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.
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.
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.
Two user secure computation of randomized functions is considered, where only one user computes the output. Both the users are semi-honest; and computation is such that no user learns any additional information about the other user's input and output other than what cannot be inferred from its own input and output. First we consider a scenario, where privacy conditions are against both the users. In perfect security setting, Kilian gave a characterization of securely computable randomized functions in [1], and we provide rate-optimal protocols for such functions. We prove that the same characterization holds in asymptotic security setting as well and give a rate-optimal protocol. In another scenario, where privacy condition is only against the user who is not computing the function, we provide rate-optimal protocols. For perfect security in both the scenarios, our results are in terms of chromatic entropies of different graphs. In asymptotic security setting, we get single-letter expressions of rates in both the scenarios.
We consider secure computation of randomized functions by two users, where both the users (Alice and Bob) have inputs, Alice sends a message to Bob over a rate-limited, noise-free link, and then Bob produces the output. We study this problem when privacy is required only against Bob, i.e., from the message, Bob must not learn any information about Alice's input other than what can be inferred by his own input and output. We give a single-letter expression for the optimal rate. We also explicitly characterize securely computable randomized functions when input has full support, which leads to a much simpler expression for the optimal rate. Recently, Data (ISIT 2016) studied the other two cases (first, when privacy is required against both the users; and second, when privacy is required only against Alice) and obtained single-letter expressions for optimal rates in both the scenarios. Yassaee, Gohari, and Aref (IEEE Transactions on Information Theory 2015) studied the case when there is no privacy requirement and obtained a single-letter expression for the optimal rate, when Alice and Bob interact for arbitrary but finite number of rounds, and both of them may produce potentially different outputs.
We consider a two-user secure computation problem in which Alice and Bob communicate interactively in order to compute some deterministic functions of the inputs. The privacy requirement is that each user should not learn any additional information about a function of the inputs other than what can be inferred from its own input and output. For the distribution-free setting, i.e., when the protocol must be correct and private for any joint input distribution, we completely characterize the set of all securely computable functions. When privacy is required only against Bob who computes a function based on a single transmission from Alice, we show that asymptotically secure computability is equivalent to perfectly secure computability. Separately, we consider an eavesdropper who has access to all the communication and should not learn any information about some function of the inputs (possibly different from the functions to be computed by the users) and show that interaction may be necessary for secure computation.
In information theoretically secure multiparty computation, several mutually distrusting users want to jointly compute a function of their private data such that users do not learn any additional information about other users' data other than what they can infer from their own data and the function value they compute. In this work we consider asymptotically secure computation - where vanishing probability of error and vanishing information leakage are allowed as block lengths become large - in a three user setting, where two users have inputs and third user securely computes the output of a function on these two inputs. We provide generic lower bounds on the amount of communication required among users and the total amount of private randomness needed to compute any function in this three-user model. We also consider some examples where our bounds are tight. Our lower bounds are derived for the honest-but-curious security model and hence also apply for the malicious model.
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.