  Private Simultaneous Messages (PSM) model is a minimal model for secure
multiparty computation. Feige, Kilian, and Naor (STOC 1994) and Ishai
(Cryptology and Information Security Series 2013) constructed PSM protocols
based on quadratic residues. In this paper, we define QR-PSM protocols as a
generalization of these protocols. A QR-PSM protocol is a PSM protocol whose
decoding function outputs the quadratic residuosity of what is computed from
messages. We design a QR-PSM protocol for any symmetric function $f: \{0,1\}^n
\rightarrow \{0,1\}$ of communication complexity $O(n^2)$. As far as we know,
it is the most efficient PSM protocol since the previously known best PSM
protocol was of $O(n^2\log n)$ (Beimel et al., CRYPTO 2014). We also study the
sizes of the underlying finite fields $\mathbb{F}_p$ in the protocols since the
communication complexity of a QR-PSM protocol is proportional to the bit length
of the prime $p$. In particular, we show that the $N$-th Peralta prime $P_N$,
which is used for general QR-PSM protocols, can be taken as at most
$(1+o(1))N^2 2^{2N-2}$, which improves the Peralta's known result (Mathematics
of Computation 1992) by a constant factor $(1+\sqrt{2})^2$.

Cryptography and Security;Number Theory