International Association
for Cryptologic Research

Messages in large-scale networks such as blockchain systems are typically disseminated using flooding protocols, in which parties send the message to a random set of peers until it reaches all parties. Optimizing the communication complexity of such protocols and, in particular, the per-party communication complexity is of primary interest since nodes in a network are often subject to bandwidth constraints. Previous flooding protocols incur a per-party communication complexity of $\Omega(l\cdot \gamma^{-1} \cdot (\log(n) + \kappa))$ bits to disseminate an $l$-bit message among $n$ parties with security parameter~$\kappa$ when it is guaranteed that a $\gamma$ fraction of the parties remain honest. In this work, we present the first flooding protocols with a per-party communication complexity of $O(l\cdot \gamma^{-1})$ bits. We further show that this is asymptotically optimal and that our protocols can be instantiated provably securely in the usual setting for proof-of-stake blockchains. To demonstrate that one of our new protocols is not only asymptotically optimal but also practical, we perform several probabilistic simulations to estimate the concrete complexity for given parameters. Our simulations show that our protocol significantly improves the per-party communication complexity over the state-of-the-art for practical parameters. Hence, for given bandwidth constraints, our results allow to, e.g., increase the block size, improving the overall throughput of a blockchain.

Distributed key generation (DKG) protocols are an essential building block for threshold cryptosystems. Many DKG protocols tolerate up to t_s<n/2 corruptions assuming a well-behaved synchronous network, but become insecure as soon as the network delay becomes unstable. On the other hand, solutions in the asynchronous model operate under arbitrary network conditions, but only tolerate t_a<n/3 corruptions, even when the network is well-behaved. In this work, we ask whether one can design a protocol that achieves security guarantees in either scenario. We show a complete characterization of _network-agnostic_ DKG protocols, showing that the tight bound is t_a + 2t_s < n. As a second contribution, we provide an optimized version of the network-agnostic MPC protocol by Blum, Liu-Zhang and Loss [CRYPTO'20] which improves over the communication complexity of their protocol by a linear factor. Moreover, using our DKG protocol, we can instantiate our MPC protocol in the _plain PKI model_, i.e., without the need to assume an expensive trusted setup. Our protocols incur comparable communication complexity as state-of-the-art DKG and MPC protocols with optimal resilience in their respective purely synchronous and asynchronous settings, thereby showing that network-agnostic security comes (almost) _for free_.

The classical “BGW protocol” (Ben-Or, Goldwasser and Wigderson, STOC 1988) shows that secure multiparty computation (MPC) among n parties can be realized with perfect full security if t < n/3 parties are corrupted. This holds against malicious adversaries in the “standard” model for MPC, where a fixed set of n parties is involved in the full execution of the protocol. However, the picture is less clear in the mobile adversary setting of Ostrovsky and Yung (PODC 1991), where the adversary may periodically “move” by uncorrupting parties and corrupting a new set of t parties. In this setting, it is unclear if full security can be achieved against an adversary that is maximally mobile, i.e., moves after every round. The question is further motivated by the “You Only Speak Once” (YOSO) setting of Gentry et al. (Crypto 2021), where not only the adversary is mobile but also each round is executed by a disjoint set of parties. Previous positive results in this model do not achieve perfect security, and either assume probabilistic corruption and a nonstandard communication model, or only realize the weaker goal of security-with-abort. The question of matching the BGW result in these settings remained open. In this work, we tackle the above two challenges simultaneously. We consider a layered MPC model, a simplified variant of the fluid MPC model of Choudhuri et al. (Crypto 2021). Layered MPC is an instance of standard MPC where the interaction pattern is defined by a layered graph of width n, allowing each party to send secret messages and broadcast messages only to parties in the next layer. We require perfect security against a malicious adversary who may corrupt at most t parties in each layer. Our main result is a perfect, fully secure layered MPC protocol with an optimal corruption threshold of t < n/3, thus extending the BGW feasibility result to the layered setting. This implies perfectly secure MPC protocols against a maximally mobile adversary.

Byzantine agreement is a fundamental primitive in cryptography and distributed computing, and minimizing its round complexity is of paramount importance. It is long known that any randomized $r$-round protocol must fail with probability at least $(c\cdot r)^{-r}$, for some constant $c$, when the number of corruptions is linear in the number of parties, $t = \theta(n)$. On the other hand, current protocols fail with probability at least $2^{-r}$. Whether we can match the lower bound agreement probability remains unknown. In this work, we resolve this long-standing open question. We present a protocol that matches the lower bound up to constant factors. Our results hold under a (strongly rushing) adaptive adversary that can corrupt up to $t = (1-\epsilon)n/2$ parties, and our protocols use a public-key infrastructure and a trusted setup for unique threshold signatures. This is the first protocol that decreases the failure probability (overall) by a \emph{super-constant} factor per round.

We investigate the communication complexity of Byzantine agreement protocols for long messages against an adaptive adversary. In this setting, prior $n$-party protocols either achieved a communication complexity of $O(nl\cdot\poly(\kappa))$ or $O(nl + n^2 \cdot \poly(\kappa))$ for $l$-bit long messages and security parameter $\kappa$. We improve the state of the art by presenting protocols with communication complexity $O(nl + n \cdot \poly(\kappa))$ in both the synchronous and asynchronous communication models. The synchronous protocol tolerates $t \le (1-\epsilon) \frac{n}{2}$ corruptions and assumes a VRF setup, while the asynchronous protocol tolerates $t \le (1-\epsilon) \frac{n}{3}$ corruptions under further cryptographic assumptions. Our protocols are very simple and combine subcommittee election with the recent approach of Nayak et al. (DISC'20). Surprisingly, the analysis of our protocols is 'all but simple' and involves an interesting new application of Mc Diarmid's inequality to obtain 'almost optimal' corruption thresholds.

In recent years, permisionless blockchains have received a lot of attention both from industry and academia, where substantial effort has been spent to develop consensus protocols that are secure under the assumption that less than half (or a third) of a given resource (e.g., stake or computing power) is controlled by corrupted parties. The security proofs of these consensus protocols usually assume the availability of a network functionality guaranteeing that a block sent by an honest party is received by all honest parties within some bounded time. To obtain an overall protocol that is secure under the same corruption assumption, it is therefore necessary to combine the consensus protocol with a network protocol that achieves this property under that assumption. In practice, however, the underlying network is typically implemented by flooding protocols that are not proven to be secure in the setting where a fraction of the considered total weight can be corrupted. This has led to many so-called eclipse attacks on existing protocols and tailor-made fixes against specific attacks. To close this apparent gap, we present the first practical flooding protocol that provably delivers sent messages to all honest parties after a logarithmic number of steps. We prove security in the setting where all parties are publicly assigned a positive weight and the adversary can corrupt parties accumulating up to a constant fraction of the total weight. This can directly be used in the proof-of-stake setting, but is not limited to it. To prove the security of our protocol, we combine known results about the diameter of Erdős–Rényi graphs with reductions between different types of random graphs. We further show that the efficiency of our protocol is asymptotically optimal. The practicality of our protocol is supported by extensive simulations for different numbers of parties, weight distributions, and corruption strategies. The simulations confirm our theoretical results and show that messages are delivered quickly regardless of the weight distribution, whereas protocols that are oblivious of the parties' weights completely fail if the weights are unevenly distributed. Furthermore, the average message complexity per party of our protocol is within a small constant factor of such a protocol.

Protocols for Byzantine agreement (BA) and secure multi-party computation (MPC) can be classified according to the underlying communication model. The two most commonly considered models are the synchronous one and the asynchronous one. Synchronous protocols typically lose their security guarantees as soon as the network violates the synchrony assumptions. Asynchronous protocols remain secure regardless of the network conditions, but achieve weaker security guarantees even when the network is synchronous. Recent works by Blum, Katz and Loss [TCC'19], and Blum, Liu-Zhang and Loss [CRYPTO'20] introduced BA and MPC protocols achieving security guarantees in both settings: security up to $t_s$ corruptions in a synchronous network, and up to $t_a$ corruptions in an asynchronous network, under the provably optimal threshold trade-offs $t_a \le t_s$ and $t_a + 2t_s < n$. However, current solutions incur a high synchronous round complexity when compared to state-of-the-art purely synchronous protocols. When the network is synchronous, the round complexity of BA protocols is linear in the number of parties, and the round complexity of MPC protocols also depends linearly on the depth of the circuit to evaluate. In this work, we provide round-efficient constructions for both primitives with optimal resilience: fixed-round and expected constant-round BA protocols, and an MPC protocol whose round complexity is independent of the circuit depth.

The goal of secure multi-party computation (MPC) is to allow a set of parties to perform an arbitrary computation task, where the security guarantees depend on the set of parties that are corrupted. The more parties are corrupted, the less is guaranteed, and typically the guarantees are completely lost when the number of corrupted parties exceeds a certain corruption bound. Early and also many recent protocols are only statically secure in the sense that they provide no security guarantees if the adversary is allowed to choose adaptively which parties to corrupt. Security against an adversary with such a strong capability is often called adaptive security and a significant body of literature is devoted to achieving adaptive security, which is known as a difficult problem. In particular, a main technical obstacle in this context is the so-called ``commitment problem'', where the simulator is unable to consistently explain the internal state of a party with respect to its pre-corruption outputs. As a result, protocols typically resort to the use of cryptographic primitives like non-committing encryption, incurring a substantial efficiency loss. This paper provides a new, clean-slate treatment of adaptive security in MPC, exploiting the specification concept of constructive cryptography (CC). A new natural security notion, called \cc-adaptive security, is proposed, which is technically weaker than standard adaptive security but nevertheless captures security against a fully adaptive adversary. Known protocol examples separating between adaptive and static security are also insecure in our notion. Moreover, our notion avoids the commitment problem and thereby the need to use non-committing or equivocal tools. We exemplify this by showing that the protocols by Cramer, Damgard and Nielsen (EUROCRYPT'01) for the honest majority setting, and (the variant without non-committing encryption) by Canetti, Lindell, Ostrovsky and Sahai (STOC'02) for the dishonest majority setting, achieve \cc-adaptive security. The latter example is of special interest since all \uc-adaptive protocols in the dishonest majority setting require some form of non-committing encryption or equivocal tools.

Secure multi-party computation (MPC) allows a set of $n$ parties to jointly compute an arbitrary computation over their private inputs. Two main variants have been considered in the literature according to the underlying communication model. Synchronous MPC protocols proceed in rounds, and rely on the fact that the communication network provides strong delivery guarantees within each round. Asynchronous MPC protocols achieve security guarantees even when the network delay is arbitrary. While the problem of MPC has largely been studied in both variants with respect to both feasibility and efficiency results, there is still a substantial gap when it comes to communication complexity of adaptively secure protocols. Concretely, while adaptively secure synchronous MPC protocols with linear communication are known for a long time, the best asynchronous protocol communicates $\mathcal{O}(n^4 \kappa)$ bits per multiplication. In this paper, we make progress towards closing this gap by providing two protocols. First, we present an adaptively secure asynchronous protocol with optimal resilience $t<n/3$ and $\mathcal{O}(n^2 \kappa)$ bits of communication per multiplication, improving over the state of the art protocols in this setting by a quadratic factor in the number of parties. The protocol has cryptographic security and follows the CDN approach [Eurocrypt'01], based on additive threshold homomorphic encryption. Second, we show an optimization of the above protocol that tolerates up to $t<(1-\epsilon)n/3$ corruptions and communicates $\mathcal{O}(n\cdot \poly(\kappa))$ bits per multiplication under stronger assumptions.

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.

Protocols for secure Multi-Party Computation (MPC) can be classified according to the underlying communication model. Two prominent communication models considered in the literature are the synchronous and asynchronous models, which considerably differ in terms of the achievable security guarantees. Synchronous MPC protocols can achieve the optimal corruption threshold $n/2$ and allow every party to give input, but become completely insecure when synchrony assumptions are violated. On the other hand, asynchronous MPC protocols remain secure under arbitrary network conditions, but can tolerate only $n/3$ corruptions and parties with slow connections unavoidably cannot give input. A natural question is whether there exists a protocol for MPC that can tolerate up to $t_s < n/2$ corruptions under a synchronous network and $t_a < n/3$ corruptions even when the network is asynchronous. We answer this question by showing tight feasibility and impossibility results. More specifically, we show that such a protocol exists if and only if $t_a + 2t_s < n$ and the number of inputs taken into account under an asynchronous network is at most $n-t_s$.

This paper proposes a simple synchronous composable security framework as an instantiation of the Constructive Cryptography framework, aiming to capture minimally, without unnecessary artefacts, exactly what is needed to state synchronous security guarantees. The objects of study are specifications (i.e., sets) of systems, and traditional security properties like consistency and validity can naturally be understood as specifications, thus unifying composable and property-based definitions. The framework's simplicity is in contrast to current composable frameworks for synchronous computation which are built on top of an asynchronous framework (e.g. the UC framework), thus not only inheriting artefacts and complex features used to handle asynchronous communication, but adding additional overhead to capture synchronous communication. As a second, independent contribution we demonstrate how secure (synchronous) multi-party computation protocols can be understood as constructing a computer that allows a set of parties to perform an arbitrary, on-going computation. An interesting aspect is that the instructions of the computation need not be fixed before the protocol starts but can also be determined during an on-going computation, possibly depending on previous outputs.

Understanding the communication complexity of Byzantine agreement (BA) is a fundamental problem in distributed computing. In particular, as protocols are run with a large number of parties (as, e.g., in the context of blockchain protocols), it is important to understand the dependence of the communication on the number of parties~$n$. Although adaptively secure BA protocols with $o(n^2)$ communication are known in the synchronous and partially synchronous settings, no such protocols are known in the fully asynchronous case. We show here an asynchronous BA protocol with subquadratic communication tolerating an adaptive adversary who can corrupt $f<(1-\epsilon)n/3$ of the parties (for any $\epsilon>0$). One variant of our protocol assumes initial setup done by a trusted dealer, after which an unbounded number of BA executions can be run; alternately, we can achieve subquadratic \emph{amortized} communication with no prior setup. We also show that some form of setup is needed for (non-amortized) subquadratic BA tolerating $\Theta(n)$ corrupted parties. As a contribution of independent interest, we show a secure-computation protocol in the same threat model that has $o(n^2)$ communication when computing no-input functionalities with short output (e.g., coin tossing).

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.

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.