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The Number Field Sieve in the Medium Prime Case

  • Antoine Joux
  • Reynald Lercier
  • Nigel Smart
  • Frederik Vercauteren
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4117)

Abstract

In this paper, we study several variations of the number field sieve to compute discrete logarithms in finite fields of the form \({\mathbb F}_{p^n}\), with p a medium to large prime. We show that when n is not too large, this yields a \(L_{p^n}(1/3)\) algorithm with efficiency similar to that of the regular number field sieve over prime fields. This approach complements the recent results of Joux and Lercier on the function field sieve. Combining both results, we deduce that computing discrete logarithms have heuristic complexity \(L_{p^n}(1/3)\) in all finite fields. To illustrate the efficiency of our algorithm, we computed discrete logarithms in a 120-digit finite field \({\mathbb F}_{p^3}\).

Keywords

Prime Ideal Discrete Logarithm Discrete Logarithm Problem Cyclic Number Coprime Integer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Antoine Joux
    • 1
    • 3
  • Reynald Lercier
    • 1
    • 2
  • Nigel Smart
    • 4
  • Frederik Vercauteren
    • 5
  1. 1.DGA 
  2. 2.CELARBruzFrance
  3. 3.PRISMUniversité de Versailles St-Quentin-en-YvelinesVersaillesFrance
  4. 4.Dept. Computer ScienceUniversity of BristolBristolUnited Kingdom
  5. 5.Department of Electrical EngineeringUniversity of LeuvenLeuven-HeverleeBelgium

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