Exponentially Fewer-Server PIR from Sparser $S$-Decoding Polynomials

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We show that under a plausible number-theoretic conjecture, for any constant $s$ there exists an $s$-server private information retrieval (PIR) protocol that on an $n$-bit database requires communication $\exp(O((\log n)^{1/s} (\log \log n)^{1-1/s}))$. This achieves a superpolynomial improvement in communication when $s \geq 4$; previous constructions attaining the same communication required $2^{O(s)}$ servers. Our number-theoretic conjecture is implied by existing conjectures, namely the gener