Abstract—We consider the Gaussian wiretap channel where a transmitter wishes to communicate a secure message to a legitimate receiver in the presence of eavesdroppers, without the eavesdroppers being able to decode the secure message. We focus on a setting that the transmitter is blind with respect to the state of channels to eavesdroppers, and only has access to delayed channel state information (CSI) of the legitimate receiver, which is referred to as “blind wiretap channel with delayed CSIT”. We then consider two scenarios: (i) the case where the secure communication is aided via a distributed jammer, (ii) the case where all nodes in the network are equipped with multiple antennas, referred to as blind MIMO wiretap channel with delayed CSIT. We completely characterize the secure Degrees of Freedom (SDoF) in both scenarios, when assuming linear coding strategies at the transmitter(s). I.
Abstract—We consider the Gaussian wiretap channel where a transmitter wishes to communicate a secure message to a legitimate receiver in the presence of eavesdroppers, without the eavesdroppers being able to decode the secure message. We focus on a setting that the transmitter is blind with respect to the state of channels to eavesdroppers, and only has access to delayed channel state information (CSI) of the legitimate receiver, which is referred to as “blind wiretap channel with delayed CSIT”. We then consider two scenarios: (i) the case where the secure communication is aided via a distributed jammer, (ii) the case where all nodes in the network are equipped with multiple antennas, referred to as blind MIMO wiretap channel with delayed CSIT. We completely characterize the secure Degrees of Freedom (SDoF) in both scenarios, when assuming linear coding strategies at the transmitter(s). I.
We study the Gaussian MIMO wiretap channel where a transmitter wishes to communicate a secure message to a legitimate receiver in the presence of eavesdroppers, while the eavesdroppers should not be able to decode the secure message. Each node in the network is equipped with arbitrary number of antennas. Furthermore, channels are time varying, and the transmitter has no channel state information (CSIT) of its channels to the eavesdroppers; and it only has access to delayed CSIT of the channel to the legitimate receiver. We completely characterize the secure degrees of freedom (SDoF) of such network, when restricted to linear encoding schemes. In particular, we strictly improve the state-of-the-art achievable schemes for this network. Furthermore, we develop a tight converse result by presenting a lemma, which states that once the transmitter(s) in a network have no CSIT of a certain receiver, and receivers have the same number of antennas, the least amount of alignment will occur at that receiver, meaning that transmit signals will occupy the maximal signal dimensions at that receiver.
We consider the Gaussian wiretap channel with a transmitter, a legitimate receiver, and k eavesdroppers (k ∈ ℕ), where the secure communication is aided via a jammer. We focus on the setting where the transmitter and the jammer are blind with respect to the state of channels to eavesdroppers, and only have access to delayed channel state information (CSI) of the legitimate receiver, which is referred to as “blind cooperative wiretap channel with delayed CSIT”. We show that a strictly positive secure Degrees of Freedom (DoF) of 1 over 3 is achievable irrespective of the number of eavesdroppers (k) in the network, and further, 1 over 3 is optimal assuming linear coding strategies at the transmitters. The converse proof is based on two key lemmas. The first lemma, named Rank Ratio Inequality, shows that if two distributed transmitters employ linear strategies, the ratio of the dimensions of received linear sub-spaces at the two receivers cannot exceed 3/2, due to delayed CSI. The second lemma implies that once the transmitters in a network have no CSI with respect to a receiver, the least amount of alignment will occur at that receiver, meaning that transmit signals will occupy the maximal signal dimensions at that receiver. Finally, we show that once the transmitter and the jammer form a single transmitter with two antennas, which we refer to as MISO wiretap channel, 1 over 2 is the optimal secure DoF when using linear schemes.
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 (SDoF) for such network has only been characterized for special cases, and is unknown in general. We completely characterize the SDoF of this network for all antenna configurations. In particular, we strictly improve the state-of-the-art achievable scheme for this network by proposing more efficient artificial noise alignment at the receivers. Furthermore, we develop a tight upper bound by utilizing 4 important inequalities that provide lower bounds on the received signal dimensions at receivers which supply delayed CSIT or no CSIT, or at a collection of receivers where some supply no CSIT. These inequalities together allow for analysis of signal dimensions in networks with asymmetric CSIT; and as a result, we present a converse proof that leads to characterization of SDoF for all possible antenna configurations.
We study the Gaussian multiple-input multiple-output multiple-eavesdropper wiretap channel where a transmitter wishes to communicate a confidential message to a legitimate receiver in the presence of eavesdroppers. Each node in the network is equipped with an 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 (SDoF) for such a network has only been characterized for special cases, and is unknown in general. We completely characterize the SDoF of this network for all antenna configurations. In particular, we strictly improve the state-of-the-art achievable scheme for this network by proposing more efficient artificial noise alignment at the receivers. Furthermore, we develop a tight upper bound by utilizing three important inequalities that provide lower bounds on the received signal dimensions at receivers which supply delayed CSIT or no CSIT. These inequalities together allow for analysis of signal dimensions in networks with heterogeneous CSIT; and as a result, we present a converse proof that leads to characterization of SDoF for all possible antenna configurations.
The multiple-input single-output (MISO) wiretap channel with K legitimate single-antenna receivers, one single-antenna eavesdropper in presence of delayed channel state information at the transmitter (CSIT) is considered. The transmitter is equipped with (K+1) antennas and has independent messages intended for each one of the K legitimate receivers. While for the case of a single receiver wiretap channel, i.e., K = 1, the optimal secure degrees of freedom (SDoF) with delayed CSIT was recently characterized in [1] and shown to be 2/3, the extension to the multi-receiver case (K > 1) is far from straightforward. We present new results and insights for the simplest non-trivial extension of this problem, i.e., for the case of K = 2 receiver MISO wiretap channel with delayed CSIT. The contribution of this paper is two fold: a) an asymptotic achievable scheme is presented for the case of two legitimate receivers which achieves a sum-SDoF of 36/37 and b) a novel converse proof is presented which shows that the sum-SDoF is upper bounded by 16/15.
The secrecy degrees of freedom (SDoF) of the Gaussian multiple-input and single-output (MISO) wiretap channel is studied under the assumption that delayed channel state information (CSI) is available at the transmitter and each receiver knows its own instantaneous channel. Such scenario is of practical interest since the legitimate receiver may send its channel states to the transmitter which is overheard by the eavesdropper. We first show that a strictly positive SDoF can be guaranteed whenever the transmitter has delayed CSI (either on the legitimate channel or/and the eavesdropper channel). In particular, in the case with delayed CSI on both channels, it is shown that the optimal SDoF is 2=3. We then generalize the result to the two-user Gaussian MISO broadcast channel with confidential messages and characterize the SDoF region when the transmitter has delayed CSI of both receivers. Interestingly, the artificial noise schemes are shown to provide the optimal SDoF region by masking the confidential message to the unintended receiver while aligning the interference at each receiver.