TCC-Day 1, session 1: MPC
The first talk of the session, Computing on Authenticated Data, defined notions for deriving signatures on messages for which a certain predicate is satisfied. There has been much progress in this area recently with work on redaction and quoting of authenticated data, homomorphic signatures and transitive signatures. This can be seen as a signature analogue of fully homomorphic encryption in some sense. One particular new property identified in this work was context hiding: a derived signature on a message m' should not leak any information on the original source message, for example a signed quote should not leak any information about the message from which it was quoted. This property can be seen as a data analogue of the hiding a signer amongst a group of users in group signatures.
The second talk focused on constructions of identifiable secret sharing schemes. This is a SSS which during the reconstruction phase either outputs the correct secret or publicly identifies the set of all cheaters. In this work they modify the reconstruct algorithm to only inform the set of honest players of the cheaters (which is arguably a more natural requirement). Under this new notion they show unconditional MPC can be realized in the presence of a dishonest majority.
The final talk of the session given by Ivan Damgaard focused on two party secure computation with low communication. In this setting two parties with private inputs wish to compute some function of those inputs with low communication costs. Famously Yao showed that any poly time two party functionality can be computed securely even against a malicious adversary. For semi-honest adversaries there exists a straightforward solution based on FHE and NIZK proofs. In this work the authors succesfully build a protocol where the communication is only polylog in the circuit size, assuming the existence of collision resistant extractable hash functions, which can be instantiated via the knowledge of exponenet assumption.