An L (1/3 + ε) Algorithm for the Discrete Logarithm Problem for Low Degree Curves

  • Andreas Enge
  • Pierrick Gaudry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4515)


The discrete logarithm problem in Jacobians of curves of high genus g over finite fields \(\mathbb {F}_q\) is known to be computable with subexponential complexity \(L_{q^g}(1/2, O(1))\). We present an algorithm for a family of plane curves whose degrees in X and Y are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of \(L_{q^g}(1/3, O(1))\), and a discrete logarithm computation takes subexponential time of \(L_{q^g}(1/3+ \varepsilon, o(1))\) for any positive ε. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms.


Prime Divisor Discrete Logarithm Hyperelliptic Curve Smith Normal Form Smooth Divisor 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Andreas Enge
    • 1
  • Pierrick Gaudry
    • 2
  1. 1.INRIA Futurs & Laboratoire d’Informatique (CNRS/UMR 7161)École polytechniquePalaiseau CedexFrance
  2. 2.LORIA (CNRS/UMR 7503), Campus ScientifiqueVandœuvre-lés-Nancy CedexFrance

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