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International Workshop on the Arithmetic of Finite Fields

WAIFI 2014: Arithmetic of Finite Fields pp 3–22Cite as

Computing Discrete Logarithms in \({\mathbb F}_{3^{6 \cdot 137}}\) and \({\mathbb F}_{3^{6 \cdot 163}}\) Using Magma

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9061))

Abstract

We show that a Magma implementation of Joux’s \(L[1/4+o(1)]\) algorithm can be used to compute discrete logarithms in the 1303-bit finite field \({\mathbb F}_{3^{6 \cdot 137}}\) and the 1551-bit finite field \({\mathbb F}_{3^{6 \cdot 163}}\) with very modest computational resources. Our \({\mathbb F}_{3^{6 \cdot 137}}\) implementation was the first to illustrate the effectiveness of Joux’s algorithm for computing discrete logarithms in small-characteristic finite fields that are not Kummer or twisted-Kummer extensions.

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Notes

  1. 1.

    We note that the supersingular elliptic curves \(y^2 = x^3 - x \pm 1\) over \({\mathbb F}_{3^n}\) have embedding degree \(6\) and were proposed for cryptographic use in several early papers on pairing-based cryptography [5, 8, 14, 19].

  2. 2.

    More generally, one could consider fields \({\mathbb F}_{q^{kn}}\) where \(n \le 2q+1\). We focus on the case \(k=3\) since our target fields are \({\mathbb F}_{3^{6n}}\) with \(n \in \{137,163\}\), which we will embed in \({\mathbb F}_{(3^4)^{3 \cdot n}}\).

  3. 3.

    For our \({\mathbb F}_{3^{6 \cdot 137}}\) and \({\mathbb F}_{3^{6 \cdot 163}}\) computations, we have \(q=3^4\) and used \(q'=3^3\), so \(s=1\) and \(\deg R = 5\).

  4. 4.

    More precisely, since \(C\) has at most 34 prime factors, each of which is greater than the ten-millionth prime \(p=179424673\), the probability that \(g\) is a generator is at least \((1-\frac{1}{p})^{34} > 0.99999981\).

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Author information

Authors and Affiliations

  1. Computer Science Department, CINVESTAV-IPN, Mexico City, Mexico

    Gora Adj, Thomaz Oliveira & Francisco Rodríguez-Henríquez

  2. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada

    Alfred Menezes

Authors
  1. Gora Adj
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  2. Alfred Menezes
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  3. Thomaz Oliveira
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  4. Francisco Rodríguez-Henríquez
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Corresponding author

Correspondence to Francisco Rodríguez-Henríquez .

Editor information

Editors and Affiliations

  1. Department of Computer Science, Univ of California, Santa Barbara, Santa Barbara, California, USA

    Çetin Kaya Koç

  2. University of Paris VIII, Paris, France

    Sihem Mesnager

  3. Sabancı University, Istanbul, Turkey

    Erkay Savaş

Appendices

A Magma Script for Verifying the \({\mathbb F}_{3^{6 \cdot 137}}\) discrete log

figure a

B Magma Script for Verifying the \({\mathbb F}_{3^{6 \cdot 163}}\) discrete log

figure b

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Adj, G., Menezes, A., Oliveira, T., Rodríguez-Henríquez, F. (2015). Computing Discrete Logarithms in \({\mathbb F}_{3^{6 \cdot 137}}\) and \({\mathbb F}_{3^{6 \cdot 163}}\) Using Magma. In: Koç, Ç., Mesnager, S., Savaş, E. (eds) Arithmetic of Finite Fields. WAIFI 2014. Lecture Notes in Computer Science(), vol 9061. Springer, Cham. https://doi.org/10.1007/978-3-319-16277-5_1

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  • DOI: https://doi.org/10.1007/978-3-319-16277-5_1

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