Acceleration analysis of the quadratic sieve method based on the online matrix solving - Vynnychuk, Stepan; Misko, Vitalii (Eastern-European Journal of Enterprise Technologies, 2018)
The algorithm for the online matrix solving is proposed. The rate of acceleration of the basic quadratic sieve method based on the online matrix solving is investigated. Acceleration of the quadratic sieve method will reduce the runtime, the complexity of the algorithm and expand the set of numbers, where this algorithm is the best. It is shown that the modified algorithm has increased the number of successful decompositions. That is, the number of cases where the basic quadratic sieve (standard sieving interval and size of the factor base) failed to form a matrix to obtain a solution was reduced. This became possible due to the fact that in the modified algorithm there is no need to obtain all La+2 B-smooth numbers prior to diagonalization of the matrix, as in the case of the basic method. Among other important characteristics of this method, it should be noted that when used, the same operations as in the basic quadratic sieve method are performed, only their order is changed. The computing complexity decreases if the set of B-smooth numbers, for which the power matrix vectors form a linearly dependent system, are found quickly. According to the data obtained, the modified QS method, based on the online matrix solving, provides an acceleration of about 5.45 percent for numbers of 10130 in size. It is shown that improvements associated with solving the matrix cannot lead to a significant increase in the sieving interval. After all, the rate of acceleration decreases with increasing number N. Further improvement to the quadratic sieve method should be related to methods aimed at a significant reduction of the sieving interval and the size of the factor base, which in relative terms should be the greater, the higher N