Paper 2023/731

Fast Exhaustive Search for Polynomial Systems over F3

Bo-Yin Yang, Academia Sinica
Wei-Jeng Wang, National Taiwan University
Shang-Yi Yang, National Taiwan University, Academia Sinica
Char-Shin Miou, Chunghwa Telecom
Chen-Mou Cheng, National Taiwan University

Solving multivariate polynomial systems over finite fields is an important problem in cryptography. For random F2 low-degree systems with equally many variables and equations, enumeration is more efficient than advanced solvers for all practical problem sizes. Whether there are others remained an open problem. We here study and propose an exhaustive-search algorithm for low degrees systems over F3 which is suitable for parallelization. We implemented it on Graphic Processing Units (GPUs) and commodity CPUs. Its optimizations and differences from the F2 case are also analyzed. We can solve 30+ quadratic equations in 30 variables on an NVIDIA GeForce GTX 980 Ti in 14 minutes; a cubic system takes 36 minutes. This well outperforms existing solvers. Using these results, we compare Gröbner Bases vs. enumeration for polynomial systems over small fields as the sizes go up.

Note: Tech report which reflects the work done for Mr. Wei-Jeng Wang’s masters thesis at National Taiwan University, 2016 and intended for submission but never appear elsewhere, the author's emails are their current ones, the affiliations those at the time of the work.

Available format(s)
Attacks and cryptanalysis
Publication info
multivariatequadraticenumerationfinite field
Contact author(s)
by @ crypto tw
willwang8129 @ hotmail com tw
nick yang @ chelpis com
mcs @ cht com tw
cheng @ btq li
2023-05-25: approved
2023-05-22: received
See all versions
Short URL
No rights reserved


      author = {Bo-Yin Yang and Wei-Jeng Wang and Shang-Yi Yang and Char-Shin Miou and Chen-Mou Cheng},
      title = {Fast Exhaustive Search for Polynomial Systems over F3},
      howpublished = {Cryptology ePrint Archive, Paper 2023/731},
      year = {2023},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.