International Association
for Cryptologic Research

We consider the graph-theoretic problem of removing (few) nodes from a directed acyclic graph in order to reduce its depth. While this problem is intractable in the general case, we provide a variety of algorithms in the case where the graph is that of a circuit of fan-in (at most) two, and explore applications of these algorithms to secure multiparty computation with low communication. Over the past few years, a paradigm for low-communication secure multiparty computation has found success based on decomposing a circuit into low-depth ``chunks''. This approach was however previously limited to circuits with a ``layered'' structure. Our graph-theoretic approach extends this paradigm to all circuits. In particular, we obtain the following contributions: 1) Fractionally linear-communication MPC in the correlated randomness model: We provide an $N$-party protocol for computing an $n$-input, $m$-output $\F$-arithmetic circuit with $s$ internal gates (over any basis of binary gates) with communication complexity $(\frac{2}{3}s + n + m)\cdot N\cdot\log |\F|$, which can be improved to $((1+\epsilon)\cdot\frac{2}{5}s+n+m)\cdot N\cdot\log |\F|$ (at the cost of increasing the computational overhead from a small constant factor to a large one). Previously, comparable protocols either used more than $s\cdot N\cdot \log |\F|$ bits of communication, required super-polynomial computation, were restricted to layered circuits, or tolerated a sub-optimal corruption threshold. 2) Sublinear-Communication MPC: Assuming the existence of $N$-party Homomorphic Secret Sharing for logarithmic depth circuits (respectively doubly logarithmic depth circuits), we show there exists sublinear-communication secure $N$-party computation for \emph{all} $\log^{1+o(1)}$-depth (resp.~$(\log\log)^{1+o(1)}$-depth) circuits. Previously, this result was limited to $(\mathcal{O}(\log))$-depth (resp.~$(\mathcal{O}(\log\log))$-depth) circuits, or to circuits with a specific structure (e.g. layered). 3) The 1-out-of-M-OT complexity of MPC: We introduce the `` 1-out-of-M-OT complexity of MPC'' of a function $f$, denoted $C_M(f)$, as the number of oracle calls required to securely compute $f$ in the 1-out-of-M-OT hybrid model. We establish the following upper bound: for every $M\geq 2$, $C_N(f) \leq (1+g(M))\cdot \frac{2 |f|}{5}$, where $g(M)$ is an explicit vanishing function. We also obtain additional contributions to reducing the amount of bootstrapping for fully homomorphic encryption, and to other types of sublinear-communication MPC protocols such as those based on correlated symmetric private information retrieval.