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IACR Communications in Cryptology

Synchronous Distributed Key Generation without Broadcasts

Nibesh Shrestha, Adithya Bhat, Aniket Kate, Kartik Nayak

Published Jul 8, 2024• 4 citations
Publisher

Abstract

Distributed key generation (DKG) is a key building block in developing many efficient threshold cryptosystems. This work initiates the study of communication complexity and round complexity of DKG protocols over a point-to-point (bounded) synchronous network. Our key result is the first synchronous DKG protocol for discrete log-based cryptosystems with O ( κ n 3 ) communication complexity ( κ denotes a security parameter) that tolerates any t < n / 2 Byzantine faults among n parties. We present two variants of the protocol: (i) a protocol with worst-case O ( κ n 3 ) communication and O ( t ) rounds, and (ii) a protocol with expected O ( κ n 3 ) communication and expected constant rounds. In the process of achieving our results, we design (1) a novel weak gradecast protocol with a communication complexity of O ( κ n 2 ) for linear-sized inputs and constant rounds, (2) a protocol called “recoverable-set-of-shares” for ensuring recovery of shared secrets, (3) an oblivious leader election protocol with O ( κ n 3 ) communication and constant rounds, and (4) a multi-valued validated Byzantine agreement (MVBA) protocol with O ( κ n 3 ) communication complexity for linear-sized inputs and expected constant rounds. Each of these primitives is of independent interest.

DOI
10.62056/ayfhsgvtw

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Distributed key generation (DKG) is a key building block in developing many efficient threshold cryptosystems. This work initiates the study of communication complexity and round complexity of DKG protocols over a point-to-point (bounded) synchronous network. Our key result is the first synchronous DKG protocol for discrete log-based cryptosystems with O ( κ n 3 ) communication complexity ( κ denotes a security parameter) that tolerates any t < n / 2 Byzantine faults among n parties. We present two variants of the protocol: (i) a protocol with worst-case O ( κ n 3 ) communication and O ( t ) rounds, and (ii) a protocol with expected O ( κ n 3 ) communication and expected constant rounds. In the process of achieving our results, we design (1) a novel weak gradecast protocol with a communication complexity of O ( κ n 2 ) for linear-sized inputs and constant rounds, (2) a protocol called “recoverable-set-of-shares” for ensuring recovery of shared secrets, (3) an oblivious leader election protocol with O ( κ n 3 ) communication and constant rounds, and (4) a multi-valued validated Byzantine agreement (MVBA) protocol with O ( κ n 3 ) communication complexity for linear-sized inputs and expected constant rounds. Each of these primitives is of independent interest.
Distributed Key Generation (DKG) is a technique to bootstrap threshold cryptosystems without a trusted third party and is a building block to decentralized protocols such as randomness beacons, threshold signatures, and general multiparty computation. Until recently, DKG protocols have assumed the synchronous model and thus are vulnerable when their underlying network assumptions do not hold. The recent advancements in asynchronous DKG protocols are insufficient as they either have poor efficiency or limited functionality, resulting in a lack of concrete implementations. In this paper, we present a simple and concretely efficient asynchronous DKG (ADKG) protocol. In a network of n nodes, our ADKG protocol can tolerate up to t<n/3t\lt n/3t<n/3 malicious nodes and have an expected O(κn3)O(\kappa n^{3})O(κn3) communication cost, where κ\kappaκ is the security parameter. Our ADKG protocol produces a field element as the secret and is thus compatible with off-the-shelf threshold cryptosystems. We implement our ADKG protocol and evaluate it using a network of up to 128 nodes in geographically distributed AWS instances. Our evaluation shows that our protocol takes as low as 3 and 9.5 seconds to terminate for 32 and 64 nodes, respectively. Also, each node sends only 0.7 Megabytes and 2.9 Megabytes of data during the two experiments, respectively.
Distributed key generation (DKG) has been studied extensively in the cryptographic literature. However, it has never been examined outside of the synchronous setting, and the known DKG protocols cannot guarantee safety or liveness over the Internet. In this work, we present the first realistic DKG protocol for use over the Internet. We propose a practical system model for the Internet and define an efficient verifiable secret sharing (VSS) scheme in it. We observe the necessity of Byzantine agreement for asynchronous DKG and analyze the difficulty of using a randomized protocol for it. Using our VSS scheme and a leader-based agreement protocol, we then design a provably secure DKG protocol. We also consider and achieve cryptographic properties such as uniform randomness of the shared secret and compare static versus adaptive adversary models. Finally, we implement our DKG protocol, and establish its efficiency and reliability by extensively testing it on the PlanetLab platform. Counter to a general non-scalability perception about asynchronous systems, our experiments demonstrate that our asynchronous DKG protocol scales well with the system size and it is suitable for realizing multiparty computation and threshold cryptography over the Internet.
In this paper, we present the first Asynchronous Distributed Key Generation (ADKG) algorithm which is also the first distributed key generation algorithm that can generate cryptographic keys with a dual (f,2f+1)-threshold (where f is the number of faulty parties). As a result, using our ADKG we remove the trusted setup assumption that the most scalable consensus algorithms make. In order to create a DKG with a dual (f,2f+1)- threshold we first answer in the affirmative the open question posed by Cachin et al. [7] on how to create an Asynchronous Verifiable Secret Sharing (AVSS) protocol with a reconstruction threshold of f+1<k łe 2f+1, which is of independent interest. Our High-threshold-AVSS (HAVSS) uses an asymmetric bivariate polynomial to encode the secret. This enables the reconstruction of the secret only if a set of k nodes contribute while allowing an honest node that did not participate in the sharing phase to recover his share with the help of f+1 honest parties. Once we have HAVSS we can use it to bootstrap scalable partially synchronous consensus protocols, but the question on how to get a DKG in asynchrony remains as we need a way to produce common randomness. The solution comes from a novelEventually Perfect Common Coin (EPCC) abstraction that enables the generation of a common coin from n concurrent HAVSS invocations. EPCC's key property is that it is eventually reliable, as it might fail to agree at most f times (even if invoked a polynomial number of times). UsingEPCC we implement anEventually Efficient Asynchronous Binary Agreement (EEABA) which is optimal when the EPCC agrees and protects safety when EPCC fails. Finally, using EEABA we construct the first ADKG which has the same overhead and expected runtime as the best partially-synchronous DKG (O(n4) words, O(f) rounds). As a corollary of our ADKG, we can also create the first Validated Asynchronous Byzantine Agreement (VABA) that does not need a trusted dealer to setup threshold signatures of degree n-f. Our VABA has an overhead of expected O(n2) words and O(1) time per instance, after an initial O(n4) words and O(f) time bootstrap via ADKG.
Verifiable secret sharing (VSS) is a vital primitive in secure distributed computing. It allows an untrusted dealer to verifiably share a secret among n parties in the presence of an adversary controlling at most t of them. VSS in the synchronous communication model has received tremendous attention in the cryptographic research community. Nevertheless, recent interest in deploying secure distributed computing over the Internet requires going beyond the synchronous communication model and thoroughly investigating VSS in the asynchronous communication model. In this work, we consider the communication complexity of asynchronous VSS in the computational setting for the optimal resilience of n = 3t + 1. The best known asynchronous VSS protocol by Cachin et al. has O(n 2) message complexity and O(κn 3) communication complexity, where κ is a security parameter corresponding to the size of the secret. We close the linear complexity gap between these two measures for asynchronous VSS by presenting two protocols with O(n 2) message complexity and O(κn 2) communication complexity. Our first protocol satisfies the standard VSS definition, and can be used in stand-alone VSS scenarios as well as in applications such as Byzantine agreement. Our second and more intricate protocol satisfies a stronger VSS definition, and is useful in all VSS applications including multiparty computation and threshold cryptography.
Abstract. Byzantine broadcast is a distributed primitive that allows a specific party (called “sender”) to consistently distribute a value v among n parties in the presence of potential misbehavior of up to t of the parties. Broadcast requires that correct parties always agree on the same value and if the sender is correct, then the agreed value is v. Broadcast without a setup (i.e., from scratch) is achievable from point-to-point channels if and only if t < n/3. In case t ≥ n/3 a trusted setup is required. The setup may be assumed to be given initially or generated by the parties in a setup phase. It is known that generating setup for protocols with cryptographic security is relatively simple and only consists of setting up a public-key infrastructure. However, generating setup for information-theoretically secure protocols is much more involved. In this paper we study the complexity of setup generation for information-theoretic protocols using point-to-point channels and temporarily available broadcast channels. We optimize the number of rounds in which the temporary broadcast channels are used while minimizing the number of bits broadcast with them. We give the first information-theoretically secure broadcast protocol tolerating t < n/2 that uses the temporary broadcast channels during only 1 round in the setup phase. Furthermore, only O(n 3) bits need to be broadcast with the temporary broadcast channels during that round, independently of the security parameter employed. The broadcast protocol presented in this paper allows to construct the first informationtheoretically secure MPC protocol which uses a broadcast channel during only one round. Additionally, the presented broadcast protocol supports refreshing, which allows to broadcast an a priori unknown number of times given a fixed-size setup.
Independence is a fundamental property needed to achieve security in fault-tolerant distributed computing. In practice, distributed communication networks are neither fully synchronous or fully asynchronous, but rather loosely synchronized. By this, we mean that in a communication protocol, messages at a given round may depend on messages from other players at the same round. These possible dependencies among messages create problems if we need n players to announce independently chosen values. This task is called simultaneous broadcast. In this paper, we present the first constant round protocol for simultaneous broadcast in a reasonable computation model (which includes a common shared random string among the players). The protocol is provably secure under general cryptographic assumptions. In the process, we develop a new and stronger formal definition for this problem. Previously known protocols for this task required either O(log n) or expected constant rounds to complete (depending on the computation model considered).