Under the assumption of perfect channel state information at the transmitters (CSIT), it is known that structured codes offer significant advantages for secure communication in an interference network, e.g., structured jamming signals based on lattice codes may allow a receiver to decode the sum of the jamming signal and the signal being jammed, even though they cannot be separately resolved due to secrecy constraints, subtract the aggregate jammed signal, and then proceed to decode desired codewords at lower power levels. To what extent are such benefits of structured codes fundamentally limited by uncertainty in CSIT? To answer this question, we explore what is perhaps the simplest setting where the question presents itself -- a Z interference channel with secure communication. Using sum-set inequalities based on Aligned Images bounds we prove that the GDoF benefits of structured codes are lost completely under finite precision CSIT. The secure GDoF region of the Z interference channel is obtained as a byproduct of the analysis.
Aligned-Images Sumset Inequalities are used in this work to study the Generalized Degrees of Freedom (GDoF) of the symmetric layered multi-hop interference channel under the robust assumption that the channel state information at the transmitters (CSIT) is limited to finite precision. First, the sum-GDoF value is characterized for the $2\times 2\times 2$ setting that is comprised of $2$ sources, $2$ relays, and $2$ destinations. It is shown that the sum-GDoF do not improve even if perfect CSIT is allowed in the first hop, as long as the CSIT in the second hop is limited to finite precision. Th…
For the $K=2$ user MISO BC, i.e., the wireless broadcast channel where a transmitter equipped with $K=2$ antennas sends independent messages to $K=2$ receivers each of which is equipped with a single antenna, the sum generalized degrees of freedom (GDoF) are characterized for arbitrary channel strength and channel uncertainty levels for each of the channel coefficients. The result is extended to $K>2$ users under additional restrictions which include the assumption of symmetry.
The generalized degrees of freedom (GDoF) region of the multiple-input multiple-output (MIMO) Gaussian Z-interference channel with an arbitrary number of antennas at each node is established under the assumption of delayed channel state information at transmitters (CSIT). The GDoF region is parameterized by $α$, which links the interference-to-noise ratio (INR) to the signal-to-noise ratio (SNR) via $INR=SNR^α$. A new outer bound for the GDoF region is established by maximizing a bound on the weighted sum-rate of the two users, which in turn is obtained by using a combination of genie-aided si…
Generalized Degrees of Freedom (GDoF) are characterized for the symmetric $K$-user Multiple Input Multiple Output (MIMO) Interference Channel (IC) under the assumption that the channel state information at the transmitters (CSIT) is limited to finite precision. In this symmetric setting, each transmitter is equipped with $M$ antennas, each receiver is equipped with $N$ antennas, each desired channel (i.e., a channel between a transmit antenna and a receive antenna belonging to the same user) has strength $\sim P$, while each undesired channel has strength $\sim P^α$, where $P$ is a nominal SNR…
The Generalized Degrees of Freedom (GDoF) of the two user interference channel are characterized for all parameter regimes under the assumption of finite precision channel state information at the transmitters (CSIT), when a limited amount of (half-duplex or full-duplex) cooperation is allowed between the transmitters in the form of $π$ DoF of shared messages. In all cases, the number of over-the-air bits that each cooperation bit buys is shown to be equal to either $0, 1, 1/2$ or $1/3$. The most interesting aspect of the result is the $1/3$ slope, which appears only under finite precision CSI…
In this paper, we investigate the generalized degrees-of-freedom (GDoF) of the asymmetric interference channel with delayed channel state information at the transmitter (CSIT), where each transmitter has two antennas, each receiver has one antenna, and the strength for each interfering link can vary. The optimal sum-GDoF is characterized by matched converse and achievability proof. Through our results, we also reveal that in our antenna setting, the symmetric GDoF lower bound in [Mohanty et. al, TIT 2019] can be elevated, and the symmetric GDoF upper bound in [Mohanty et. al, TIT 2019] is tigh…
The generalized degrees of freedom (GDoF) characterization of the symmetric K-user interference channel is obtained under finite precision channel state information at the transmitters (CSIT). The symmetric setting is where each cross channel is capable of carrying degrees of freedom (DoF) while each direct channel is capable of carrying 1 DoF. Remarkably, under finite precision CSIT the symmetric K-user interference channel loses all the GDoF benefits of interference alignment. The GDoF per user diminish with the number of users everywhere except in the very strong (optimal for every receiver…
In this work, we study the generalized degrees-of-freedom (GDoF) of downlink and uplink cellular networks, modeled as Gaussian interfering broadcast channels (IBC) and Gaussian interfering multiple access channels (IMAC), respectively. We focus on regimes of low inter-cell interference, where single-cell transmission with power control and treating inter-cell interference as noise (mc-TIN) is GDoF optimal. Recent works have identified two relevant regimes in this context: one in which the GDoF region achieved through mc-TIN for both the IBC and IMAC is a convex polyhedron without the need for …
We study the generalized degrees-of-freedom (GDoF) of cellular networks under finite precision channel state information at the transmitters (CSIT). We consider downlink settings modeled by the interfering broadcast channel (IBC) under no multi-cell cooperation, and the overloaded multiple-input-single-output broadcast channel (MISO-BC) under full multi-cell cooperation. We focus on three regimes of interest: the mc-TIN regime, where a scheme based on treating inter-cell interference as noise (mc-TIN) was shown to be GDoF optimal for the IBC; the mc-CTIN regime, where the GDoF region achievabl…
We study the $K=3$ user multiple input single output (MISO) broadcast channel (BC) with $M=3$ antennas at the transmitter and $1$ antenna at each receiver, from the generalized degrees of freedom (GDoF) perspective, under the assumption that the channel state information at the transmitter (CSIT) is limited to finite precision. In particular, our goal is to identify a parameter regime where a simple layered superposition (SLS) coding scheme achieves the entire GDoF region. With $α_{ij}$ representing the channel strength parameter for the link from the $j^{th}$ antenna of the transmitter to the…
We consider the cache-aided MISO broadcast channel (BC) in which a multi-antenna transmitter serves $K$ single-antenna receivers, each equipped with a cache memory. The transmitter has access to partial knowledge of the channel state information. For a symmetric setting, in terms of channel strength levels, partial channel knowledge levels and cache sizes, we characterize the generalized degrees of freedom (GDoF) up to a constant multiplicative factor. The achievability scheme exploits the interplay between spatial multiplexing gains and coded-multicasting gain. On the other hand, a cut-set-ba…
This work analyses the Generalized Degrees-of-Freedom (GDoF) of the 2-User Multiple-Input Single-Output (MISO) Broadcast Channel (BC) in the so-called Distributed CSIT regime, with application to decentralized wireless networks. This regime differs from the classical limited CSIT one in that the CSIT is not just noisy but also imperfectly shared across the transmitters (TXs). Hence, each TX precodes data on the basis of local CSIT and statistical quality information at other TXs. We derive the GDoF result and obtain the surprising outcome that by specific accounting of the pathloss information…
Significant progress has been made recently in Generalized Degrees of Freedom (GDoF) characterizations of wireless interference channels (IC) and broadcast channels (BC) under the assumption of finite precision channel state information at the transmitters (CSIT), especially for smaller or highly symmetric network settings. A critical barrier in extending these results to larger and asymmetric networks is the inherent combinatorial complexity of such networks. Motivated by other fields such as extremal combinatorics and extremal graph theory, we explore the possibility of an extremal network t…
We study a symmetric Gaussian wiretap channel with a helper, where a confidential message is sent from a transmitter to a legitimate receiver, in the presence of a helper and an eavesdropper, under a weak notion of secrecy constraint. For this setting, we characterize the optimal secure generalized degrees-of-freedom (GDoF). The result reveals that, adding a helper can significantly increase the secure GDoF of the wiretap channel. The result is supported by a new converse and a new scheme. In the proposed scheme, the helper sends a cooperative jamming signal at a specific power level and direc…
The generalized degrees of freedom (GDoF) of the two user symmetric multiple input multiple output (MIMO) interference channel (IC) are characterized as a function of the channel strength levels and the level of channel state information at the transmitters (CSIT). In this symmetric setting, each transmitter is equipped with M antennas, each receiver is equipped with N antennas, and both cross links have the same strength parameter $α$ and the same channel uncertainty parameter $β$. The main challenge resides in the proof of the outer bound which is accomplished by a generalization of the alig…
In many communication channels, secrecy constraints usually incur a penalty in capacity, as well as generalized degrees-of-freedom (GDoF). In this work we show an interesting observation that, adding a helper can totally remove the penalty in sum GDoF, for a two-user symmetric Gaussian interference channel. For the interference channel where each transmitter sends a message to an intended receiver without secrecy constraints, the sum GDoF is a well-known "W" curve, characterized by Etkin-Tse-Wang in 2008. If the secrecy constraints are imposed on this interference channel, where the message of…
We investigate the problem of secure transmission over a two-user multi-input multi-output (MIMO) X-channel in which channel state information is provided with one-unit delay to both transmitters (CSIT), and each receiver feeds back its channel output to a different transmitter. We refer to this model as MIMO X-channel with asymmetric output feedback and delayed CSIT. The transmitters are equipped with M-antennas each, and the receivers are equipped with N-antennas each. For this model, accounting for both messages at each receiver, we characterize the optimal sum secure degrees of freedom (SD…
In this paper, the sum secure degrees of freedom (SDoF) of the $K$-user Multiple Input/Single Output (MISO) Broadcast Channel with Confidential Messages (BCCM) and alternating Channel State Information at the Transmitter (CSIT) is investigated. In the MISO BCCM, a $K$-antenna transmitter (TX) communicates toward $K$ single-antenna receivers (RXs), so that message for RX $k$ is kept secret from RX $j$ with $j<k$. For this model, we consider the scenario in which the CSI of the RXs from $2$ to $K$ is instantaneously known at the transmitter while CSI of RX $1$ is known at the transmitter (i) ins…
We study the problem of secure transmission over a Gaussian two-user multi-input single-output (MISO) broadcast channel under the assumption that links connecting the transmitter to the two receivers may have unequal strength statistically. In addition to this, the state of the channel to each receiver is conveyed in a strictly causal manner to the transmitter. We focus on a two state topological setting of strong v.s. weak links. Under these assumptions, we first consider the MISO wiretap channel and establish bounds on generalized secure degrees of freedom (GSDoF). Next, we extend this model…
In this paper, we investigate the secure rate-splitting for the two-user multiple-input multiple-output (MIMO) broadcast channel with imperfect channel state information at the transmitter (CSIT) and a multiple-antenna jammer, where each receiver has an equal number of antennas and the jammer has perfect channel state information (CSI). Specifically, we design a secure rate-splitting multiple-access strategy, where the security of split private and common messages is ensured by precoder design with joint nulling and aligning the leakage information, regarding different antenna configurations. …
In this paper, we study the secure degrees-of-freedom (SDoF) characterization for the multiple-input multiple-output (MIMO) X channel with confidential messages and delayed CSIT. In particular, we propose a transmission scheme, which can be regarded as a generalization of the state-of-the-art scheme without security and with delayed CSIT. The key of this generalization is performing the security analysis, by which we derive the optimal duration of the artificial noise transmission phase. As a result, we derive the sum-SDoF lower bound. Furthermore, we reveal that if the number of receive anten…
We consider a secure communication over a two-user Gaussian interference channel, where each transmitter sends a confidential message to its legitimate receiver. For this setting, we identify a regime where the simple scheme of using Gaussian wiretap codebook at each transmitter (without cooperative jamming) and treating interference as noise at each intended receiver (in short, GWC-TIN scheme) achieves the optimal secure sum capacity to within a constant gap. For the symmetric case, this simple scheme is optimal when the interference-to-signal ratio (all link strengths in decibel scale) is no…
The two user multiple-input single-output (MISO) broadcast channel with confidential messages (BCCM) is studied in which the nature of channel state information at the transmitter (CSIT) from each user can be of the form $I_{i}$, $i=1,2$ where $I_{1}, I_{2}\in \{\mathsf{P}, \mathsf{D}, \mathsf{N}\}$, and the forms $\mathsf{P}$, $\mathsf{D}$ and $\mathsf{N}$ correspond to perfect and instantaneous, completely delayed, and no CSIT, respectively. Thus, the overall CSIT can alternate between $9$ possible states corresponding to all possible values of $I_{1}I_{2}$, with each state occurring for $λ_…
We study the problem of secure transmission over a Gaussian multi-input single-output (MISO) two receiver channel with an external eavesdropper, under the assumption that the state of the channel which is available to each receiver is conveyed either perfectly ($P$) or with delay ($D$) to the transmitter. Denoting by $S_1$, $S_2$, and $S_3$ the channel state information at the transmitter (CSIT) of user 1, user 2, and eavesdropper, respectively, the overall CSIT can then alternate between eight possible states, i.e., $(S_1,S_2,S_3) \in \{P,D\}^3$. We denote by $λ_{S_1 S_2 S_3}$ the fraction of…
We consider three channel models: the wiretap channel with $M$ helpers, the $K$-user multiple access wiretap channel, and the $K$-user interference channel with an external eavesdropper, when no eavesdropper's channel state information (CSI) is available at the transmitters. In each case, we establish the optimal sum secure degrees of freedom (s.d.o.f.) by providing achievable schemes and matching converses. We show that the unavailability of the eavesdropper's CSIT does not reduce the s.d.o.f. of the wiretap channel with helpers. However, there is loss in s.d.o.f. for both the multiple access…
We study the Gaussian MIMOME wiretap channel where a transmitter wishes to communicate a confidential message to a legitimate receiver in the presence of eavesdroppers, while the eavesdroppers should not be able to decode the confidential message. Each node in the network is equipped with arbitrary number of antennas. Furthermore, channels are time varying, and there is no channel state information available at the transmitter (CSIT) with respect to eavesdroppers' channels; and transmitter only has access to delayed CSIT of the channel to the legitimate receiver. The secure degrees of freedom …
In this paper, the $K$-user interference channel with secrecy constraints is considered with delayed channel state information at transmitters (CSIT). We propose a novel secure retrospective interference alignment scheme in which the transmitters carefully mix information symbols with artificial noises to ensure confidentiality. Achieving positive secure degrees of freedom (SDoF) is challenging due to the delayed nature of CSIT, and the distributed nature of the transmitters. Our scheme works over two phases: phase one in which each transmitter sends information symbols mixed with artificial n…
In this paper we consider the secure transmission over the fast fading multiple antenna Gaussian broadcast channels with confidential messages (FMGBC-CM), where a multiple-antenna transmitter sends independent confidential messages to two users with information theoretic secrecy and only the statistics of the receivers' channel state information are known at the transmitter. We first use the same marginal property of the FMGBC-CM to classify the non-trivial cases, i.e., those not degraded to the common wiretap channels. We then derive the achievable rate region for the FMGBC-CM by solving the …
In this work, the role of unidirectional limited rate transmitter cooperation is studied in the case of the 2-user Z interference channel (Z-IC) with secrecy constraints at the receivers, on achieving two conflicting goals simultaneously: mitigating interference and ensuring secrecy. First, the problem is studied under the linear deterministic model. The achievable schemes for the deterministic model use a fusion of cooperative precoding and transmission of a jamming signal. The optimality of the proposed scheme is established for the deterministic model for all possible parameter settings usi…
We consider the secure transmission in ergodic fast-Rayleigh fading multiple-input single-output single-antennaeavesdropper (MISOSE) wiretap channels. We assume that the statistics of both the legitimate and eavesdropper channels is the only available channel state information at the transmitter (CSIT). By introducing a new secrecy capacity upper bound, we prove that the secrecy capacity is achieved by Gaussian input without prefixing. To attain this, we form another MISOSE channel for upper-bounding, and tighten the bound by finding the worst correlations between the legitimate and eavesdropp…
In this paper, we consider secure transmissions in ergodic Rayleigh fast-faded multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channels with only statistical channel state information at the transmitter (CSIT). When the legitimate receiver has more (or equal) antennas than the eavesdropper, we prove the first MIMOME secrecy capacity with partial CSIT by establishing a new secrecy capacity upper-bound. The key step is to form an MIMOME degraded channel by dividing the legitimate receiver's channel matrix into two submatrices, and setting one of the submatrices to b…
We consider two fundamental multi-user channel models: the multiple-input multiple-output (MIMO) wiretap channel with one helper (WTH) and the MIMO multiple access wiretap channel (MAC-WT). In each case, the eavesdropper has $K$ antennas while the remaining terminals have $N$ antennas each. We consider a fast fading channel where the channel state information (CSI) of the legitimate receiver is available at the transmitters but no channel state information at the transmitters (CSIT) is available for the eavesdropper's channel. We determine the optimal sum secure degrees of freedom (s.d.o.f.) f…
In wiretap channels the eavesdropper's channel state information (CSI) is commonly assumed to be known at transmitter, fully or partially. However, under perfect secrecy constraint the eavesdropper may not be motivated to feedback any correct CSI. In this paper we consider a more feasible problem for the transmitter to have eavesdropper's CSI. That is, the fast fading multiple-antenna Gaussian broadcast channels (FMGBC-CM) with confidential messages, where both receivers are legitimate users such that they both are willing to feedback accurate CSI to maintain their secure transmission, and not…
We consider compound multi-input multi-output (MIMO) wiretap channels where minimal channel state information at the transmitter (CSIT) is assumed. Code construction is given for the special case of isotropic mutual information, which serves as a conservative strategy for general cases. Using the flatness factor for MIMO channels, we propose lattice codes universally achieving the secrecy capacity of compound MIMO wiretap channels up to a constant gap (measured in nats) that is equal to the number of transmit antennas. The proposed approach improves upon existing works on secrecy coding for MI…
In this paper, the role of secret key with finite rate is studied to enhance the secrecy performance of the system when users are operating in interference limited scenarios. To address this problem, a 2-user Gaussian Z-IC with secrecy constraint at the receiver is considered. One of the fundamental problems here is how to use the secret key as a part of the encoding process. The paper proposes novel achievable schemes, where the schemes differ from each other based on how the key has been used in the encoding process. The first achievable scheme uses one part of the key for one-time pad and r…
