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Towards Practical Non-Interactive Public-Key Cryptosystems Using Non-Maximal Imaginary Quadratic Orders

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Abstract

We present a new non-interactive public-key distribution system based on the class group of a non-maximal imaginary quadratic order Cl( Δ p ). The main advantage of our system over earlier proposals based on (Z/nZ)* [25,27] is that embedding id information into group elements in a cyclic subgroup of the class group is easy (straight-forward embedding into prime ideals suffices) and secure, since the entire class group is cyclic with very high probability. Computational results demonstrate that a key generation center (KGC) with modest computational resources can set up a key distribution system using reasonably secure public system parameters. In order to compute discrete logarithms in the class group, the KGC needs to know the prime factorization of Δ p =Δ1 p 2. We present an algorithm for computing discrete logarithms in Cl(Δ p ) by reducing the problem to computing discrete logarithms in Cl(Δ1) and either F* p or F* p 2. Our algorithm is a specific case of the more general algorithm used in the setting of ray class groups [5]. We prove—for arbitrary non-maximal orders—that this reduction to discrete logarithms in the maximal order and a small number of finite fields has polynomial complexity if the factorization of the conductor is known.

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Authors and Affiliations

  1. secunet Security Networks AG, Sudetenstrasse 16, D-96247, Michelau, Germany

    Detlef Hühnlein

  2. Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4

    Michael J. Jr. Jacobson

  3. Fachhochschule Trier, Schneidershof, D-54293, Trier, Germany

    Damian Weber

Authors
  1. Detlef Hühnlein
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  2. Michael J. Jr. Jacobson
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  3. Damian Weber
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Hühnlein, D., Jacobson, M.J.J. & Weber, D. Towards Practical Non-Interactive Public-Key Cryptosystems Using Non-Maximal Imaginary Quadratic Orders. Designs, Codes and Cryptography 30, 281–299 (2003). https://doi.org/10.1023/A:1025746127771

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