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IEEE International Symposium on Information Theory (ISIT)

Common randomness for secure computing

Prakash Narayan, Himanshu Tyagi, Shun Watanabe

Published Jun 1, 2015• 11 citations
Publisher

Abstract

We revisit A.C. Yao's classic problem of secure function computation by interactive communication, in an information theoretic setting. Our approach, based on examining the underlying common randomness, provides a new proof of the characterization of a securely computable function by deterministic protocols. This approach also yields a characterization of the minimum communication needed for secure computability.

DOI
10.1109/isit.2015.7282595

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We revisit A.C. Yao's classic problem of secure function computation by interactive communication, in an information theoretic setting. Our approach, based on examining the underlying common randomness, provides a new proof of the characterization of a securely computable function by deterministic protocols. This approach also yields a characterization of the minimum communication needed for secure computability.
We consider interactive computation of randomized functions between two users with the following privacy requirement: the interaction should not reveal to either user any extra information about the other user's input and output other than what can be inferred from the user's own input and output. We also consider the case where privacy is required against only one of the users. For both cases, we give single-letter expressions for feasibility and optimal rates of communication. Then we discuss the role of common randomness and interaction in both privacy settings. We also study perfectly secure non-interactive computation when only one of the users computes a randomized function based on a single transmission from the other user. We characterize randomized functions which can be perfectly securely computed in this model and obtain tight bounds on the optimal message lengths in all the privacy settings.
This dissertation concerns the secure processing of distributed data by multi-ple terminals, using interactive public communication among themselves, in order to accomplish a given computational task. In the setting of a probabilistic multiterminal source model in which several terminals observe correlated random signals, we analyze secure distributed data processing protocols that harness the correlation in the data. The specific tasks considered are: computing functions of the data under secrecy requirements; generating secretly shared bits with minimal rate of public communication; and securely sharing bits in presence of a querying eavesdropper. In studying these various secure distributed processing tasks, we adopt a unified approach that entails examining the form of underlying common randomness (CR) that is generated at the terminals during distributed processing. We make the case that the exact form of established CR is linked inherently to the data processing task at hand, and its characterization can lead to a structural understanding of the associated algorithms. An identification of the underlying CR and its decomposition into independent components, each with a different operational significance, is
In secure multiparty computation (MPC), mutually distrusting users collaborate to compute a function of their private data without revealing any additional information about their data to the other users. While it is known that information theoretically secure MPC is possible among n users having access to private randomness and are pairwise connected by secure, noiseless, and bidirectional links against the collusion of less than n/2 users (in the honest-but-curious model; the threshold is n/3 in the malicious model), relatively little is known about the communication and randomness complexity of secure computation, i.e., the amount of communication and randomness required to compute securely. In this paper, we employ information theoretic techniques to obtain lower bounds on communication and randomness complexity of secure MPC. We restrict ourselves to a concrete interactive setting involving three users under which all functions are securely computable against corruption of individual users in the honest-but-curious model. We derive lower bounds for both the perfect security case (i.e., zero-error and no leakage of information) and asymptotic security (where the probability of error and information leakage vanish as block-length goes to ∞). Our techniques include the use of a data processing inequality for residual information (i.e., the gap between mutual information and Gács-Körner common information), a new information inequality for three-user 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. Our lower bounds are shown to be tight for various functions of interest. In particular, we show concrete functions which have communication-ideal protocols, i.e., 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 et al. (26th Annual ACM Symposium on Theory of Computing, 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.
In the secure two-party computation problem, two parties wish to compute a (possibly randomized) function of their inputs via an interactive protocol, while ensuring that neither party learns more than what can be inferred from only their own input and output. For semi-honest parties and information-theoretic security guarantees, it is well-known that, if only noise-less communication is available, only a limited set of functions can be securely computed; however, if interaction is also allowed over general communication primitives (multi-input/output channels), there are “complete” primitives that enable any function to be securely computed. The general set of complete primitives was characterized recently by Maji, Prabhakaran, and Rosulek leveraging an earlier specialized characterization by Kilian. Our contribution in this paper is a simple, self-contained, alternative derivation using elementary information-theoretic tools.
We consider interactive computation of randomized functions between two users with the following privacy requirement: the interactive communication should not reveal to either user any extra information about the other user's input and output other than what can be inferred from the user's own input and output. We also consider the case where privacy is required against only one of the users. For both cases, we give single-letter expressions for feasibility and optimal rates of communication. Then we discuss the role of common randomness and interaction in both privacy settings.
Secure multiparty computation (MPC\mathsf {MPC}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\mathsf {MPC}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\mathsf {MPC}MPC protocols known to date.