A Study on Optimization of Sparse and Dense Linear System Solver Over GF(2) on GPUs

Abstract

There are various crypt-analytic techniques where solving a large dense or sparse system of linear equations over finite field becomes a challenge due to high computation. For instance, the problem like NFS for factorization of large integers, symmetric ciphers for crypt-analytic problem, discrete log problem, and algebraic attacks involves solving large sparse or dense linear systems over finite field. Here, we consider GF(2) finite field. Gaussian elimination is the popular and relevant method for solving large dense systems while Block Lanczos and Block Wiedemann algorithms are well known for solving large sparse systems. However, the time complexity of such popular method makes it reluctant and hence, the concept of parallelism is made compulsory for such methods. In addition, the availability of high end parallel processors and accelerators such as general-purpose graphics processing units (GPGPUs) solves computationally intensive problems in reasonable time. The accelerators with thousand of cores available today explore the bandwidth of memory and take advantage of multi-level parallelism on multi-node and multi-GPU units. Here, we consider Nvidia GPUs like Keplar, Pascal, and Volta along CUDA and MPI. Also, CUDA-aware MPI leverages GPU-Direct RDMA and P2P for inter- and intranode communication.