针对指标-3型积分代数方程的数值解,研究其配置边值方法,基于插值多项式,利用未计算的近似值,通过将原方程进行离散化构造了指标-3型积分代数方程的配置边值方法,并分析了该方法的可解性和收敛性,证明了利用该方法求解指标-3型积分代数方程可达到较高收敛阶,最后通过数值实验验证了方法的有效性。Regarding the numerical solution of the index-3 integral algebraic equation, the collocation boundary value method was investigated. Based on the interpolation polynomial and the utilization of uncomputed approximate values, the collocation boundary value method for the index-3 integral algebraic equation was constructed by discretizing the original equation. The solvability and convergence of this method were analyzed. It was demonstrated that the application of this method in solving the index-3 integral algebraic equation can achieve a relatively high convergence order. Finally, the validity of the method was verified through numerical experiments.
由于现有协议的安全性为基于某种安全假设的计算安全,依赖于敌手的计算能力,因此,本文针对恶意敌手模型,使用矩阵伪装技术对方程的系数矩阵进行隐藏,结合矩阵的LU分解(lower-upper decomposition)算法,提出一种新的信息论安全外包求解线性代数方程组(information-theoretically secure outsourcing of linear algebraic equations,ITS-OutsLAE)方法 .与之前的研究相比,在保持计算和通信复杂度与现有最优方案保持一致的同时,首次将方程组唯一解的安全性提升至信息论安全(完美保密).给出了形式化的安全性证明,并通过理论分析和实验证明了所提方法的实用性.
