, cryptography, discrete logarithm problem In this paper we present an efficient, polynomial-time method to perform calculations in the divisor class group of a curve which has a single point on its normalization above infinity. In particular, we provide a unique representation of divisor classes and an algorithm for reducing a divisor on such a curve to its corresponding representative. Such curves include the case of elliptic, odd-degree hyperelliptic and superelliptic curves.In the case when the curve is defined over a finite field, the divisor class group is a finite group which can be used for implementing discrete logarithm based public key cryptosystems. This paper therefore provides a new class of groups for cryptography.On the other hand, we present a method to solve the discrete logarithm problem in these groups. This method is sub-exponential when the degree of the defining equation of the curve is large.
Abstract. A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor's algorithm. We show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved. Furthermore, we show how the two representations are connected and compare the computational complexity.
Abstract. We introduce a new cryptosystem with trapdoor decryption based on the difficulty of computing discrete logarithms in the class group of the non-maximal imaginary quadratic order OAq, where Alq = Aq 2, A square-free and q prime. The trapdoor information is the conductor q. Knowledge of this trapdoor information enables one to switch to and from the class group of the maximal order (_OA, where the representatives of the ideal classes have smaller coefficients. Thus, the decryption procedure may be performed in the class group of Oa rather than in the class group of the public (.9Aq, which is much more efficient. We show that inverting our proposed cryptosystem is computationally equivalent to factoring the non-fundamental discriminant Aq, which is intractable for a suitable choice of A and q. We also describe how signature schemes in OAq may be set up using this trapdoor information. Furthermore, we illustrate how one may embed key escrow capability into classical imaginary quadratic field cryptosystems.
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