On the Communication Complexity of Secure Computation

Information theoretically secure multi-party computation (MPC) is a central primitive of modern cryptography. However, relatively little is known about the communication complexity of this primitive. In this work, we develop powerful information theoretic tools to prove lower bounds on the communication complexity of MPC. We restrict ourselves to a 3-party setting in order to bring out the power of these tools without introducing too many complications. Our techniques include the use of a data processing inequality for residual information - i.e., the gap between mutual information and G\'acs-K\"orner common information, a new information inequality for 3-party protocols, and the idea of distribution switching by which lower bounds computed under certain worst-case scenarios can be shown to apply for the general case. Using these techniques we obtain tight bounds on communication complexity by MPC protocols for various interesting functions. In particular, we show concrete functions that have "communication-ideal" protocols, which achieve the minimum communication simultaneously on all links in the network. Also, we obtain the first explicit example of a function that incurs a higher communication cost than the input length in the secure computation model of Feige, Kilian and Naor (1994), who had shown that such functions exist. We also show that our communication bounds imply tight lower bounds on the amount of randomness required by MPC protocols for many interesting functions.Comment: 37 page

On the Communication Complexity of Secret Key Generation in the Multiterminal Source Model

Communication complexity refers to the minimum rate of public communication required for generating a maximal-rate secret key (SK) in the multiterminal source model of Csiszar and Narayan. Tyagi recently characterized this communication complexity for a two-terminal system. We extend the ideas in Tyagi's work to derive a lower bound on communication complexity in the general multiterminal setting. In the important special case of the complete graph pairwise independent network (PIN) model, our bound allows us to determine the exact linear communication complexity, i.e., the communication complexity when the communication and SK are restricted to be linear functions of the randomness available at the terminals.Comment: A 5-page version of this manuscript will be submitted to the 2014 IEEE International Symposium on Information Theory (ISIT 2014

On The Multiparty Communication Complexity of Testing Triangle-Freeness

In this paper we initiate the study of property testing in simultaneous and non-simultaneous multi-party communication complexity, focusing on testing triangle-freeness in graphs. We consider the $\textit{coordinator}$ model, where we have $k$ players receiving private inputs, and a coordinator who receives no input; the coordinator can communicate with all the players, but the players cannot communicate with each other. In this model, we ask: if an input graph is divided between the players, with each player receiving some of the edges, how many bits do the players and the coordinator need to exchange to determine if the graph is triangle-free, or $\textit{far}$ from triangle-free? For general communication protocols, we show that $\tilde{O}(k(nd)^{1/4}+k^2)$ bits are sufficient to test triangle-freeness in graphs of size $n$ with average degree $d$ (the degree need not be known in advance). For $\textit{simultaneous}$ protocols, where there is only one communication round, we give a protocol that uses $\tilde{O}(k \sqrt{n})$ bits when $d = O(\sqrt{n})$ and $\tilde{O}(k (nd)^{1/3})$ when $d = \Omega(\sqrt{n})$; here, again, the average degree $d$ does not need to be known in advance. We show that for average degree $d = O(1)$, our simultaneous protocol is asymptotically optimal up to logarithmic factors. For higher degrees, we are not able to give lower bounds on testing triangle-freeness, but we give evidence that the problem is hard by showing that finding an edge that participates in a triangle is hard, even when promised that at least a constant fraction of the edges must be removed in order to make the graph triangle-free.Comment: To Appear in PODC 201

Exponential Separation of Quantum Communication and Classical Information

We exhibit a Boolean function for which the quantum communication complexity is exponentially larger than the classical information complexity. An exponential separation in the other direction was already known from the work of Kerenidis et. al. [SICOMP 44, pp. 1550-1572], hence our work implies that these two complexity measures are incomparable. As classical information complexity is an upper bound on quantum information complexity, which in turn is equal to amortized quantum communication complexity, our work implies that a tight direct sum result for distributional quantum communication complexity cannot hold. The function we use to present such a separation is the Symmetric k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057], whose classical communication complexity is exponentially larger than its classical information complexity. In this paper, we show that the quantum communication complexity of this function is polynomially equivalent to its classical communication complexity. The high-level idea behind our proof is arguably the simplest so far for such an exponential separation between information and communication, driven by a sequence of round-elimination arguments, allowing us to simplify further the approach of Rao and Sinha. As another application of the techniques that we develop, we give a simple proof for an optimal trade-off between Alice's and Bob's communication while computing the related Greater-Than function on n bits: say Bob communicates at most b bits, then Alice must send n/exp(O(b)) bits to Bob. This holds even when allowing pre-shared entanglement. We also present a classical protocol achieving this bound.Comment: v1, 36 pages, 3 figure

Computational Efficiency Requires Simple Taxation

We characterize the communication complexity of truthful mechanisms. Our departure point is the well known taxation principle. The taxation principle asserts that every truthful mechanism can be interpreted as follows: every player is presented with a menu that consists of a price for each bundle (the prices depend only on the valuations of the other players). Each player is allocated a bundle that maximizes his profit according to this menu. We define the taxation complexity of a truthful mechanism to be the logarithm of the maximum number of menus that may be presented to a player. Our main finding is that in general the taxation complexity essentially equals the communication complexity. The proof consists of two main steps. First, we prove that for rich enough domains the taxation complexity is at most the communication complexity. We then show that the taxation complexity is much smaller than the communication complexity only in "pathological" cases and provide a formal description of these extreme cases. Next, we study mechanisms that access the valuations via value queries only. In this setting we establish that the menu complexity -- a notion that was already studied in several different contexts -- characterizes the number of value queries that the mechanism makes in exactly the same way that the taxation complexity characterizes the communication complexity. Our approach yields several applications, including strengthening the solution concept with low communication overhead, fast computation of prices, and hardness of approximation by computationally efficient truthful mechanisms