On the Capacity of Private Monomial Computation
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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 F_q, where q=p^k is a power of a prime p and k > 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.
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