Recent Progress in Private Simultaneous Messages Protocols

The private simultaneous messages (PSM) model is a simple variant of the secure multiparty computation (MPC). In the k-party PSM model, each party $P_{\imath}$ has a private input $x_{i}$ for $i=1, \ldots, k$. For a function f, each $\lt p\gt P_{i}$ encrypts $x_{i}$ into a message $m_{i}$ with a random string r shared among $\lt p\gt P_{1}, \ldots, P_{k}$, and sends $m_{i}$ to the referee $R. R$ computes $f\left(x_{1}, \ldots, x_{k}\right)$ from their respective messages $m_{1}, \ldots, m_{k}$. Then, R learns nothing from $m_{1}, \ldots, m_{k}$ except for the output value $f\left(x_{1}, \ldots, x_{k}\right)$. This simple model provides interesting cryptographic applications and is essential for understanding the intrinsic costs (e.g., of communication $\left|m_{1}\right|+\ldots+\left|m_{k}\right|$ and randomness $|r|$) to achieve MPC. This study surveys recent results associated with the PSM and closely related models.