Abstract We revisit the following open problem in information-theoretic cryptography: Does the communication complexity of unconditionally secure computation depend on the computational complexity of the function being computed? For instance, can computationally unbounded players compute an arbitrary function of their inputs with polynomial communication complexity and a linear threshold of unconditional privacy? Can this be done using a constant number of communication rounds? We provide an explanation for the difficulty of resolving these questions by showing that they are closely related to the problem of obtaining efficient protocols for (information-theoretic) private information retrieval and hence also to the problem of constructing short locally-decodable error-correcting codes. The latter is currently considered to be among the most intriguing open problems in complexity theory. Keywords. Information-theoretic cryptography, secure multiparty computation, private information retrieval, locally decodable codes. 1 Introduction In STOC 1990, Beaver, Micali, and Rogaway [5] posed the following question: Is there a constant-round protocol that allows k computationally unbounded players to defeat a computationally unbounded adversary, while using only a polynomial amount of communication in the total length of their inputs?
A major open problem in information-theoretic cryptography is to obtain a super-polynomial lower bound for the communication complexity of basic cryptographic tasks. This question is wide open even for very powerful non-interactive primitives such as private information retrieval (or locally-decodable codes), general secret sharing schemes, conditional disclosure of secrets, and fully-decomposable randomized encoding (or garbling schemes). In fact, for all these primitives we do not even have super-linear lower bounds. Furthermore, it is unknown how to relate these questions to each other or to other complexity-theoretic questions.In this note, we relate all these questions to the classical topic of query/space trade-offs, lifted to the setting of interactive proof systems. Specifically, we consider the following Advisor-Verifier-Prover (AVP) game: First, a function f is given to the advisor who computes an advice a. Next, an input x is given to the verifier and to the prover who claims that
f(x)=1. The verifier should check this claim via a single round of interaction based on the private advice a and without having any additional information on f. We focus on the case where the prover is laconic and communicates only a constant number of bits, and, mostly restrict the attention to the simplest, purely information-theoretic setting, where all parties are allowed to be computationally unbounded. The goal is to minimize the total communication complexity which is dominated by the length of the advice plus the length of the verifier’s query.As our main result, we show that a super-polynomial lower bound for AVPs implies a super-polynomial lower bound for a wide range of information-theoretic cryptographic tasks. In particular, we present a communication-efficient transformation from any of the above primitives into an AVP protocol. Interestingly, each primitive induces some additional property over the resulting protocol. Thus AVP games form a new common yardstick that highlights the differences between all the above primitives.Equipped with this view, we revisit the existing (somewhat weak) lower bounds for the above primitives, and show that many of these lower bounds can be unified by proving a single counting-based lower bound on the communication of AVPs, whereas some techniques are inherently limited to specific domains. The latter is shown by proving the first polynomial separations between the complexity of secret-sharing schemes and conditional disclosure of secrets and between the complexity of randomized encodings and conditional disclosure of secrets.
Abstract. In this paper, we re-visit the problem of unconditionally secure multiparty set intersection in information theoretic model. Li et.al [24] have proposed a protocol for n-party set intersection problem, which players are corrupted by an active adversary having unbounded computing power. Moreover, they have claimed that their protocol takes six rounds of communication and incurs a communication complexity of O(n 4 m 2), where each player has a set of size m. However, we show that the round complexity and communication complexity of the protocol in [24] is much more than what is claimed in [24]. We then propose a novel unconditionally secure protocol for multiparty set intersection problem with n> 3t players, which significantly improves the ”actual ” round and communication complexity (as shown in this paper) of the protocol given in [24]. To design our protocol, we use several tools which are of independent interest. provides unconditional security when t < n 3
We show that any multiparty functionality can be evaluated using a 2-round protocol with perfect correctness and perfect semihonest security, provided that the majority of parties are honest. This settles the round complexity of information-theoretic semihonest multiparty computation, resolving a longstanding open question [Y. Ishai and E. Kushilevitz, Randomizing polynomials: A new representation with applications to round-efficient secure computation, in Proceedings of the 41st Annual Symposium on Foundations of Computer Science FOCS 2000, IEEE Computer Society, 2000, pp. 294--304]. The protocol is efficient for
NC1 functionalities. Furthermore, given black-box access to a one-way function, the protocol can be made efficient for any polynomial functionality, at the cost of only guaranteeing computational security. Our results are based on a new notion of multiparty randomized encoding which extends and relaxes the standard notion of randomized encoding of functions [Y. Ishai and E. Kushilevitz, Randomizing polynomials: A new representation with applications to round-efficient secure computation, in Proceedings of the 41st Annual Symposium on Foundations of Computer Science FOCS 2000, IEEE Computer Society, 2000, pp. 294--304]. The property of a multiparty randomized encoding (MPRE) is that if the functionality
g is an encoding of the functionality
f, then for any (permitted) coalition of players, their respective outputs and inputs in
g allow them to simulate their respective inputs and outputs in
f, without learning anything else, including the other outputs of
f. We further introduce a new notion of effective degree, and show that the round complexity of a functionality
f is characterized by the degree of its MPRE. We construct degree-2 MPREs for general functionalities in several settings under different assumptions, and use these constructions to obtain 2-round protocols. Our constructions also give rise to new protocols in the client-server model with optimal round complexity.
Abstract. Substantial efforts have been spent on characterizing the round complexity of various cryptographic tasks. In this work we study the round complexity of secure multiparty computation in the presence of an active (Byzantine) adversary, assuming the availability of secure point-to-point channels and a broadcast primitive. It was recently shown that in this setting three rounds are sufficient for arbitrary secure computation tasks, with a linear security threshold, and two rounds are sufficient for certain nontrivial tasks. This leaves open the question whether every function can be securely computed in two rounds. We show that the answer to this question is “no”: even some very simple functions do not admit secure 2-round protocols (independently of their communication and time complexity) and thus 3 is the exact round complexity of general secure multiparty computation. Yet, we also present some positive results by identifying a useful class of functions which can be securely computed in two rounds. Our results apply both to the information-theoretic and to the computational notions of security.
Each of two parties $P_\mathcal{X} $ and $P_\matcal{Y} $ holds an n-bit input, x and y, respectively. They wish to privately compute the value of
f(x,y). That is, $P_\mathcal{X} $ should not learn any additional information about y (in the information-theoretic sense) other than what follows from its input x and the function value
f(x,y), and similarly, $P_\mathcal{Y} $ should not learn any additional information about x. In this paper, the two following basic questions in the theory of private computations are considered: 1. Which functions can be privately computed? 2. What is the communication complexity of protocols that privately compute a function f (in the case that such protocols exist)? A complete combinatorial characterization of privately computable functions is given. This characterization is used to derive tight bounds on the rounds complexity of any privately computable function and to design optimal private protocols that compute these functions. It is shown that for every
1≤g(n)≤2⋅(2n−1) there are functions that can be privately computed with
g(n)-rounds of communication, but not with
(g(n)−1)-rounds of communication. This implies that the communication costs of private protocols can be exponentially higher than the communication costs of nonprivate protocols. Interestingly, randomization helps neither to increase the set of privately computable functions, nor to improve the rounds complexity of these functions.
Secure multiparty computation (
MPC) addresses the challenge of evaluating functions on secret inputs without compromising their privacy. A central question in multiparty computation is to understand the amount of communication needed to securely evaluate a circuit of size s. In this work, we revisit this fundamental question in the setting of information-theoretically secure
MPC in the correlated randomness model, where a trusted dealer distributes correlated random coins, independent of the inputs, to all parties before the start of the protocol. This setting is of strong theoretical interest, and has led to the most practically efficient
MPC protocols known to date.