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Applebaum, Benny, Kachlon, Eliran, Patra, Arpita.  2020.  The Round Complexity of Perfect MPC with Active Security and Optimal Resiliency. 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS). :1277—1284.
In STOC 1988, Ben-Or, Goldwasser, and Wigderson (BGW) established an important milestone in the fields of cryptography and distributed computing by showing that every functionality can be computed with perfect (information-theoretic and error-free) security at the presence of an active (aka Byzantine) rushing adversary that controls up to n/3 of the parties. We study the round complexity of general secure multiparty computation in the BGW model. Our main result shows that every functionality can be realized in only four rounds of interaction, and that some functionalities cannot be computed in three rounds. This completely settles the round-complexity of perfect actively-secure optimally-resilient MPC, resolving a long line of research. Our lower-bound is based on a novel round-reduction technique that allows us to lift existing three-round lower-bounds for verifiable secret sharing to four-round lower-bounds for general MPC. To prove the upper-bound, we develop new round-efficient protocols for computing degree-2 functionalities over large fields, and establish the completeness of such functionalities. The latter result extends the recent completeness theorem of Applebaum, Brakerski and Tsabary (TCC 2018, Eurocrypt 2019) that was limited to the binary field.
Iwamoto, M., Ohta, K., Shikata, J..  2018.  Security Formalizations and Their Relationships for Encryption and Key Agreement in Information-Theoretic Cryptography. IEEE Transactions on Information Theory. 64:654–685.
This paper analyzes the formalizations of information-theoretic security for the fundamental primitives in cryptography: symmetric-key encryption and key agreement. Revisiting the previous results, we can formalize information-theoretic security using different methods, by extending Shannon's perfect secrecy, by information-theoretic analogues of indistinguishability and semantic security, and by the frameworks for composability of protocols. We show the relationships among the security formalizations and obtain the following results. First, in the case of encryption, there are significant gaps among the formalizations, and a certain type of relaxed perfect secrecy or a variant of information-theoretic indistinguishability is the strongest notion. Second, in the case of key agreement, there are significant gaps among the formalizations, and a certain type of relaxed perfect secrecy is the strongest notion. In particular, in both encryption and key agreement, the formalization of composable security is not stronger than any other formalizations. Furthermore, as an application of the relationships in encryption and key agreement, we simultaneously derive a family of lower bounds on the size of secret keys and security quantities required under the above formalizations, which also implies the importance and usefulness of the relationships.
Liu, Tianren, Vaikuntanathan, Vinod.  2018.  Breaking the Circuit-Size Barrier in Secret Sharing. Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. :699-708.
We study secret sharing schemes for general (non-threshold) access structures. A general secret sharing scheme for n parties is associated to a monotone function F:\0,1\n$\rightarrow$\0,1\. In such a scheme, a dealer distributes shares of a secret s among n parties. Any subset of parties T $\subseteq$ [n] should be able to put together their shares and reconstruct the secret s if F(T)=1, and should have no information about s if F(T)=0. One of the major long-standing questions in information-theoretic cryptography is to minimize the (total) size of the shares in a secret-sharing scheme for arbitrary monotone functions F. There is a large gap between lower and upper bounds for secret sharing. The best known scheme for general F has shares of size 2n-o(n), but the best lower bound is $Ømega$(n2/logn). Indeed, the exponential share size is a direct result of the fact that in all known secret-sharing schemes, the share size grows with the size of a circuit (or formula, or monotone span program) for F. Indeed, several researchers have suggested the existence of a representation size barrier which implies that the right answer is closer to the upper bound, namely, 2n-o(n). In this work, we overcome this barrier by constructing a secret sharing scheme for any access structure with shares of size 20.994n and a linear secret sharing scheme for any access structure with shares of size 20.999n. As a contribution of independent interest, we also construct a secret sharing scheme with shares of size 2Õ($\surd$n) for 2n n/2 monotone access structures, out of a total of 2n n/2$\cdot$ (1+O(logn/n)) of them. Our construction builds on recent works that construct better protocols for the conditional disclosure of secrets (CDS) problem.