The second lower bound applies to the perfect maliciously secure setting with $n=3t+1$ parties. We show that for any $n$ and all large enough $S$, there exists a reactive functionality $F_S$ taking an $S$-bit string as input (and with short output) such that any protocol implementing $F_S$ with perfect malicious security must communicate $\Omega(nS)$ bits. Since the functionalities we study can be implemented with linear size circuits, the result can equivalently be stated as follows: for any $n$ and all large enough $g \in \mathbb{N}$ there exists a reactive functionality $F_C$ doing computation specified by a Boolean circuit $C$ with $g$ gates, where any perfectly secure protocol implementing $F_C$ must communicate $\Omega(n g)$ bits. The results easily extends to constructing similar functionalities defined over any fixed finite field. Using known techniques, we also show an upper bound that matches the lower bound up to a constant factor (existing upper bounds are a factor $\log n$ off for Boolean circuits).
Both results also extend to the case where the threshold $t$ is suboptimal. Namely if $n=kt+s$ the bound is weakened by a factor $O(s)$, which corresponds to known optimizations via packed secret-sharing.
Category / Keywords: foundations / secure multiparty computation, lower bound, communication complexity Date: received 4 Mar 2021 Contact author: ivan at cs au dk,805864812@qq com,nis@cs au dk Available format(s): PDF | BibTeX Citation Version: 20210304:133149 (All versions of this report) Short URL: ia.cr/2021/278