Secure Multi-Player Protocols: Fundamentals, Generality, and Efficiency
While classically cryptography is concerned with the problem of private communication among two entities, say players, in modern cryptography multi-player protocols play an important role. And among these, it is probably fair to say that secret sharing, and its stronger version verifiable secret sharing (VSS), as well as multi-party computation (MPC) belong to the most appealing and/or useful ones. The former two are basic tools to achieve better robustness of cryptographic schemes against malfunction or misuse by ``decentralizing'' the security from one single to a whole group of individuals (captured by the term threshold cryptography). The latter allows--at least in principle--to execute any collaboration among a group of players in a secure way that guarantees the correctness of the outcome but simultaneously respects the privacy of the participants.
In this work, we study three aspects of secret sharing, VSS and MPC, which we denote by fundamentals, generality, and efficiency. By fundamentals we mean the quest for understanding why a protocol works and is secure in abstract (and hopefully simple) mathematical terms. By generality we mean generality with respect to the underlying mathematical structure, in other words, minimizing the mathematical axioms required to do some task. And efficiency of course deals with the improvement of protocols with respect to some meaningful complexity measure.
We briefly summarize our main results. (1) We give a complete characterization of black-box secret sharing in terms of simple algebraic conditions on the integer sharing coefficients, and we propose a black-box secret sharing scheme with minimal expansion factor. Note that, in contrast to the classical field-based secret sharing schemes, a black-box secret sharing scheme allows to share a secret sampled from an arbitrary Abelian group and requires only black-box access to the group operations and to random group elements. Such a scheme may be very useful in the construction of threshold cryptosystems based on groups with secret order (like RSA). (2) We show that without loss of efficiency, MPC can be based on arbitrary finite rings. This is in sharp contrast to the literature where essentially all MPC protocols require a much stronger mathematical structure, namely a field. Apart from its theoretical value, this can lead to efficiency improvements since it allows a greater freedom in the (mathematical) representation of the task that needs to be securely executed. (3) We propose a unified treatment of perfectly secure linear VSS and distributed commitments (a weaker version of the former), and we show that the security of such a scheme can be reduced to a linear algebra condition. The security of all known schemes follows as corollaries whose proofs are pure linear algebra arguments, in contrast to some hybrid arguments used in the literature. (4) We construct a new unconditionally secure VSS scheme with minimized reconstruction complexity in the setting of a dishonest minority. This improves on the reconstruction complexity of the previously best known scheme without affecting the (asymptotic) complexity of the sharing phase. In the context of MPC with pre-processing, our scheme allows to improve the computation phase of earlier protocols without increasing the complexity of the pre-processing. (5) Finally, we consider commitment multiplication proofs, which allow to prove a multiplicative relation among three commitments, and which play a crucial role in computationally secure MPC. We study a non-interactive solution which works in a distributed-verifier setting and essentially consists of a few executions of Pedersen's VSS. We simplify and improve the protocol, and we point out a (previously overlooked) property which helps to construct non-interactive proofs of partial knowledge in this setting. This allows for instance to prove the knowledge of out of given secrets, without revealing which ones. We also construct a non-interactive distributed-verifier proof of circuit satisfiability, which--in principle--allows to prove anything that can be proven without giving away the proof.