The Multiple Number Field Sieve with Conjugation and Generalized Joux-Lercier Methods

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9056)


In this paper, we propose two variants of the Number Field Sieve (NFS) to compute discrete logarithms in medium characteristic finite fields. We consider algorithms that combine two ideas, namely the Multiple variant of the Number Field Sieve (MNFS) taking advantage of a large number of number fields in the sieving phase, and two recent polynomial selections for the classical Number Field Sieve. Combining MNFS with the Conjugation Method, we design the best asymptotic algorithm to compute discrete logarithms in the medium characteric case. The asymptotic complexity of our improved algorithm is \(L_{p^n} (1/3, (8 (9+4 \sqrt{6})/15)^{1/3}) \approx L_{p^n}(1/3, 2.156) \), where \({\mathbb F}_{p^n}\) is the target finite field. This has to be compared with the complexity of the previous state-of-the-art algorithm for medium characteristic finite fields, NFS with Conjugation Method, that has a complexity of approximately \(L_{p^n}(1/3, 2.201)\). Similarly, combining MNFS with the Generalized Joux-Lercier method leads to an improvement on the asymptotic complexities in the boundary case between medium and high characteristic finite fields.


Commutative Diagram Discrete Logarithm Problem Linear Polynomial Irreducible Factor Boundary Case 
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Copyright information

© International Association for Cryptologic Research 2015

Authors and Affiliations

  1. 1.CNRS and Direction Générale de l’ArmementRennesFrance
  2. 2.Laboratoire d’Informatique de Paris 6UPMC/Sorbonnes-UniversitésParisFrance

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