Abstract. Consider a scenario where an l-bit secret has been distributed among n players by an honest dealer using some secret sharing scheme. Then, if all players behave honestly, the secret can be reconstructed in one round with zero error probability, and by broadcasting nl bits. We ask the following question: how close to this ideal can we get if up to t players (but not the dealer) are corrupted by an adaptive, active adversary with unbounded computing power?- and where in addition we of course require that the adversary does not learn the secret ahead of reconstruction time. It is easy to see that t = ⌊(n − 1)/2 ⌋ is the maximal value of t that can be tolerated, and furthermore, we show that the best we can hope for is a one-round reconstruction protocol where every honest player outputs the correct secret or “failure”. For any such protocol with failure probability at most 2 −Ω(k) , we show a lower bound of Ω(nl + kn 2) bits on the information communicated. We further show that this is tight up to a constant factor. The lower bound trivially applies as well to VSS schemes, where also the dealer may be corrupt. Using generic methods, the scheme establishing the upper bound can be turned into a VSS with efficient reconstruction. However, the distribution phase becomes very inefficient. Closing this gap, we present a new VSS protocol where the distribution complexity matches that of the previously best known VSS, but where the reconstruction phase meets our lower bound up to a constant factor. The reconstruction is a factor of n better than previous VSS protocols. We show an application of this to multi-party computation with pre-processing, improving the complexity of earlier similar protocols by a factor of n. 1
Abstract. Consider a scenario where an l-bit secret has been distributed among n players by an honest dealer using some secret sharing scheme. Then, if all players behave honestly, the secret can be reconstructed in one round with zero error probability, and by broadcasting nl bits. We ask the following question: how close to this ideal can we get if up to t players (but not the dealer) are corrupted by an adaptive, active adversary with unbounded computing power?- and where in addition we of course require that the adversary does not learn the secret ahead of reconstruction time. It is easy to see that t = ⌊(n − 1)/2 ⌋ is the maximal value of t that can be tolerated, and furthermore, we show that the best we can hope for is a one-round reconstruction protocol where every honest player outputs the correct secret or “failure”. For any such protocol with failure probability at most 2 −Ω(k) , we show a lower bound of Ω(nl + kn 2) bits on the information communicated. We further show that this is tight up to a constant factor. The lower bound trivially applies as well to VSS schemes, where also the dealer may be corrupt. Using generic methods, the scheme establishing the upper bound can be turned into a VSS with efficient reconstruction. However, the distribution phase becomes very inefficient. Closing this gap, we present a new VSS protocol where the distribution complexity matches that of the previously best known VSS, but where the reconstruction phase meets our lower bound up to a constant factor. The reconstruction is a factor of n better than previous VSS protocols. We show an application of this to multi-party computation with pre-processing, improving the complexity of earlier similar protocols by a factor of n. 1
Broadcast and verifiable secret sharing (VSS) are central building blocks for secure multi-party computation. These protocols are required to be resilient against a Byzantine adversary who controls at most t out of the n parties running the protocol. In this dissertation, we consider the design of fault-tolerant protocols for broadcast and verifiable secret sharing with stronger security guarantees and improved round complexity. Broadcast allows a party to send the same message to all parties, and all parties are assured they have received identical messages. Given a public-key infrastructure (PKI) and digital signatures, it is possible to construct broadcast protocols tolerating any number of corrupted parties. We address two important issues related to broadcast: (1) Almost all existing protocols do not distinguish between corrupted parties (who do not follow the protocol) and honest parties whose secret (signing) keys have been compromised (but who continue to behave honestly); (2) all existing protocols for broadcast are insecure against an adaptive adversary who can choose which parties to corrupt as the protocol progresses. We propose new security models that capture these issues, and present tight feasibility and impossibility results. In the problem of verifiable secret sharing, there is a designated player who shares a secret during an initial sharing phase such that the secret is hidden from an adversary that corrupts at most t parties. In a subsequent reconstruction phase of the protocol, a unique secret, well-defined by the view of honest players in the sharing phase, is reconstructed. The round complexity of VSS protocols is a very important metric of their efficiency. We show two improvements regarding the round complexity of information-theoretic VSS. First, we construct an efficient perfectly secure VSS protocol tolerating t < n/3 corrupted parties that is simultaneously optimal in both the number of rounds and the number of invocations of broadcast. Second, we construct a statistically secure VSS protocol tolerating t < n/2 corrupted parties that has optimal round complexity, and an efficient statistical VSS protocol tolerating t < n/2 corrupted parties that requires one additional round.
We first study the problem of doing Verifiable Secret Sharing (VSS) information theoretically secure for a general access structure. We do it in the model where private channels between players and a broadcast channel is given, and where an active, adaptive adversary can corrupt any set of players not in the access structure. In particular, we consider the complexity of protocols for this problem, as a function of the access structure and the number of players. For all access structures where VSS is possible at all, we show that, up to a polynomial time black-box reduction, the complexity of adaptively secure VSS is the same as that of ordinary secret sharing (SS), where security is only required against a passive, static adversary. Previously, such a connection was only known for linear secret sharing and VSS schemes. We then show an impossibility result indicating that a similar equivalence does not hold for Multiparty Computation (MPC): we show that even if protocols are given blac...
In a t-out-of-n robust secret sharing scheme, a secret message is shared among n parties who can reconstruct the message by combining their shares. An adversary can adaptively corrupt up to t of the parties, get their shares, and modify them arbitrarily. The scheme should satisfy privacy, meaning that the adversary cannot learn anything about the shared message, and robustness, meaning that the adversary cannot cause the reconstruction procedure to output an incorrect message. Such schemes are only possible in the case of an honest majority, and here we focus on unconditional security in the maximal corruption setting where n = 2t+ 1. In this scenario, to share an m-bit message with a reconstruction failure probability of at most 2−k, a known lower-bound shows that the share size must be at least m + k bits. On the other hand, all prior constructions have share size that scales linearly with the number of parties n, and the prior state-of-the-art scheme due to Cevallos et al. (EUROCRYPT ’12) achieves m+ Õ(k + n). In this work, we construct the first robust secret sharing scheme in the maximal corruption setting with n = 2t+ 1, that avoids the linear dependence between share size and the number of
The round complexity of verifiable secret sharing (VSS) schemes has been studied extensively for threshold adversaries. In particular, Fitzi et al. showed an efficient 3-round VSS for n ≥ 3t+1 [4], where an infinitely powerful adversary can corrupt t (or less) parties out of n parties. This paper shows that for non-threshold adversaries, 1. Two round VSS is possible iff the underlying adversary structure satisfies Q 4 condition; 2. Three round VSS is possible iff the underlying adversary structure satisfies Q 3 condition. Further as a special case of our three round protocol, we can obtain a more efficient 3-round VSS than the VSS of Fitzi et al. for n = 3t + 1. More precisely, the communication complexity of the reconstruction phase is reduced from O(n 3) to O(n 2). We finally point out a flaw in the reconstruction phase of VSS of Fitzi et al., and show how to fix it.
We first study the problem of doing Verifiable Secret Sharing (VSS) information theoretically secure for a general access structure.We do it in the model where private channels between players and a broadcast channel is given, and where an active, adaptive adversary can corrupt any set of players not in the access structure.In particular, we consider the complexity of protocols for this problem, as a function of the access structure and the number of players.For all access structures where VSS is possible at all, we show that, up to a polynomial time black-box reduction, the complexity of adaptively secure VSS is the same as that of ordinary secret sharing (SS), where security is only required against a passive, static adversary.Previously, such a connection was only known for linear secret sharing and VSS schemes.We then show an impossibility result indicating that a similar equivalence does hot hold for Multiparty Computation (MPC): we show that even if protocols are given black-box access for free to an idealized secret sharing scheme secure for the access structure in question, it is not possible to handle all relevant access structures efficiently, not even if the adversary is passive and static.In other words, general MPC can only be black-box reduced efficiently to secret sharing if extra properties of the secret sharing scheme used (such as linearity) are assumed.
Verifiable secret sharing (VSS) is a fundamental cryptographic primitive, lying at the core of secure multi-party computation (MPC) and, as the distributed analogue of a commitment functionality, used in numerous applications. In this paper we focus on unconditionally secure VSS protocols with honest majority. In this setting it is typically assumed that parties are connected pairwise by authenticated, private channels, and that in addition they have access to a “broadcast ” channel. Because broadcast cannot be simulated on a point-to-point network when a third or more of the parties are corrupt, it is impossible to construct VSS (and more generally, MPC) protocols in this setting without using a broadcast channel (or some equivalent addition to the model). A great deal of research has focused on increasing the efficiency of VSS, primarily in terms of round complexity. In this work we consider a refinement of the round complexity of VSS, by adding a measure we term broadcast complexity. We view the broadcast channel as an expensive resource and seek to minimize the number of rounds in which it is invoked as well. We construct a (linear) VSS protocol which uses the broadcast channel only twice in the sharing phase, while running in an overall constant number of rounds. 1