In two-sampling testing, one observes two independent sequences of independent and identically distributed random variables distributed according to the distributions $P_1$ and $P_2$ and wishes to decide whether $P_1=P_2$ (null hypothesis) or $P_1\neq P_2$ (alternative hypothesis). The Gutman test for this problem compares the empirical distributions of the observed sequences and decides on the null hypothesis if the Jensen-Shannon (JS) divergence between these empirical distributions is below a given threshold. This paper proposes a generalization of the Gutman test, termed \emph{divergence test}, which replaces the JS divergence by an arbitrary divergence. For this test, the exponential decay of the type-II error probability for a fixed type-I error probability is studied. First, it is shown that the divergence test achieves the optimal first-order exponent, irrespective of the choice of divergence. Second, it is demonstrated that divergence tests with invariant divergences achieve the same second-order asymptotics as the Gutman test. In addition, a connection between two-sample testing and robust goodness-of-fit testing is established.

Consider a composite hypothesis testing problem where the test has access to the null hypothesisPbut not to the alternative hypothesisQ. The generalized likelihood-ratio test (GLRT) for this problem is the Hoeffding test, which acceptsPif the Kullback-Leibler (KL) divergence between the empirical distribution ofZnandPis below some threshold. This paper proposes a generalization of the Hoeffding test, termed divergence test, for which the KL divergence is replaced by an arbitrary divergence. For this test, the first and second-order terms of the type-II error probability for a fixed type-I error probability are characterized and compared with the error terms of the Neyman-Pearson test, which is the optimal test when bothPandQare known. It is demonstrated that, irrespective of the divergence, divergence tests achieve the first-order term of the Neyman-Pearson test. In contrast, the second-order term of divergence tests is strictly worse than that of the Neyman-Pearson test. It is further demonstrated that divergence tests with an invariant divergence achieve the same second-order term as the Hoeffding test, but divergence tests with a non-invariant divergence may outperform the Hoeffding test for some alternative hypothesesQ. This implies that the GLRT may have a second-order asymptotic performance that is strictly suboptimal.

We consider a binary statistical hypothesis testing problem, where n independent and identically distributed random variables Znare either distributed according to the null hypothesis P or the alternate hypothesis Q, and only P is known. For this problem, a well-known test is the Hoeffding test, which accepts P if the Kullback-Leibler (KL) divergence between the empirical distribution of Znand P is below some threshold. In this paper, we consider Hoeffding-like tests, where the KL divergence is replaced by other divergences, and characterize, for a large class of divergences, the first and second-order terms of the type-II error for a fixed type-I error. Since the considered class includes the KL divergence, we obtain the second-order term of the Hoeffding test as a special case.

We consider the coded caching problem with an additional privacy constraint that a user should not get any information about the demands of the other users. We first show that a demand-private scheme for$N$files and$K$users can be obtained from a non-private scheme that serves only a subset of the demands for the$N$files and$NK$users problem. We further use this fact to construct a demand-private scheme for$N$files and$K$users from a particular known non-private scheme for$N$files and$NK-K+1$users. It is then demonstrated that, the memory-rate pair$(M,\min \{N,K\}(1-M/N))$, which is achievable for non-private schemes with uncoded transmissions, is also achievable under demand privacy. We further propose a scheme that improves on these ideas by removing some redundant transmissions. The memory-rate trade-off achieved using our schemes is shown to be within a multiplicative factor of 3 from the optimal when$K < N$and of 8 when$N \leq K$. Finally, we give the exact memory-rate trade-off for demand-private coded caching problems with$N\geq K=2$.

We study the fundamental problem of index coding under an additional privacy constraint that requires each receiver to learn nothing more about the collection of messages beyond its demanded messages from the server and what is available to it as side information. To enable such private communication, we allow the use of a collection of independent secret keys, each of which is shared amongst a subset of users and is known to the server. The goal is to study properties of the key access structures that make the problem feasible and then design encoding and decoding schemes efficient in the size of the server transmission as well as the sizes of the secret keys. We call this theprivate index codingproblem. We begin by characterizing the key access structures that make private index coding feasible. We also give conditions to check if a given linear scheme is a valid private index code. For up to three users, we characterize the rate region of feasible server transmission and key rates, and show that all feasible rates can be achieved using scalar linear coding and time sharing; we also show that scalar linear codes are sub-optimal for four receivers. The outer bounds used in the case of three users are extended to arbitrary number of users and seen as a generalized version of the well-known polymatroidal bounds for the standard non-private index coding. We also show that the presence of common randomness and private randomness does not change the rate region. Furthermore, we study the case where the server has the ability to multicast to any subset of users, and demonstrate how this flexibility can be used to provide privacy and characterize the minimum number of server multicasts required.

This paper considers a Gaussian multiple-access channel with random user activity where the total number of users$\ell _{n}$and the average number of active users$k_{n}$may grow with the blocklength$n$. For this channel, it studies the maximum number of bits that can be transmitted reliably per unit-energy as a function of$\ell _{n}$and$k_{n}$. When all users are active with probability one, i.e.,$\ell _{n} = k_{n}$, it is demonstrated that, if$k_{n}$is of an order strictly below$n/\log n$, then each user can achieve the single-user capacity per unit-energy$(\log e)/N_{0}$(where$N_{0}/ 2$is the noise power) by using an orthogonal-access scheme. In contrast, if$k_{n}$is of an order strictly above$n/\log n$, then the users cannot achieve any positive rate per unit-energy. Consequently, there is a sharp transition between orders of growth where interference-free communication is feasible and orders of growth where reliable communication at a positive rate per unit-energy is infeasible. It is further demonstrated that orthogonal-access schemes in combination with orthogonal codebooks, which achieve the capacity per unit-energy when the number of users is bounded, can be strictly suboptimal. When the user activity is random, i.e., when$\ell _{n}$and$k_{n}$are different, it is demonstrated that, if$k_{n}\log \ell _{n}$is sublinear in$n$, then each user can achieve the single-user capacity per unit-energy$(\log e)/N_{0}$. Conversely, if$k_{n}\log \ell _{n}$is superlinear in$n$, then the users cannot achieve any positive rate per unit-energy. Consequently, there is again a sharp transition between orders of growth where interference-free communication is feasible and orders of growth where reliable communication at a positive rate is infeasible that depends on the asymptotic behaviors of both$\ell _{n}$and$k_{n}$. It is further demonstrated that orthogonal-access schemes, which are optimal when all users are active with probability one, can be strictly suboptimal in general.

We consider a Gaussian multiple-access channel with random user activity where the total number of users ℓnand the average number of active users knmay be unbounded. For this channel, we characterize the maximum number of bits that can be transmitted reliably per unit-energy in terms of ℓnand kn. We show that if knlog ℓnis sublinear in n, then each user can achieve the single-user capacity per unit-energy. Conversely, if knlog ℓnis superlinear in n, then the capacity per unit-energy is zero. We further demonstrate that orthogonal-access schemes, which are optimal when all users are active with probability one, can be strictly suboptimal.

We consider the demand-private coded caching problem in a noiseless broadcast network. It is known from past works that a demand-private scheme for N files and K users can be obtained from a non-private scheme for N files and NK users. We first propose a scheme that improves on this idea by removing some redundant transmissions. The memory- rate trade-off achieved using this scheme is shown to be within a multiplicative factor of 3 from the optimal for all the memory regimes when KK = 2.

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.

We consider a Gaussian multiple-access channel where the number of transmitters grows with the blocklength n. For this setup, the maximum number of bits that can be transmitted reliably per unit-energy is analyzed. We show that if the number of users is of an order strictly above n/log n, then the users cannot achieve any positive rate per unit-energy. In contrast, if the number of users is of order strictly below n/log n, then each user can achieve the single-user capacity per unit-energy (log e)/N0(where N0/2 is the noise power) by using an orthogonal access scheme such as time division multiple access. We further demonstrate that orthogonal codebooks, which achieve the capacity per unit-energy when the number of users is bounded, can be strictly suboptimal.

We consider a function computation problem in a three-node wireless network. Nodes A and B observe two correlated sources X and Y, respectively, and want to compute a function f (X, Y). To achieve this, nodes A and B send messages to a relay node C at rates RA and RB, respectively. The relay C then broadcasts a message to A and B at rate RC. We allow block coding and study the achievable region of rate triples under both zero-error and E-error. As a preparation, we first consider a broadcast network from the relay to A and B. A and B have side information X and Y, respectively. The relay node C observes both X and Y and broadcasts an encoded message to A and B. We want to obtain the optimal broadcast rate such that A and B can recover the function f (X, Y) from the received message and their individual side information X and Y, respectively. For this problem, we show equivalence between E-error and zero-error computations-this gives a rate characterization for zero-error computation. As a corollary, this also gives a rate characterization for the relay network under zero error for a class of functions called component-wise one-to-one functions when the support set of pXY is full. For the relay network, the zero-error rate region for arbitrary functions is characterized in terms of graph coloring of some suitably defined probabilistic graphs. We then give a single-letter inner bound to this rate region. Furthermore, we extend the graph theoretic ideas to address the E-error problem and obtain a single-letter inner bound.

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 study the problem of index coding under the privacy requirement that receivers do not learn anything more than the messages they already have as side information and the message they want from the server. To achieve this private index coding, we consider the use of secret keys that are shared among various subsets of users and the server. We characterize key access structures that allow private index coding. For up to three receivers, we characterize the rate region of transmission and key rates and show that scalar coding is optimal; we also show that scalar linear codes are sub-optimal for four receivers. Furthermore, when no keys are available, we consider a weaker notion of privacy analogous to weak security. Finally, for a different setting in which the server is allowed to send messages exclusively to a subset of users, we study the number of transmissions required to achieve error-free decoding and privacy.

We consider the problem of oblivious transfer (OT) over OFDM and MIMO wireless communication systems where only the receiver knows the channel state information. The sender and receiver also have unlimited access to a noise-free real channel. Using a physical layer approach, based on the properties of the noisy fading channel, we propose a scheme for honest-but-curious parties that enables the transmitter to send obliviously one-of-two files, i.e., without knowing which one has been actually requested by the receiver, while also ensuring that the receiver does not get any information about the other file.

We consider zero error function computation in a three node wireless network. Nodes A and B observe X and Y respectively, and want to compute a function f(X, Y ) with zero error. To achieve this, nodes A and B send messages to a relay node C at rates RAand RBrespectively. The relay C then broadcasts a message to A and B at rate RCto help them compute f(X, Y ) with zero error. We allow block coding, and study the region of rate-triples (RA, RB, RC) that are feasible. The rate region is characterized in terms of graph coloring of some suitably defined probabilistic graphs. We give single letter inner and outer bounds which meet for some simple examples. We provide a sufficient condition on the joint distribution pXYunder which the relay can also compute f(X, Y ) if A and B can compute it with zero error.

We consider the problem of oblivious transfer (OT) over OFDM and MIMO wireless communication systems where only the receiver knows the channel state information. The sender and receiver also have unlimited access to a noise-free real channel. Using a physical layer approach, based on the properties of the noisy fading channel, we propose a scheme that enables the transmitter to send obliviously one-of-two files, i.e, without knowing which one has been actually requested by the receiver, while also ensuring that the receiver does not get any information about the other file.