Fast matrix decomposition in F2F2

• Matrices with entries in the field F2 are stored in bits of integers.
• Operations are parallelized by using integer arithmetic and logical operators.
• Columns exchange are completely avoided by an efficient use of pseudoinverse.
• The asymptotically fast algorithm combines the “four Russian” and Strassen algorithms.
• Tuned parameters for the algorithm are deduced and verified experimentally.

In this work an efficient algorithm to perform a block decomposition for large dense rectangular matrices with entries in F2F2 is presented. Matrices are stored as column blocks of row major matrices in order to facilitate rows operation and matrix multiplications with blocks of columns. One of the major bottlenecks of matrix decomposition is the pivoting involving both rows and column exchanges. Since row swaps are cheap and column swaps are order of magnitude slower, the number of column swaps should be reduced as much as possible. Here an algorithm that completely avoids the column permutations is presented. An asymptotically fast algorithm is obtained by combining the four Russian algorithm and the recursion with the Strassen algorithm for matrix–matrix multiplication. Moreover optimal parameters for the tuning of the algorithm are theoretically estimated and then experimentally verified. A comparison with the state of the art public domain software SAGE shows that the proposed algorithm is generally faster.