Reducing logarithms in totally non-maximal imaginary quadratic orders to logarithms in finite fields

Detlef Huhnlein, Tsuyoshi Takagi

Research output: Chapter in Book/Report/Conference proceedingConference contribution

8 Citations (Scopus)

Abstract

We discuss the discrete logarithm problem over the class group Cl(Δ) of an imaginary quadratic order OΔ, which was proposed as a public-key cryptosystem by Buchmann and Williams [8]. While in the meantime there has been found a subexponential algorithm for the computation of discrete logarithms in Cl(Δ) [16], this algorithm only has running time LΔ[1/2, c] and is far less efficient than the number field sieve with Lp[1/3, c] to compute logarithms in IF*p. Thus one can choose smaller parameters to obtain the same level of security. It is an open question whether there is an LΔ[1/3, c] algorithm to compute discrete logarithms in arbitrary Cl(Δ). In this work we focus on the special case of totally non-maximal imaginary quadratic orders OΔpsuch that Δp = Δ1p2 and the class number of the maximal order h(Δ1) = 1, and we will show that there is an LΔp[1/3, c] lgorithm to compute discrete logarithms over the class group Cl(Δp). The logarithm problem in Cl(Δp) can be reduced in (expected) O(log3 p) bit operations to the logarithm problem in IF*p (if (Δ1/ p) = 1) or IF *p2 (if (Δ1/ p) = -1) respectively. This result implies that the recently proposed efficient DSA-analogue in totally non-maximal imaginary quadratic order OΔp [21] are only as secure as the original DSA scheme based on finite fields and hence loose much of its attractiveness.

Original languageEnglish
Title of host publicationAdvances in Cryptology - ASIACRYPT 1999 - International Conference on the Theory and Application of Cryptology and Information Security, Proceedings
EditorsKwok Yan Lam, Eiji Okamoto, Chaoping Xing
PublisherSpringer Verlag
Pages220-231
Number of pages12
ISBN (Print)3540666664, 9783540666660
DOIs
Publication statusPublished - 1999
Event5th International Conference on the Theory and Applications of Cryptology and Information Security, ASIACRYPT 1999 - Singapore, Singapore
Duration: Nov 14 1999Nov 18 1999

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume1716
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other5th International Conference on the Theory and Applications of Cryptology and Information Security, ASIACRYPT 1999
CountrySingapore
CitySingapore
Period11/14/9911/18/99

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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    Huhnlein, D., & Takagi, T. (1999). Reducing logarithms in totally non-maximal imaginary quadratic orders to logarithms in finite fields. In K. Y. Lam, E. Okamoto, & C. Xing (Eds.), Advances in Cryptology - ASIACRYPT 1999 - International Conference on the Theory and Application of Cryptology and Information Security, Proceedings (pp. 220-231). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1716). Springer Verlag. https://doi.org/10.1007/978-3-540-48000-6_18