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

Abstract

AbstractIn 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.KeywordsMultivariate quadratic polynomialsSolving systems of equationsExhaustive searchParallelizationField-Programmable Gate Arrays (FPGAs)