Detect, Pack and Batch: Perfectly-Secure MPC with Linear Communication and Constant Expected Time

We prove that perfectly-secure optimally-resilient secure Multi-Party Computation (MPC) for a circuit with $C$ gates and depth $D$ can be obtained in $O((Cn+n^4 + Dn^2)\log n)$ communication complexity and $O(D)$ expected time. For $D \ll n$ and $C\geq n^3$, this is the first perfectly-secure optimal-resilient MPC protocol with linear communication complexity per gate and constant expected time complexity per layer. Compared to state-of-the-art MPC protocols in the player elimination framework [Beerliova and Hirt TCC\u2708, and Goyal, Liu, and Song CRYPTO\u2719], for $C>n^3$ and $D \ll n$, our results significantly improve the run time from $\Omega(n+D)$ to expected $O(D)$ while keeping communication complexity at $O(Cn\log n)$. Compared to state-of-the-art MPC protocols that obtain an expected $O(D)$ time complexity [Abraham, Asharov, and Yanai TCC\u2721], for $C>n^3$, our results significantly improve the communication complexity from $O(Cn^4\log n)$ to $O(Cn\log n)$ while keeping the expected run time at $O(D)$. One salient part of our technical contribution is centered around a new primitive we call detectable secret sharing . It is perfectly-hiding, weakly-binding, and has the property that either reconstruction succeeds or $O(n)$ parties are (privately) detected. On the one hand, we show that detectable secret sharing is sufficiently powerful to generate multiplication triplets needed for MPC. On the other hand, we show how to share $p$ secrets via detectable secret sharing with communication complexity of just $O(n^4\log n+p \log n)$. When sharing $p\geq n^4$ secrets, the communication cost is amortized to just $O(1)$ field elements per secret. Our second technical contribution is a new Verifiable Secret Sharing protocol that can share $p$ secrets at just $O(n^4\log n+pn\log n)$ word complexity. When sharing $p\geq n^3$ secrets, the communication cost is amortized to just $O(n)$ filed elements per secret. The best prior required $\Omega(n^3)$ communication per secret

Secure Computation with Constant Communication Overhead using Multiplication Embeddings

Secure multi-party computation (MPC) allows mutually distrusting parties to compute securely over their private data. The hardness of MPC, essentially, lies in performing secure multiplications over suitable algebras. Parties use diverse cryptographic resources, like computational hardness assumptions or physical resources, to securely compute these multiplications. There are several cryptographic resources that help securely compute one multiplication over a large finite field, say $\mathbb{G}\mathbb{F}[2^n]$, with linear communication complexity. For example, the computational hardness assumption like noisy Reed-Solomon codewords are pseudorandom. However, it is not known if we can securely compute, say, a linear number of AND-gates from such resources, i.e., a linear number of multiplications over the base field $\mathbb{G}\mathbb{F}[2]$. Before our work, we could only perform $o(n)$ secure AND-evaluations. This example highlights the general inefficiency of multiplying over the base field using one multiplication over the extension field. Our objective is to remove this hurdle and enable secure computation of boolean circuits while incurring a constant communication overhead based on more diverse cryptographic resources. Technically, we construct a perfectly secure protocol that realizes a linear number of multiplication gates over the base field using one multiplication gate over a degree-$n$ extension field. This construction relies on the toolkit provided by algebraic function fields. Using this construction, we obtain the following results. If we can perform one multiplication over $\mathbb{G}\mathbb{F}[2^n]$ with linear communication using a particular cryptographic resource, then we can also evaluate linear-size boolean circuits with linear communication using the same cryptographic resource. In particular, we provide the first construction that computes a linear number of oblivious transfers with linear communication complexity from the computational hardness assumptions like noisy Reed-Solomon codewords are pseudorandom, or arithmetic-analogues of LPN-style assumptions. Next, we highlight the potential of our result for other applications to MPC by constructing the first correlation extractor that has $1/2$ resilience and produces a linear number of oblivious transfers

Efficient MPC with a Mixed Adversary

Over the past 20 years, the efficiency of secure multi-party protocols has been greatly improved. While the seminal protocols from the late 80’s require a communication of Ω(n⁶) field elements per multiplication among n parties, recent protocols offer linear communication complexity. This means that each party needs to communicate a constant number of field elements per multiplication, independent of n. However, these efficient protocols only offer active security, which implies that at most t<n/3 (perfect security), respectively t<n/2 (statistical or computational security) parties may be corrupted. Higher corruption thresholds (i.e., t≥ n/2) can only be achieved with degraded security (unfair abort), where one single corrupted party can prevent honest parties from learning their outputs. The aforementioned upper bounds (t<n/3 and t<n/2) have been circumvented by considering mixed adversaries (Fitzi et al., Crypto' 98), i.e., adversaries that corrupt, at the same time, some parties actively, some parties passively, and some parties in the fail-stop manner. It is possible, for example, to achieve perfect security even if 2/3 of the parties are faulty (three quarters of which may abort in the middle of the protocol, and a quarter may even arbitrarily misbehave). This setting is much better suited to many applications, where the crash of a party is more likely than a coordinated active attack. Surprisingly, since the presentation of the feasibility result for the mixed setting, no progress has been made in terms of efficiency: the state-of-the-art protocol still requires a communication of Ω(n⁶) field elements per multiplication. In this paper, we present a perfectly-secure MPC protocol for the mixed setting with essentially the same efficiency as the best MPC protocols for the active-only setting. For the first time, this allows to tolerate faulty majorities, while still providing optimal efficiency. As a special case, this also results in the first fully-secure MPC protocol secure against any number of crashing parties, with optimal (i.e., linear in n) communication. We provide simulation-based proofs of our construction.ISSN:1868-896

MPC for MPC: Secure Computation on a Massively Parallel Computing Architecture

Massively Parallel Computation (MPC) is a model of computation widely believed to best capture realistic parallel computing architectures such as large-scale MapReduce and Hadoop clusters. Motivated by the fact that many data analytics tasks performed on these platforms involve sensitive user data, we initiate the theoretical exploration of how to leverage MPC architectures to enable efficient, privacy-preserving computation over massive data. Clearly if a computation task does not lend itself to an efficient implementation on MPC even without security, then we cannot hope to compute it efficiently on MPC with security. We show, on the other hand, that any task that can be efficiently computed on MPC can also be securely computed with comparable efficiency. Specifically, we show the following results: - any MPC algorithm can be compiled to a communication-oblivious counterpart while asymptotically preserving its round and space complexity, where communication-obliviousness ensures that any network intermediary observing the communication patterns learn no information about the secret inputs; - assuming the existence of Fully Homomorphic Encryption with a suitable notion of compactness and other standard cryptographic assumptions, any MPC algorithm can be compiled to a secure counterpart that defends against an adversary who controls not only intermediate network routers but additionally up to 1/3 - ? fraction of machines (for an arbitrarily small constant ?) - moreover, this compilation preserves the round complexity tightly, and preserves the space complexity upto a multiplicative security parameter related blowup. As an initial exploration of this important direction, our work suggests new definitions and proposes novel protocols that blend algorithmic and cryptographic techniques

How to Securely Compute the Modulo-Two Sum of Binary Sources

In secure multiparty computation, mutually distrusting users in a network want to collaborate to compute functions of data which is distributed among the users. The users should not learn any additional information about the data of others than what they may infer from their own data and the functions they are computing. Previous works have mostly considered the worst case context (i.e., without assuming any distribution for the data); Lee and Abbe (2014) is a notable exception. Here, we study the average case (i.e., we work with a distribution on the data) where correctness and privacy is only desired asymptotically. For concreteness and simplicity, we consider a secure version of the function computation problem of K\"orner and Marton (1979) where two users observe a doubly symmetric binary source with parameter p and the third user wants to compute the XOR. We show that the amount of communication and randomness resources required depends on the level of correctness desired. When zero-error and perfect privacy are required, the results of Data et al. (2014) show that it can be achieved if and only if a total rate of 1 bit is communicated between every pair of users and private randomness at the rate of 1 is used up. In contrast, we show here that, if we only want the probability of error to vanish asymptotically in block length, it can be achieved by a lower rate (binary entropy of p) for all the links and for private randomness; this also guarantees perfect privacy. We also show that no smaller rates are possible even if privacy is only required asymptotically.Comment: 6 pages, 1 figure, extended version of submission to IEEE Information Theory Workshop, 201

The Crypto-democracy and the Trustworthy

In the current architecture of the Internet, there is a strong asymmetry in terms of power between the entities that gather and process personal data (e.g., major Internet companies, telecom operators, cloud providers, ...) and the individuals from which this personal data is issued. In particular, individuals have no choice but to blindly trust that these entities will respect their privacy and protect their personal data. In this position paper, we address this issue by proposing an utopian crypto-democracy model based on existing scientific achievements from the field of cryptography. More precisely, our main objective is to show that cryptographic primitives, including in particular secure multiparty computation, offer a practical solution to protect privacy while minimizing the trust assumptions. In the crypto-democracy envisioned, individuals do not have to trust a single physical entity with their personal data but rather their data is distributed among several institutions. Together these institutions form a virtual entity called the Trustworthy that is responsible for the storage of this data but which can also compute on it (provided first that all the institutions agree on this). Finally, we also propose a realistic proof-of-concept of the Trustworthy, in which the roles of institutions are played by universities. This proof-of-concept would have an important impact in demonstrating the possibilities offered by the crypto-democracy paradigm.Comment: DPM 201