Extension and optimization of the FIND algorithm: computing Green's and less-than Green's functions (with technical appendix)

The FIND algorithm is a fast algorithm designed to calculate certain entries of the inverse of a sparse matrix. Such calculation is critical in many applications, e.g., quantum transport in nano-devices. We extended the algorithm to other matrix inverse related calculations. Those are required for example to calculate the less-than Green's function and the current density through the device. For a 2D device discretized as an N_x x N_y mesh, the best known algorithms have a running time of O(N_x^3 N_y), whereas FIND only requires O(N_x^2 N_y). Even though this complexity has been reduced by an order of magnitude, the matrix inverse calculation is still the most time consuming part in the simulation of transport problems. We could not reduce the order of complexity, but we were able to significantly reduce the constant factor involved in the computation cost. By exploiting the sparsity and symmetry, the size of the problem beyond which FIND is faster than other methods typically decreases from a 130x130 2D mesh down to a 40x40 mesh. These improvements make the optimized FIND algorithm even more competitive for real-life applications

An ideal multi-secret sharing scheme based on minimal privileged coalitions

How to construct an ideal multi-secret sharing scheme for general access structures is difficult. In this paper, we solve an open problem proposed by Spiez et al.recently [Finite Fields and Their Application, 2011(17) 329-342], namely to design an algorithm of privileged coalitions of any length if such coalitions exist. Furthermore, in terms of privileged coalitions, we show that most of the existing multi-secret sharing schemes based on Shamir threshold secret sharing are not perfect by analyzing Yang et al.'s scheme and Pang et al.'s scheme. Finally, based on the algorithm mentioned above, we devise an ideal multi-secret sharing scheme for families of access structures, which possesses more vivid authorized sets than that of the threshold scheme.Comment: 13page