Vinod M. Prabhakaran

Gowtham R. Kurri, Varun Narayanan, Vinod M. Prabhakaran et al.

In distributed hypothesis testing, a central server performs hypothesis testing based on information received from distributed sensors/clients. We study a secure variant of this problem in which the central server determines the hypothesis class of an underlying distribution without learning any additional information about the distribution itself. We prove that, in its standard form, this is impossible to achieve, even for simple and highly restricted cases. To bypass this impossibility, we augment the model with a shared secret key available to clients but hidden from the server. We show that a single-bit secret key enables perfectly secure testing for simple classes by reducing the test distributions to a symmetric, canonical instance. Finally, for arbitrary hypothesis classes over finite domains, we establish a reduction to standard hypothesis testing using Private Simultaneous Messages (PSM) protocols, achieving polynomial communication and key lengths.

Malhar A. Managoli, Vinod M. Prabhakaran

We study adversarial binary hypothesis testing under memory constraints. The test is a time-invariant randomized finite state machine (FSM) with S states. Associated with each hypothesis is a set of distributions. Given the hypothesis, the distribution of each sample is chosen from the set associated with the hypothesis by an adversary who has access to past samples and the history of states of the FSM so far. We obtain upper and lower bounds on the minimax asymptotic probability of error as a function of S. The bounds have the same exponential behaviour in S and match for a class of problems.

Gunank Jakhar, Gowtham R. Kurri, Suryajith Chillara et al.

Dembo, Cover, and Thomas (1991) developed an elegant information-theoretic framework for proving determinantal inequalities for positive definite matrices, which relies on the structural inequalities of differential entropy. Submodular functions, which subsume entropy, inherently satisfy these structural inequalities because they obey generalized forms of the fundamental properties of entropy -- a chain rule and the property that conditioning reduces the function's value (under an appropriate definition of conditioning). Applying subadditivity, Han's inequality (1978), and partition subadditivity (i.e., subadditivity over a partition) yields Hadamard's, Szász's, and Fischer's inequalities, respectively. Furthermore, this framework recovers Ky Fan's inequality (1955), a strengthening of Hadamard's inequality. This improvement fundamentally arises because conditional subadditivity yields a tighter upper bound on the joint entropy than the one obtained via unconditional subadditivity. In this paper, we establish conditional strengthenings of Han's inequality and partition subadditivity in the general setting of submodular functions. We derive equality conditions for these strengthened bounds and characterize when they strictly improve their unconditional counterparts. We specialize these results to differential entropy and apply them to establish strengthened versions of Szász's and Fischer's inequalities. The strengthening of Szász's inequality recovers Ky Fan's inequality as a special case, and is strictly stronger than the classical Szász's inequality for any non-diagonal positive definite matrix. We also derive an inequality concerning eigenvalues, which generalizes and strictly strengthens a corresponding eigenvalue inequality of Ky Fan. We provide numerical examples to explicitly illustrate the tightness of our proposed matrix determinantal bounds.

Malhar A. Managoli, Vinod M. Prabhakaran, Suhas Diggavi

Federated learning with heterogeneous data and personalization has received significant recent attention. Separately, robustness to corrupted data in the context of federated learning has also been studied. In this paper we explore combining personalization for heterogeneous data with robustness, where a constant fraction of the clients are corrupted. Motivated by this broad problem, we formulate a simple instantiation which captures some of its difficulty. We focus on the specific problem of personalized mean estimation where the data is drawn from a Gaussian mixture model. We give an algorithm whose error depends almost linearly on the ratio of corrupted to uncorrupted samples, and show a lower bound with the same behavior, albeit with a gap of a constant factor.

Manoj Mishra, Bikash Kumar Dey, Vinod M. Prabhakaran et al.

In this paper, we study the problem of obtaining $1$-of-$2$ string oblivious transfer (OT) between users Alice and Bob, in the presence of a passive eavesdropper Eve. The resource enabling OT in our setup is a noisy broadcast channel from Alice to Bob and Eve. Apart from the OT requirements between the users, Eve is not allowed to learn anything about the users' inputs. When Alice and Bob are honest-but-curious and the noisy broadcast channel is made up of two independent binary erasure channels (connecting Alice-Bob and Alice-Eve), we derive the $1$-of-$2$ string OT capacity for both $2$-privacy (when Eve can collude with either Alice or Bob) and $1$-privacy (when no such collusion is allowed). We generalize these capacity results to $1$-of-$N$ string OT and study other variants of this problem. When Alice and/or Bob are malicious, we present a different scheme based on interactive hashing. This scheme is shown to be optimal for certain parameter regimes. We present a new formulation of multiple, simultaneous OTs between Alice-Bob and Alice-Cathy. For this new setup, we present schemes and outer bounds that match in all but one regime of parameters. Finally, we consider the setup where the broadcast channel is made up of a cascade of two independent binary erasure channels (connecting Alice-Bob and Bob-Eve) and $1$-of-$2$ string OT is desired between Alice and Bob with $1$-privacy. For this setup, we derive an upper and lower bound on the $1$-of-$2$ string OT capacity which match in one of two possible parameter regimes.

Deepesh Data, Vinod M. Prabhakaran, Manoj M. Prabhakaran

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 other users. While it is known that information theoretically secure MPC is possible among $n$ users (connected by secure and noiseless links and have access to private randomness) against the collusion of less than $n/2$ users in the honest-but-curious model, relatively less is known about the communication and randomness complexity of secure computation. In this work, we employ information theoretic techniques to obtain lower bounds on the amount of communication and randomness required for secure MPC. We restrict ourselves to a concrete interactive setting involving 3 users under which all functions are securely computable against corruption of a single user 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 $\infty$). 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 3-user protocols, and the idea of distribution switching. 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, and also use minimum amount of randomness. 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 [STOC, 1994]", who had shown that such functions exist.

Manoj Mishra, Bikash Kumar Dey, Vinod M. Prabhakaran et al.

We study oblivious transfer (OT) between Alice and Bob in the presence of an eavesdropper Eve over a degraded wiretapped binary erasure channel from Alice to Bob and Eve. In addition to the privacy goals of oblivious transfer between Alice and Bob, we require privacy of Alice and Bob's private data from Eve. In previous work we derived the OT capacity (in the honest-but-curious model) of the wiretapped binary independent erasure channel where the erasure processes of Bob and Eve are independent. Here we derive a lower bound on the OT capacity in the same secrecy model when the wiretapped binary erasure channel is degraded in favour of Bob.

Manoj Mishra, Tanmay Sharma, Bikash K. Dey et al.

We study the following private data transfer problem: Alice has a database of files. Bob and Cathy want to access a file each from this database (which may or may not be the same file), but each of them wants to ensure that their choices of file do not get revealed even if Alice colludes with the other user. Alice, on the other hand, wants to make sure that each of Bob and Cathy does not learn any more information from the database than the files they demand (the identities of which will be unknown to her). Moreover, they should not learn any information about the other files even if they collude. It turns out that it is impossible to accomplish this if Alice, Bob, and Cathy have access only to private randomness and noiseless communication links. We consider this problem when a binary erasure broadcast channel with independent erasures is available from Alice to Bob and Cathy in addition to a noiseless public discussion channel. We study the file-length-per-broadcast-channel-use rate in the honest-but-curious model. We focus on the case when the database consists of two files, and obtain the optimal rate. We then extend to the case of larger databases, and give upper and lower bounds on the optimal rate.

Deepesh Data, Bikash Kumar Dey, Manoj Mishra et al.

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 Körner 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.

K. Sankeerth Rao, Vinod M. Prabhakaran

We derive a new upper bound on the string oblivious transfer capacity of discrete memoryless channels. The main tool we use is the tension region of a pair of random variables introduced in Prabhakaran and Prabhakaran (2014) where it was used to derive upper bounds on rates of secure sampling in the source model. In this paper, we consider secure computation of string oblivious transfer in the channel model. Our bound is based on a monotonicity property of the tension region in the channel model. We show that our bound strictly improves upon the upper bound of Ahlswede and Csiszár (2013).

Manoj Mishra, Bikash Kumar Dey, Vinod M. Prabhakaran et al.

We consider oblivious transfer between Alice and Bob in the presence of an eavesdropper Eve when there is a broadcast channel from Alice to Bob and Eve. In addition to the secrecy constraints of Alice and Bob, Eve should not learn the private data of Alice and Bob. When the broadcast channel consists of two independent binary erasure channels, we derive the oblivious transfer capacity for both 2-privacy (where the eavesdropper may collude with either party) and 1-privacy (where there are no collusions).

Deepesh Data, Vinod M. Prabhakaran, Manoj M. Prabhakaran

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 Gács-Körner 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.

Vinod M. Prabhakaran, Manoj M. Prabhakaran

In this paper we generalize the notion of common information of two dependent variables introduced by Gács & Körner. They defined common information as the largest entropy rate of a common random variable two parties observing one of the sources each can agree upon. It is well-known that their common information captures only a limited form of dependence between the random variables and is zero in most cases of interest. Our generalization, which we call the Assisted Common Information system, takes into account almost-common information ignored by Gács-Körner common information. In the assisted common information system, a genie assists the parties in agreeing on a more substantial common random variable; we characterize the trade-off between the amount of communication from the genie and the quality of the common random variable produced using a rate region we call the region of tension. We show that this region has an application in deriving upperbounds on the efficiency of secure two-party sampling, which is a special case of secure multi-party computation, a central problem in modern cryptography. Two parties desire to produce samples of a pair of jointly distributed random variables such that neither party learns more about the other's output than what its own output reveals. They have access to a set up - correlated random variables whose distribution is different from the desired distribution - and noiseless communication. We present an upperbound on the rate at which a given set up can be used to produce samples from a desired distribution by showing a monotonicity property for the region of tension: a protocol between two parties can only lower the tension between their views. Then, by calculating the bounds on the region of tension of various pairs of correlated random variables, we derive bounds on the rate of secure two-party sampling.