We consider the problem of private computation (PC) in a distributed storage system. In such a setting a user wishes to compute a function of $f$ messages replicated across $n$ noncolluding databases, while revealing no information about the desired function to the databases. We provide an information-theoretically accurate achievable PC rate, which is the ratio of the smallest desired amount of information and the total amount of downloaded information, for the scenario of nonlinear computation. For a large message size the rate equals the PC capacity, i.e., the maximum achievable PC rate, when the candidate functions are the $f$ independent messages and one arbitrary nonlinear function of these. When the number of messages grows, the PC rate approaches an outer bound on the PC capacity. As a special case, we consider private monomial computation (PMC) and numerically compare the achievable PMC rate to the outer bound for a finite number of messages.

翻译：我们考虑分布式存储系统中的私有计算（Private Computation, PC）问题。在该场景下，用户希望计算在$n$个非共谋数据库间复制的$f$条消息的某个函数，同时不向数据库泄露任何关于所需函数的信息。针对非线性计算场景，我们提出一种信息论意义上精确的可实现PC速率，即最小期望信息量与总下载信息量之比。当候选函数为$f$条独立消息及其任意一个非线性函数时，在大消息长度条件下该速率等于PC容量（即可实现PC速率的最大值）。随着消息数量增长，该PC速率趋近于PC容量的一个外边界。作为特例，我们研究了私有单项式计算（Private Monomial Computation, PMC），并通过有限消息数量的数值实验将可实现PMC速率与外界限进行了对比。