International Association
for Cryptologic Research

The field of distributed certification is concerned with certifying properties of distributed networks, where the communication topology of the network is represented as an arbitrary graph; each node of the graph is a separate processor, with its own internal state. To certify that the network satisfies a given property, a prover assigns each node of the network a certificate, and the nodes then communicate with one another and decide whether to accept or reject. We require soundness and completeness: the property holds if and only if there exists an assignment of certificates to the nodes that causes all nodes to accept. Our goal is to minimize the length of the certificates, as well as the communication between the nodes of the network. Distributed certification has been extensively studied in the distributed computing community, but it has so far only been studied in the information-theoretic setting, where the prover and the network nodes are computationally unbounded. In this work we introduce and study computationally bounded distributed certification: we define locally verifiable distributed SNARG (LVDSNARGs), which are an analog of SNARGs for distributed networks, and are able to circumvent known hardness results for information-theoretic distributed certification by requiring both the prover and the verifier to be computationally efficient (namely, PPT algorithms). We give two LVDSNARG constructions: the first allows us to succinctly certify any network property in P, using a global prover that can see the entire network; the second construction gives an efficient distributed prover, which succinctly certifies the execution of any efficient distributed algorithm. Our constructions rely on non-interactive batch arguments for NP (BARGs) and on RAM SNARGs, which have recently been shown to be constructible from standard cryptographic assumptions.

Topology-hiding broadcast ( THB ) enables parties communicating over an incomplete network to broadcast messages while hiding the topology from within a given class of graphs. THB is a central tool underlying general topology-hiding secure computation ( THC ) (Moran et al. TCC’15). Although broadcast is a privacy-free task, it was recently shown that THB for certain graph classes necessitates computational assumptions, even in the semi-honest setting, and even given a single corrupted party. In this work, we investigate the minimal assumptions required for topology-hiding communication: both Broadcast or Anonymous Broadcast (where the broadcaster’s identity is hidden). We develop new techniques that yield a variety of necessary and sufficient conditions for the feasibility of THB / THAB in different cryptographic settings: information theoretic, given existence of key agreement, and given existence of oblivious transfer. Our results show that feasibility can depend on various properties of the graph class, such as connectivity , and highlight the role of different properties of topology when kept hidden, including direction , distance , and/or distance-of-neighbors to the broadcaster. An interesting corollary of our results is a dichotomy for THC with a public number of at least three parties, secure against one corruption: information-theoretic feasibility if all graphs are 2-connected; necessity and sufficiency of key agreement otherwise.

A distributed computation in which nodes are connected by a partial communication graph is called topology hiding if it does not reveal information about the graph beyond what is revealed by the output of the function. Previous results have shown that topology-hiding computation protocols exist for graphs of constant degree and logarithmic diameter in the number of nodes (Moran–Orlov–Richelson, TCC’15; Hirt et al., Crypto’16) as well as for other graph families, such as cycles, trees, and low circumference graphs (Akavia–Moran, Eurocrypt’17), but the feasibility question for general graphs was open. In this work, we positively resolve the above open problem: we prove that topology-hiding computation is feasible for all graphs under either the decisional Diffie–Hellman or quadratic residuosity assumption. Our techniques employ random or deterministic walks to generate paths covering the graph, upon which we apply the Akavia–Moran topology-hiding broadcast for chain graphs (paths). To prevent topology information revealed by the random walk, we design multiple graph-covering sequences that, together, are locally identical to receiving at each round a message from each neighbor and sending back a processed message from some neighbor (in a randomly permuted order).

Topology-Hiding Computation (THC) allows a set of parties to securely compute a function over an incomplete network without revealing information on the network topology. Since its introduction in TCC’15 by Moran et al., the research on THC has focused on reducing the communication complexity, allowing larger graph classes, and tolerating stronger corruption types. All of these results consider a fully synchronous model with a known upper bound on the maximal delay of all communication channels. Unfortunately, in any realistic setting this bound has to be extremely large, which makes all fully synchronous protocols inefficient. In the literature on multi-party computation, this is solved by considering the fully asynchronous model. However, THC is unachievable in this model (and even hard to define), leaving even the definition of a meaningful model as an open problem. The contributions of this paper are threefold. First, we introduce a meaningful model of unknown and random communication delays for which THC is both definable and achievable. The probability distributions of the delays can be arbitrary for each channel, but one needs to make the (necessary) assumption that the delays are independent. The existing fully-synchronous THC protocols do not work in this setting and would, in particular, leak information about the topology. Second, in the model with trusted stateless hardware boxes introduced at Eurocrypt’18 by Ball et al., we present a THC protocol that works for any graph class. Third, we explore what is achievable in the standard model without trusted hardware and present a THC protocol for specific graph types (cycles and trees) secure under the DDH assumption. The speed of all protocols scales with the actual (unknown) delay times, in contrast to all previously known THC protocols whose speed is determined by the assumed upper bound on the network delay.

An incompressible encoding can probabilistically encode some data $m$ into a codeword $c$, which is not much larger. Anyone can decode $c$ to recover the original data $m$. However, $c$ cannot be efficiently compressed, even if the original data $m$ is given to the decompression procedure for free. In other words, $c$ is a representation of $m$, yet is computationally incompressible even given $m$. An incompressible encoding is composable if many encodings cannot be simultaneously compressed into anything sufficiently smaller than their concatenation. A recent work of Damgard, Ganesh and Orlandi (CRYPTO '19) defined a variant of incompressible encodings and gave an applications to ``proofs of replicated storage''. They constructed incompressible encodings in an ideal permutation model over a structured domain, but it was left open if they can be constructed under standard assumptions, or even in the more basic random-oracle model. In this work, we give new constructions, negative results and applications of incompressible encodings: * We construct incompressible encodings in the common random string (CRS) model under the Decisional Composite Residuosity (DCR) or Learning with Errors (LWE) assumptions. However, the construction has several drawbacks: (1) it is not composable, (2) it only achieves selective security, and (3) the CRS is as long as the data $m$. * We leverage the above construction to also get a scheme in the random-oracle model, under the same assumptions, that avoids all of the above drawbacks. Furthermore, it is significantly more efficient than the prior ideal-model construction. * We give black-box separations, showing that incompressible encodings in the plain model cannot be proven secure under any standard hardness assumption, and incompressible encodings in the CRS model must inherently suffer from all of the drawbacks above. * We give a new application to ``big-key cryptography in the bounded-retrieval model'', where secret keys are made intentionally huge to make them hard to exfiltrate. Using incompressible encodings, we can get all the security benefits of a big key without wasting storage space, by having the key to encode useful data.

Topology-hiding broadcast (THB) enables parties communicating over an incomplete network to broadcast messages while hiding the topology from within a given class of graphs. THB is a central tool underlying general topology-hiding secure computation (THC) (Moran et al. TCC’15). Although broadcast is a privacy-free task, it was recently shown that THB for certain graph classes necessitates computational assumptions, even in the semi-honest setting, and even given a single corrupted party. In this work we investigate the minimal assumptions required for topology-hiding communication—both Broadcast or Anonymous Broadcast (where the broadcaster’s identity is hidden). We develop new techniques that yield a variety of necessary and sufficient conditions for the feasibility of THB/THAB in different cryptographic settings: information theoretic, given existence of key agreement, and given existence of oblivious transfer. Our results show that feasibility can depend on various properties of the graph class, such as connectivity, and highlight the role of different properties of topology when kept hidden, including direction, distance, and/or distance-of-neighbors to the broadcaster. An interesting corollary of our results is a dichotomy for THC with a public number of at least three parties, secure against one corruption: information-theoretic feasibility if all graphs are 2-connected; necessity and sufficiency of key agreement otherwise.

Two paradigms for secure MPC are synchronous and asynchronous protocols. While synchronous protocols tolerate more corruptions and allow every party to give its input, they are very slow because the speed depends on the conservatively assumed worst-case delay $\Delta$ of the network. In contrast, asynchronous protocols allow parties to obtain output as fast as the actual network allows, a property called \emph{responsiveness}, but unavoidably have lower resilience and parties with slow network connections cannot give input. It is natural to wonder whether it is possible to leverage synchronous MPC protocols to achieve responsiveness, hence obtaining the advantages of both paradigms: full security with responsiveness up to t corruptions, and 'extended' security (full security or security with unanimous abort) with no responsiveness up to a larger threshold T of corruptions. We settle the question by providing matching feasibility and impossibility results: -For the case of unanimous abort as extended security, there is an MPC protocol if and only if T + 2t < n. -For the case of full security as extended security, there is an MPC protocol if and only if T < n/2 and T + 2t < n. In particular, setting t = n/4 allows to achieve a fully secure MPC for honest majority, which in addition benefits from having substantial responsiveness.

We introduce a new cryptographic primitive: Proofs of Space-Time (PoSTs) and construct an extremely simple, practical protocol for implementing these proofs. A PoST allows a prover to convince a verifier that she spent a “space-time” resource (storing data—space—over a period of time). Formally, we define the PoST resource as a trade-off between CPU work and space-time (under reasonable cost assumptions, a rational user will prefer to use the lower-cost space-time resource over CPU work).Compared to a proof-of-work, a PoST requires less energy use, as the “difficulty” can be increased by extending the time period over which data is stored without increasing computation costs. Our definition is very similar to “Proofs of Space” [ePrint 2013/796, 2013/805] but, unlike the previous definitions, takes into account amortization attacks and storage duration. Moreover, our protocol uses a very different (and much simpler) technique, making use of the fact that we explicitly allow a space-time tradeoff, and doesn’t require any non-standard assumptions (beyond random oracles). Unlike previous constructions, our protocol allows incremental difficulty adjustment, which can gracefully handle increases in the price of storage compared to CPU work. In addition, we show how, in a crypto-currency context, the parameters of the scheme can be adjusted using a market-based mechanism, similar in spirit to the difficulty adjustment for PoW protocols.

Topology-hiding computation (THC) is a form of multi-party computation over an incomplete communication graph that maintains the privacy of the underlying graph topology. Existing THC protocols consider an adversary that may corrupt an arbitrary number of parties, and rely on cryptographic assumptions such as DDH.In this paper we address the question of whether information-theoretic THC can be achieved by taking advantage of an honest majority. In contrast to the standard MPC setting, this problem has remained open in the topology-hiding realm, even for simple “privacy-free” functions like broadcast, and even when considering only semi-honest corruptions.We uncover a rich landscape of both positive and negative answers to the above question, showing that what types of graphs are used and how they are selected is an important factor in determining the feasibility of hiding topology information-theoretically. In particular, our results include the following. We show that topology-hiding broadcast (THB) on a line with four nodes, secure against a single semi-honest corruption, implies key agreement. This result extends to broader classes of graphs, e.g., THB on a cycle with two semi-honest corruptions.On the other hand, we provide the first feasibility result for information-theoretic THC: for the class of cycle graphs, with a single semi-honest corruption. Given the strong impossibilities, we put forth a weaker definition of distributional-THC, where the graph is selected from some distribution (as opposed to worst-case). We present a formal separation between the definitions, by showing a distribution for which information theoretic distributional-THC is possible, but even topology-hiding broadcast is not possible information-theoretically with the standard definition.We demonstrate the power of our new definition via a new connection to adaptively secure low-locality MPC, where distributional-THC enables parties to “reuse” a secret low-degree communication graph even in the face of adaptive corruptions.

Topology-hiding communication protocols allow a set of parties, connected by an incomplete network with unknown communication graph, where each party only knows its neighbors, to construct a complete communication network such that the network topology remains hidden even from a powerful adversary who can corrupt parties. This communication network can then be used to perform arbitrary tasks, for example secure multi-party computation, in a topology-hiding manner. Previously proposed protocols could only tolerate passive corruption. This paper proposes protocols that can also tolerate fail-corruption (i.e., the adversary can crash any party at any point in time) and so-called semi-malicious corruption (i.e., the adversary can control a corrupted party’s randomness), without leaking more than an arbitrarily small fraction of a bit of information about the topology. A small-leakage protocol was recently proposed by Ball et al. [Eurocrypt’18], but only under the unrealistic set-up assumption that each party has a trusted hardware module containing secret correlated pre-set keys, and with the further two restrictions that only passively corrupted parties can be crashed by the adversary, and semi-malicious corruption is not tolerated. Since leaking a small amount of information is unavoidable, as is the need to abort the protocol in case of failures, our protocols seem to achieve the best possible goal in a model with fail-corruption.Further contributions of the paper are applications of the protocol to obtain secure MPC protocols, which requires a way to bound the aggregated leakage when multiple small-leakage protocols are executed in parallel or sequentially. Moreover, while previous protocols are based on the DDH assumption, a new so-called PKCR public-key encryption scheme based on the LWE assumption is proposed, allowing to base topology-hiding computation on LWE. Furthermore, a protocol using fully-homomorphic encryption achieving very low round complexity is proposed.