In the above article[1], the authors introduced a PIR scheme for the Additive White Gaussian Noise (AWGN) Multiple Access Channel (MAC), both with and without fading. The authors utilized the additive nature of the channel and leveraged the linear properties and structure of lattice codes to retrieve the desired message without the servers acquiring any knowledge about the retrieved message’s index. Theorems 3 and 4 in[1]contain an error arising from the incorrect usage of the modulo operator. Moreover, the proofs assume a one-to-one mapping function,$\phi (\cdot)$, between a message$W_{j}\in \mathbb {F}_{p}^{L}$and the elements of$\mathcal { C}$, mistakenly suggesting that the user possesses all the required information in advance. To deal with that, we defined$\phi (\cdot)$as a one-to-one mapping function between a vector oflinformation bits and a lattice point$\lambda \in {\mathcal { C}}$. Herein, we present the corrected versions of these theorems.

Covert communication conceals transmission of messages between Alice and Bob from an adversary, Willie, who tries to determine if a transmission took place or not. While covert communication in a basic, standard setting where all variables are known to Willie, results in the well-known square-root law, when a jammer is present and assists Alice by creating uncertainty in Willie's decoder, a strictly positive transmission rate is possible. In this work, we analyze the case where the jammer is equipped with multiple antennas. Specifically, we analyze the effect of multiple antennas at the jammer on Alice's transmission power and consequently on the transmission rate. We consider both the case where the channel knowledge of Willie is known as well as the case where it is unknown. We formulate several optimization problems for the transmission strategies of the jammer to maximize his assistance to Alice, in terms of maximizing Bob's received SNR and consequently the covert rate. When the channel information is known to the jammer, we show that under an achievable covertness scheme, the optimal strategy of the jammer is to perform beamforming towards a single direction with all his available power. This direction though, is not trivial, since it reflects a tradeoff point between minimizing the interference at Bob and maximizing the interference at Willie. When the channel knowledge is unknown, we show that the optimal strategy of the jammer is either to transmit isotropically to all directions or to the null-space of Bob, where this choice depends on certain channel conditions. This is in contrast to current schemes in the literature. Furthermore, we extend the optimization problems to the case where Bob is also equipped with multiple antennas, and provide insightful results, shown to be asymptotically optimal, accompanied by simulations.

Compute and Forward (CF) is a coding scheme which enables receivers to decode linear combinations of simultaneously transmitted messages while exploiting the linear properties of lattice codes and the additive nature of a shared medium. The scheme was originally designed for relay networks, yet, it was found useful in other communication problems, such as MIMO communication. Works in the current literature assume a fixed number of transmitters and receivers in the system. However, following the increase in communication networks density, it is interesting to investigate the performance of CF when the number of transmitters is large. In this work, we show that as the number of transmitters, L, grows, CF becomes degenerated, in the sense that a relay prefers to decode only one (strongest) user instead of any other linear combination of the transmitted codewords, treating the other users as noise. Moreover, the system's sum-rate tends to zero as well. This makes scheduling necessary in order to maintain the superior abilities CF provides. We thus examine the problem of scheduling for CF. We start with insights on why good scheduling opportunities can be found. Then, we provide an asymptotically optimal, polynomial-time scheduling algorithm and analyze its performance. We conclude that with proper scheduling, CF is not merely non-degenerated, but, in fact, provides a gain for the system sum-rate, up to the optimal scaling law of O(loglogL).

Consider the problem of Private Information Retrieval (PIR) where a user wishes to retrieve a single message from N non-communicating and non-colluding databases (servers). All servers store the same set of M messages and they respond to the user through a block fading Gaussian Multiple Access Channel (MAC). The goal in this setting is to keep the index of the required message private from the servers while minimizing the overall communication overhead.This work provides joint privacy-channel coding retrieval schemes for the AWGN MAC with and without fading. The schemes exploit the linearity of the channel while using the Compute and Forward (CF) coding scheme. Consequently, single-user encoding and decoding are performed to retrieve the private message. The achievable retrieval rates are shown to outperform a separation-based scheme for which the retrieval and the channel coding are designed separately. Moreover, these rates are asymptotically optimal as the SNR grows and are up to a constant gap of 2 bits per channel use for every SNR.

Covert communication conceals transmission of messages from Alice to Bob out of a watchful adversary, Willie, which tries to determine if a transmission took place or not. While covert communication in a basic, vanilla settings where all variables are known to Willie results in the well known square-root law, when a jammer is present and assists Alice by creating uncertainty in Willie's decoder, this transmission may have a positive rate.In this work, we analyze the case where the jammer is equipped with multiple antennas and obtain the optimal transmission strategy of the jammer in order to maximize his assistance to Alice, in terms of maximizing a ratio between Willie's and Bob's noise variance. We show that the optimal strategy of the jammer is to perform beamforming towards a single direction with all his available power. This direction though, is not trivial, since it reflects an optimal tradeoff point between minimizing the interference at Bob and maximizing the interference at Willie.

Consider a Compute and Forward (CF) relay network with L users and a single relay. The relay tries to decode a linear function of the transmitted signals. For such a network, letting all L users transmit simultaneously, especially when L is large, causes a significant degradation in the rate in which the relay is able to decode. In fact, the rate goes to zero very fast with L. Therefore, in each transmission phase only a fixed number of users should transmit, i.e., users should be scheduled. In this work, we examine the problem of scheduling for CF and lay the foundations for identifying the optimal schedule which, to date, lacks a clear understanding. Specifically, we start with insights why when the number of users is large, good scheduling opportunities can be found. Then, we provide an asymptotically optimal, polynomial time scheduling algorithm and analyze it's performance. We conclude that scheduling under CF provides a gain in the system sum-rate, up to the optimal scaling law of O(log log L).

Consider the problem of a multiple access channel with a large number of users. In such a system, mostly due to practical constraints (e.g., decoding complexity), not all users can be scheduled together, and usually only one user may transmit at any given time. Assuming a distributed, opportunistic scheduling algorithm, we analyze the system's properties, such as delay, QoS, and capacity scaling laws. Specifically, we start with analyzing the performance while assuming the users are not necessarily fully backlogged, focusing on the queuing problem and, especially, on the strong dependence between the queues. We first extend a known queuing model by Ephremides and Zhu, to give new results on the convergence of the probability of collision to its average value (as the number of users grows), and hence for the ensuing system performance metrics, such as throughput and delay. This model, however, is limited in the number of users one can analyze. We thus suggest a new model, which is much simpler yet can accurately describe the system behavior when the number of users is large. We then proceed to the analysis of this system under the assumption of time dependent channels. Specifically, we assume each user experiences a different channel state sequence, expressing different channel fluctuations (specifically, the Gilbert-Elliott model). The system performance under this setting is analyzed, along with the channel capacity scaling laws.

Compute and Forward (CF) is a promising relaying scheme which, instead of decoding single messages or forwarding/amplifying information at the relay, decodes linear combinations of the simultaneously transmitted messages. The current literature includes several coding schemes and results on the degrees of freedom in CF, yet for systems with a fixed number of transmitters and receivers. It is unclear, however, how CF behaves at the limit of a large number of transmitters. In this paper, we investigate the performance of CF in that regime. Specifically, we show that as the number of transmitters grows, CF becomes degenerated, in the sense that a relay prefers to decode only one (strongest) user instead of any other linear combination of the transmitted codewords, treating the other users as noise. Moreover, the sum-rate tends to zero as well. This makes scheduling necessary in order to maintain the superior abilities CF provides. Indeed, under scheduling, we show that non-trivial linear combinations are chosen, and the sum-rate does not decay, even without state information at the transmitters and without interference alignment.