New Complexity Trade-Offs for the (Multiple) Number Field Sieve Algorithm in Non-Prime Fields

  • Palash Sarkar
  • Shashank SinghEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9665)


The selection of polynomials to represent number fields crucially determines the efficiency of the Number Field Sieve (NFS) algorithm for solving the discrete logarithm in a finite field. An important recent work due to Barbulescu et al. builds upon existing works to propose two new methods for polynomial selection when the target field is a non-prime field. These methods are called the generalised Joux-Lercier (GJL) and the Conjugation methods. In this work, we propose a new method (which we denote as \(\mathcal {A}\)) for polynomial selection for the NFS algorithm in fields \(\mathbb {F}_{Q}\), with \(Q=p^n\) and \(n>1\). The new method both subsumes and generalises the GJL and the Conjugation methods and provides new trade-offs for both n composite and n prime. Let us denote the variant of the (multiple) NFS algorithm using the polynomial selection method “X” by (M)NFS-X. Asymptotic analysis is performed for both the NFS-\(\mathcal {A}\) and the MNFS-\(\mathcal {A}\) algorithms. In particular, when \(p=L_Q(2/3,c_p)\), for \(c_p\in [3.39,20.91]\), the complexity of NFS-\(\mathcal {A}\) is better than the complexities of all previous algorithms whether classical or MNFS. The MNFS-\(\mathcal {A}\) algorithm provides lower complexity compared to NFS-\(\mathcal {A}\) algorithm; for \(c_p\in (0, 1.12] \cup [1.45,3.15]\), the complexity of MNFS-\(\mathcal {A}\) is the same as that of the MNFS-Conjugation and for \(c_p\notin (0, 1.12] \cup [1.45,3.15]\), the complexity of MNFS-\(\mathcal {A}\) is lower than that of all previous methods.


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Copyright information

© International Association for Cryptologic Research 2016

Authors and Affiliations

  1. 1.Applied Statistics UnitIndian Statistical InstituteKolkataIndia

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