Paper 2010/313

Fast Exhaustive Search for Polynomial Systems in $F_2$

Charles Bouillaguet, Chen-Mou Cheng, Tony (Tung) Chou, Ruben Niederhagen, Adi Shamir, and Bo-Yin Yang


We analyze how fast we can solve general systems of multivariate equations of various low degrees over \GF{2}; this is a well known hard problem which is important both in itself and as part of many types of algebraic cryptanalysis. Compared to the standard exhaustive-search technique, our improved approach is more efficient both asymptotically and practically. We implemented several optimized versions of our techniques on CPUs and GPUs. Modern graphic cards allows our technique to run more than 10 times faster than the most powerful CPU available. Today, we can solve 48+ quadratic equations in 48 binary variables on a NVIDIA GTX 295 video card (USD 500) in 21 minutes. With this level of performance, solving systems of equations supposed to ensure a security level of 64 bits turns out to be feasible in practice with a modest budget. This is a clear demonstration of the power of GPUs in solving many types of combinatorial and cryptanalytic problems.

Available format(s)
Publication info
Published elsewhere. Will be an extended version of our paper at CHES 2010
multivariate polynomialssystem-solvingparallelizationGraphic Processing Units (GPUs)
Contact author(s)
by @ crypto tw
2010-05-26: revised
2010-05-25: received
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      author = {Charles Bouillaguet and Chen-Mou Cheng and Tony (Tung) Chou and Ruben Niederhagen and Adi Shamir and Bo-Yin Yang},
      title = {Fast Exhaustive Search for Polynomial Systems in $F_2$},
      howpublished = {Cryptology ePrint Archive, Paper 2010/313},
      year = {2010},
      note = {\url{}},
      url = {}
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