The problem of $X$-secure $T$-private linear computation with graph based replicated storage (GXSTPLC) is to enable the user to retrieve a linear combination of messages privately from a set of $N$ distributed servers where every message is only allowed to store among a subset of servers subject to an $X$-security constraint, i.e., any groups of up to $X$ colluding servers must reveal nothing about the messages. Besides, any groups of up to $T$ servers cannot learn anything about the coefficients of the linear combination retrieved by the user. In this work, we completely characterize the asymptotic capacity of GXSTPLC, i.e., the supremum of average number of desired symbols retrieved per downloaded symbol, in the limit as the number of messages $K$ approaches infinity. Specifically, it is shown that a prior linear programming based upper bound on the asymptotic capacity of GXSTPLC due to Jia and Jafar is tight by constructing achievability schemes. Notably, our achievability scheme also settles the exact capacity (i.e., for finite $K$) of $X$-secure linear combination with graph based replicated storage (GXSLC). Our achievability proof builds upon an achievability scheme for a closely related problem named asymmetric $\mathbf{X}$-secure $\mathbf{T}$-private linear computation with graph based replicated storage (Asymm-GXSTPLC) that guarantees non-uniform security and privacy levels across messages and coefficients. In particular, by carefully designing Asymm-GXSTPLC settings for GXSTPLC problems, the corresponding Asymm-GXSTPLC schemes can be reduced to asymptotic capacity achieving schemes for GXSTPLC. In regard to the achievability scheme for Asymm-GXSTPLC, interesting aspects of our construction include a novel query and answer design which makes use of a Vandermonde decomposition of Cauchy matrices, and a trade-off among message replication, security and privacy thresholds.

翻译：基于图复制存储的$X$-安全$T$-私有线性计算问题(GXSTPLC)旨在使用户能够从一组$N$个分布式服务器中私有地检索消息的线性组合，其中每条消息仅允许存储在服务器子集中，并需满足$X$-安全约束，即任意最多$X$个合谋的服务器不能获取关于消息的任何信息。此外，任意最多$T$个服务器不能获知用户检索的线性组合系数。本文完整刻画了GXSTPLC的渐近容量——即当消息数量$K$趋于无穷大时，每个下载符号中检索到的期望符号数量的上确界。具体而言，通过构造可达性方案，证明了Jia与Jafar先前基于线性规划得到的GXSTPLC渐近容量上界是紧的。值得注意的是，我们的可达性方案还解决了基于图复制存储的$X$-安全线性组合(GXSLC)的精确容量(即针对有限$K$)。可达性证明基于一个密切相关的问题——基于图复制存储的非对称$\mathbf{X}$-安全$\mathbf{T}$-私有线性计算(Asymm-GXSTPLC)的可达性方案，该方案保证了消息与系数间非均匀的安全与隐私级别。具体来说，通过为GXSTPLC问题精心设计Asymm-GXSTPLC设置，相应的Asymm-GXSTPLC方案可简化为GXSTPLC的渐近容量可达方案。关于Asymm-GXSTPLC的可达性方案，我们构造的亮点包括：利用柯西矩阵的范德蒙德分解设计新型查询与应答，以及消息复制、安全阈值与隐私阈值之间的权衡。