The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Implementation of Strassen's algorithm for matrix multiplication
Supercomputing '96 Proceedings of the 1996 ACM/IEEE conference on Supercomputing
Hacker's Delight
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Algorithms for solving linear and polynomial systems of equations over finite fields, with applications to cryptanalysis
The M4RIE library for dense linear algebra over small fields with even characteristic
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Fast matrix decomposition in F2
Journal of Computational and Applied Mathematics
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We describe an efficient implementation of a hierarchy of algorithms for multiplication of dense matrices over the field with two elements (F2). In particular we present our implementation—in the M4RI library—of Strassen-Winograd matrix multiplication and the “Method of the Four Russians for Multiplication” (M4RM) and compare it against other available implementations. Good performance is demonstrated on AMD's Opteron processor and particulary good performance on Intel's Core 2 uo processor. The open-source M4RI library is available as a stand-alone package as well as part of the Sage mathematics system. In machine terms, addition in F2 is logical-XOR, and multiplication is logical-AND, thus a machine word of 64 bits allows one to operate on 64 elements of F2 in parallel: at most one CPU cycle for 64 parallel additions or multiplications. As such, element-wise operations over F2 are relatively cheap. In fact, in this paper, we conclude that the actual bottlenecks are memory reads and writes and issues of data locality. We present our empirical findings in relation to minimizing these and give an analysis thereof.