SPDZ refers to a family of protocols for Secure Multi-Party Computation (MPC) that lie at the foundation of very popular software frameworks for MPC, such as SCALE-MAMBA and MP-SPDZ. SPDZ provides good efficiency while guaranteeing security even when all but one of the participants are corrupted. This seemingly optimal property comes at a price: the protocol only offers security with abort, meaning that even a single cheating participant can force the protocol to abort, leaving honest participants with no clue on what the correct output is, or who cheated. This is especially problematic since cheating participants are able to obtain the correct output of the computation, effectively `stealing' it. We propose a *hybrid secure* adaptation to SPDZ, which retains the existing security guarantees, but in case the number of cheating players is less than half of the total, we achieve *fairness*, meaning that either all players obtain the correct output of the computation, or no player does. The `less than half' threshold of corrupted players has been proven to be a tight bound to achieve fairness. Aside from the description of the protocol and its security proof, we also present a proof-of-concept implementation, and evaluate its practical performance, thereby demonstrating that our solution has negligible overhead compared to standard SPDZ in most application scenarios.

In the model that has become known as “Perfectly Secure Message Transmission” (PSMT), a sender Alice is connected to a receiver Bob through n parallel two-way channels. A computationally unbounded adversary Eve controls t of these channels, meaning she can acquire and alter any data that is transmitted over these channels. The sender Alice wishes to communicate a secret message to Bob privately and reliably, i.e. in such a way that Eve gains no information about the message while Bob is able to recover it completely. We focus on PSMT protocols that work in two transmission rounds for n = 2t + 1. We break from previous work by following a conceptually simpler blueprint. This has two consequences: first, we obtain improved efficiency, namely, we reduce the previously best-known communication complexity, i.e. the number of transmitted bits necessary to communicate a 1-bit secret, from O(n3log n) to O(n2log n). Our solution also reaches optimal transmission rate for a secret of size O(n log n), thus answering the hitherto open question of attaining a threshold below O(n2log n) bits. Second, our construction can be adapted to more general scenarios relevant to Network Coding, where the adversary is given more power.

In the model of Secure Network Coding, a sender is connected to several receivers by a network, i.e. a directed graph with a single source node and several destination nodes, where each node can perform operations on the values received via the incoming edges and sends the results via the outbound edges. An active adversary controls some of the edges; this means that he can read every symbol transmitted over the edges under his control and replace them with symbols of his choice. The goal of Secure Network Coding is to design protocols that allow transmission of a secret message from the sender to all receivers in a private and reliable way. Classically, only one-way communication (from sender to receivers) has been studied; in this setting, security can be guaranteed as long as the number of edges controlled by the adversary is less than one third of the network connectivity. In this paper, we present a procedure where receivers are allowed to send feedback to the sender; with this feature, security is guaranteed against a stronger adversary: namely, the number of corrupted edges only needs to be smaller than one half of the connectivity. Furthermore, like previous state-of-the-art work on the single-round scenario, our scheme is universal, i.e. it does not require knowledge of the network code.