Fast Exhaustive Search for Quadratic Systems in \(\mathbb {F}_{2}\) on FPGAs

  • Charles Bouillaguet
  • Chen-Mou Cheng
  • Tung Chou
  • Ruben Niederhagen
  • Bo-Yin Yang
Conference paper

DOI: 10.1007/978-3-662-43414-7_11

Volume 8282 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Bouillaguet C., Cheng CM., Chou T., Niederhagen R., Yang BY. (2014) Fast Exhaustive Search for Quadratic Systems in \(\mathbb {F}_{2}\) on FPGAs. In: Lange T., Lauter K., Lisoněk P. (eds) Selected Areas in Cryptography -- SAC 2013. SAC 2013. Lecture Notes in Computer Science, vol 8282. Springer, Berlin, Heidelberg

Abstract

In 2010, Bouillaguet et al. proposed an efficient solver for polynomial systems over \(\mathbb {F}_{2}\) that trades memory for speed [BCC+10]. As a result, 48 quadratic equations in 48 variables can be solved on a graphics processing unit (GPU) in 21 min. The research question that we would like to answer in this paper is how specifically designed hardware performs on this task. We approach the answer by solving multivariate quadratic systems on reconfigurable hardware, namely Field-Programmable Gate Arrays (FPGAs). We show that, although the algorithm proposed in [BCC+10] has a better asymptotic time complexity than traditional enumeration algorithms, it does not have a better asymptotic complexity in terms of silicon area. Nevertheless, our FPGA implementation consumes 20–25 times less energy than its GPU counterpart. This is a significant improvement, not to mention that the monetary cost per unit of computational power for FPGAs is generally much cheaper than that of GPUs.

Keywords

Multivariate quadratic polynomials Solving systems of equations Exhaustive search Parallelization Field-Programmable Gate Arrays (FPGAs) 

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Charles Bouillaguet
    • 1
  • Chen-Mou Cheng
    • 2
  • Tung Chou
    • 3
  • Ruben Niederhagen
    • 4
  • Bo-Yin Yang
    • 4
  1. 1.Université de LilleLilleFrance
  2. 2.National Taiwan UniversityTaipeiTaiwan
  3. 3.Technische Universiteit EindhovenEindhovenThe Netherlands
  4. 4.Academia SinicaTaipeiTaiwan