Improved Capacity Outer Bound for Private Quadratic Monomial Computation
Karen M. Dæhli, Sarah A Obead, Hsuan-Yin Lin, Eirik Rosnes
TL;DR
The paper tackles the capacity outer bound for private quadratic nonparallel monomial computation (PQNMC), a nonlinear extension of private computation, where the bound depends on the order of the $\mu=\binom{f}{2}$ candidate quadratic monomials. It proposes three ordering strategies—enhanced edge-coloring (E-EC), longest-distance first (LDF), and entropy-based greedy (EBG)—and evaluates them against exhaustive search for $f<6$ and directed random search for larger $f$, revealing that EBGreedy often yields the smallest gap to the best order. For $f<6$ the optimal order is found exactly, independent of the number of databases $n$, while for $6\le f\le 12$ the comparative results show a trade-off between computational complexity and bound tightness, with LDF and EBG offering strong performance and E-EC providing a low-complexity alternative. The work provides practical methods to tighten the capacity outer bound in PQNMC and has potential implications for broader private computation scenarios and nonlinear function evaluation in distributed systems.
Abstract
In private computation, a user wishes to retrieve a function evaluation of messages stored on a set of databases without revealing the function's identity to the databases. Obead \emph{et al.} introduced a capacity outer bound for private nonlinear computation, dependent on the order of the candidate functions. Focusing on private \emph{quadratic monomial} computation, we propose three methods for ordering candidate functions: a graph edge-coloring method, a graph-distance method, and an entropy-based greedy method. We confirm, via an exhaustive search, that all three methods yield an optimal ordering for $f < 6$ messages. For $6 \leq f \leq 12$ messages, we numerically evaluate the performance of the proposed methods compared with a directed random search. For almost all scenarios considered, the entropy-based greedy method gives the smallest gap to the best-found ordering.