Adaptively secure, universally composable, multi-party computation in constant rounds

Cryptographic protocols with adaptive security ensure that security holds against an adver-sary who can dynamically determine which parties to corrupt as the protocol progresses—or even after the protocol is finished. In the setting where all parties may potentially be corrupted, and secure erasure is not assumed, it has been a long-standing open question to design secure-computation protocols with adaptive security running in constant rounds. Here, we show a constant-round, universally composable protocol for computing any functionality, tolerating a malicious, adaptive adversary corrupting any number of parties. Interest-ingly, our protocol can compute all functionalities, not just adaptively well-formed ones