Paper 2019/885

On the alpha value of polynomials in the tower number field sieve algorithm

Aurore Guillevic and Shashank Singh


In this paper, we provide a notable step towards filling the gap between theory (estimates of running-time) and practice (a discrete logarithm record computation) for the Tower Number Field Sieve (TNFS) algorithm. We propose a generalisation of ranking formula for selecting the polynomials used in the very first step of TNFS algorithm. For this we provide a definition and an exact implementation (Magma and SageMath) of the alpha function. This function measures the bias in the smoothness probability of norms in number fields compared to random integers of the same size. We use it to estimate the yield of polynomials, that is the expected number of relations, as a generalisation of Murphy's E function, and finally the total amount of operations needed to compute a discrete logarithm with TNFS algorithm in the targeted fields. This is an improvement of the earlier work of Barbulescu and Duquesne on estimating the running-time of the algorithm. We apply our estimates to a wide size range of finite fields GF(p^n), for small composite n = 12, 16, 18, 24, that are target fields of pairing-friendly curves.

Note: Implementation available at Published paper available at

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Mathematical Cryptology
number field sievediscrete logarithmpairing-friendly curve
Contact author(s)
aurore guillevic @ inria fr
shashank @ iiserb ac in
2021-02-22: last of 2 revisions
2019-08-01: received
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      author = {Aurore Guillevic and Shashank Singh},
      title = {On the alpha value of polynomials in the tower number field sieve algorithm},
      howpublished = {Cryptology ePrint Archive, Paper 2019/885},
      year = {2019},
      note = {\url{}},
      url = {}
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