Direct solution of the (11,9,8)-MinRank problem by the block Wiedemann algorithm in magma with a tesla GPU

Abstract

We show how some very large multivariate polynomial systems over finite fields can be solved by Gröbner basis techniques coupled with the Block Wiedemann algorithm, thus extending the Wiedemann-based 'Sparse FGLM' approach of Faugère and Mou. The main components of our approach are a dense variant of the Faugère F4 Gröbner basis algorithm and the Block Wiedemann algorithm, which have been implemented within the Magma Computer Algebra System (released in version V2.20 in late 2014). A major feature of the algorithms is that they map much of the computation to dense matrix multiplication, and this allows dramatic speedups to be achieved for large examples when an Nvidia Tesla GPU is available. As a result, the Magma implementation can directly solve a 16-bit random instance of the Courtois (11,9,8)-MinRank Challenge C in about 15.1 hours with a single Intel Sandybridge CPU core coupled with an Nvidia Tesla K40 GPU.