On the Capacity of Private Monomial Computation

TL;DR

This paper investigates the maximum efficiency of private monomial computation over replicated databases, deriving capacity bounds and proposing a capacity-achieving scheme for large finite fields.

Contribution

It derives the capacity of private monomial computation under certain conditions and introduces a novel scheme that achieves this capacity asymptotically.

Findings

Derived the capacity of PMC for large finite fields.

Proposed a capacity-achieving PMC scheme for arbitrary q.

Provided formulas for entropy of multivariate monomials.

Abstract

In this work, we consider private monomial computation (PMC) for replicated noncolluding databases. In PMC, a user wishes to privately retrieve an arbitrary multivariate monomial from a candidate set of monomials in $f$ messages over a finite field $\mathbb F_q$, where $q=p^k$ is a power of a prime $p$ and $k \ge 1$, replicated over $n$ databases. We derive the PMC capacity under a technical condition on $p$ and for asymptotically large $q$. The condition on $p$ is satisfied, e.g., for large enough $p$. Also, we present a novel PMC scheme for arbitrary $q$ that is capacity-achieving in the asymptotic case above. Moreover, we present formulas for the entropy of a multivariate monomial and for a set of monomials in uniformly distributed random variables over a finite field, which are used in the derivation of the capacity expression.