We improve the communication complexity in the Private Simultaneous Messages (PSM) model, which is a minimal model of non-interactive information-theoretic multi-party computation. The state-of-the-art PSM protocols were recently constructed by Beimel, Kushilevitz and Nissim (EUROCRYPT 2018).

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We present new constructions of $k$-party PSM protocols. The new protocols match the previous upper bounds when $k=2$ or $3$ and improve the upper bounds for larger $k$. We also construct $2$-party PSM protocols with unbalanced communication complexity.
More concretely,

– For infinitely many $k$ (including all $k leq 20$), we construct $k$-party PSM protocols for arbitrary functionality $f:[N]^kto{0,1}$, whose communication complexity is $O_k(N^{frac{k-1}{2}})$. This improves the former best known upper bounds of $O_k(N^{frac{k}{2}})$ for $kgeq 6$, $O(N^{7/3})$ for $k=5$, and $O(N^{5/3})$ for $k=4$.

– For all rational $0

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