Achieving a log(n) Speed Up for Boolean Matrix Operations and Calculating the Complexity of the Dense Linear Algebra step of Algebraic Stream Cipher Attacks and of Integer Factorization Methods

The purpose of this paper is to calculate the running time of dense boolean matrix operations, as used in stream cipher cryptanalysis and integer factorization. Several variations of Gaussian Elimination, Strassen\u27s Algorithm and the Method of Four Russians are analyzed. In particular, we demonstrate that Strassen\u27s Algorithm is actually slower than the Four Russians algorithm for matrices of the sizes encountered in these problems. To accomplish this, we introduce a new model for tabulating the running time, tracking matrix reads and writes rather than field operations, and retaining the coefficients rather than dropping them. Furthermore, we introduce an algorithm known heretofore only orally, a ``Modified Method of Four Russians\u27\u27, which has not appeared in the literature before. This algorithm is $\log n$ times faster than Gaussian Elimination for dense boolean matrices. Finally we list rough estimates for the running time of several recent stream cipher cryptanalysis attacks