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The Special Number Field Sieve in \(\mathbb{F}_{p^{n}}\)

Application to Pairing-Friendly Constructions

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Book cover Pairing-Based Cryptography – Pairing 2013 (Pairing 2013)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 8365))

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Abstract

In this paper, we study the discrete logarithm problem in finite fields related to pairing-based curves. We start with a precise analysis of the state-of-the-art algorithms for computing discrete logarithms that are suitable for finite fields related to pairing-friendly constructions. To improve upon these algorithms, we extend the Special Number Field Sieve to compute discrete logarithms in \(\mathbb{F}_{p^{n}}\), where p has an adequate sparse representation. Our improved algorithm works for the whole range of applicability of the Number Field Sieve.

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Author information

Authors and Affiliations

  1. CryptoExperts, France

    Antoine Joux

  2. Chaire de Cryptologie de la Fondation de l’UPMC, 4 place Jussieu, 75005, Paris, France

    Antoine Joux

  3. Laboratoire PRISM, Université de Versailles Saint-Quentin, 45 avenue des États-Unis, 78000, Versailles, France

    Cécile Pierrot

  4. UPMC, Univ Paris 06, LIP6, France

    Antoine Joux & Cécile Pierrot

  5. PolSys Project, INRIA, Paris-Rocquencourt Center, France

    Antoine Joux & Cécile Pierrot

  6. CNRS, UMR 7606, LIP6, France

    Antoine Joux & Cécile Pierrot

Authors
  1. Antoine Joux
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  2. Cécile Pierrot
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Editor information

Editors and Affiliations

  1. School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, No. 800, Dongchuan Road, 200240, Shanghai, China

    Zhenfu Cao

  2. School of Information Science and Technology, Sun Yat-sen University, No. 135, Xingang Xi Road, 510275, Guangzhou, China

    Fangguo Zhang

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Joux, A., Pierrot, C. (2014). The Special Number Field Sieve in \(\mathbb{F}_{p^{n}}\) . In: Cao, Z., Zhang, F. (eds) Pairing-Based Cryptography – Pairing 2013. Pairing 2013. Lecture Notes in Computer Science, vol 8365. Springer, Cham. https://doi.org/10.1007/978-3-319-04873-4_3

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  • DOI: https://doi.org/10.1007/978-3-319-04873-4_3

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-04872-7

  • Online ISBN: 978-3-319-04873-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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