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Nonuniform Polynomial Time Algorithm to Solve Decisional Diffie-Hellman Problem in Finite Fields under Conjecture

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Topics in Cryptology — CT-RSA 2002 (CT-RSA 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2271))

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Abstract

In this paper, we show that curves which are defined over a number field of small degree but have a large torsion group over the number field have considerable cryptographic significance. If those curves exist and the heights of torsions are small, they can serve as a bridge for prime shifting, which results a nonuniform polynomial time algorithm to solve DDH on finite fields and a nonuniform subexpontial time algorithm to solve elliptic curve discrete logarithm problem. At this time we are unable to prove the existence of those curves. To the best of our knowledge, this is the first attempt to apply the ideas related to the Uniform Boundedness Theorem(UBT), formerly known as Uniform Boundedness Conjecture, in cryptography.

Part of the research was done while the first author was a student in the University of Southern California and the second author was visiting there. The first author was partially support by NSF grant CCR-9820778.

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Author information

Authors and Affiliations

  1. School of Computer Science, University of Oklahoma, 73019, Norman, OK, USA

    Qi Cheng

  2. NTT Laboratories, 1-1 Hikarinooka, 239-0847, Yokosuka-shi, Japan

    Shigenori Uchiyama

Authors
  1. Qi Cheng
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  2. Shigenori Uchiyama
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, 3001, Leuven-Heverlee, Belgium

    Bart Preneel

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Cheng, Q., Uchiyama, S. (2002). Nonuniform Polynomial Time Algorithm to Solve Decisional Diffie-Hellman Problem in Finite Fields under Conjecture. In: Preneel, B. (eds) Topics in Cryptology — CT-RSA 2002. CT-RSA 2002. Lecture Notes in Computer Science, vol 2271. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45760-7_20

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  • DOI: https://doi.org/10.1007/3-540-45760-7_20

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43224-1

  • Online ISBN: 978-3-540-45760-2

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