Li Jin, Liu Nan, Kang Wei
National Mobile Communications Research Laboratory, Southeast University, Nanjing 211189, China.
School of Information Science and Engineering, Southeast University, Nanjing 211189, China.
Entropy (Basel). 2024 May 8;26(5):407. doi: 10.3390/e26050407.
This paper studies the problem of minimizing the total cost, including computation cost and communication cost, in the system of two-sided secure distributed matrix multiplication (SDMM) under an arbitrary collusion pattern. In order to perform SDMM, the two input matrices are split into some blocks, blocks of random matrices are appended to protect the security of the two input matrices, and encoded copies of the blocks are distributed to all computing nodes for matrix multiplication calculation. Our aim is to minimize the total cost, overall matrix splitting factors, number of appended random matrices, and distribution vector, while satisfying the security constraint of the two input matrices, the decodability constraint of the desired result of the multiplication, the storage capacity of the computing nodes, and the delay constraint. First, a strategy of appending zeros to the input matrices is proposed to overcome the divisibility problem of matrix splitting. Next, the optimization problem is divided into two subproblems with the aid of alternating optimization (AO), where a feasible solution can be obtained. In addition, some necessary conditions for the problem to be feasible are provided. Simulation results demonstrate the superiority of our proposed scheme compared to the scheme without appending zeros and the scheme with no alternating optimization.
本文研究了在任意勾结模式下，双边安全分布式矩阵乘法（SDMM）系统中最小化包括计算成本和通信成本在内的总成本的问题。为了执行SDMM，将两个输入矩阵拆分为若干块，附加随机矩阵块以保护两个输入矩阵的安全，并将块的编码副本分发给所有计算节点进行矩阵乘法计算。我们的目标是在满足两个输入矩阵的安全约束、乘法期望结果的可解码性约束、计算节点的存储容量以及延迟约束的同时，最小化总成本、整体矩阵拆分因子、附加随机矩阵的数量和分布向量。首先，提出了一种向输入矩阵附加零的策略，以克服矩阵拆分的可除性问题。其次，借助交替优化（AO）将优化问题分为两个子问题，从而可以获得可行解。此外，还提供了该问题可行的一些必要条件。仿真结果表明，与不附加零的方案和无交替优化的方案相比，我们提出的方案具有优越性。
