We study the secure degrees of freedom (d.o.f.) of the MISO compound wiretap channel. The transmitter has M antennas, whereas the legitimate receiver and the eavesdropper each have one antenna and the channel vectors take one of finitely many values. If the number of states of either the legitimate receiver or the eavesdropper channel is less than M, then then we achieve full 1 d.o.f. If however the number of states of both the legitimate receiver and the eavesdropper channel are at-least equal to M, then we establish that the d.o.f. is strictly less than 1. Our upper bound is, to our knowledge, the first bound which is strictly tighter than the ¿pairwise upper bound¿. Lower bounds that combine ideas based on time-sharing, noise transmission, signal alignment and multi-level coding schemes are also provided.
We study the secure degrees of freedom (d.o.f.) of the MISO compound wiretap channel. The transmitter has M antennas, whereas the legitimate receiver and the eavesdropper each have one antenna and the channel vectors take one of finitely many values. If the number of states of either the legitimate receiver or the eavesdropper channel is less than M, then then we achieve full 1 d.o.f. If however the number of states of both the legitimate receiver and the eavesdropper channel are at-least equal to M, then we establish that the d.o.f. is strictly less than 1. Our upper bound is, to our knowledge, the first bound which is strictly tighter than the ¿pairwise upper bound¿. Lower bounds that combine ideas based on time-sharing, noise transmission, signal alignment and multi-level coding schemes are also provided.
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 secure degrees of freedom (s.d.o.f.) of a multiantenna Gaussian wiretap channel with N antennas at the transmitter and receiver and an arbitrary number of antennas, M, at the wiretapper is characterized when a multiantenna cooperative jammer (CJ) is available as a helper. This generalizes our previous result that assumed the same number of antennas at the eavesdropper as the legitimate parties. In particular, for arbitrary values of N andM, the s.d.o.f. is derived for all possible values of the number of antenna at the CJ, K. The achievability is based on a variety of signalling, beamforming, and alignment techniques which vary according to the value of K, whether M is larger than, smaller than, or equal to N, and whether the s.d.o.f. is integer valued or not an integer. The converse is based on combining an upper bound for the s.d.o.f. which allows for cooperation between the transmitter and CJ and holds for some values of K, with another upper bound which exploits the secrecy and reliability constraints and holds for other values of K.
This paper considers the compound wiretap channel, which generalizes Wyner's wiretap model to allow the channels to the (legitimate) receiver and to the eavesdropper to take a number of possible states. No matter which states occur, the transmitter guarantees that the receiver decodes its message and that the eavesdropper is kept in full ignorance about the message. The compound wiretap channel can also be viewed as a multicast channel with multiple eavesdroppers, in which the transmitter sends information to all receivers and keeps the information secret from all eavesdroppers. For the discrete memoryless channel, lower and upper bounds on the secrecy capacity are derived. The secrecy capacity is established for the degraded channel and the semideterministic channel with one receiver. The parallel Gaussian channel is further studied. The secrecy capacity and the secrecy degree of freedom ( ) are derived for the degraded case with one receiver. Schemes to achieve the for the case with two receivers and two eavesdroppers are constructed to demonstrate the necessity of a prefix channel in encoder design. Finally, the multi-antenna (i.e., MIMO) compound wiretap channel is studied. The secrecy capacity is established for the degraded case and an achievable is given for the general case.
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.
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 a wiretap channel model where the sender has transmit antennas and there are two groups consisting of and receivers respectively. Each receiver has a single an- tenna. We consider two scenarios. First we consider the compound wiretap model — group 1 constitutes the set of legitimate receivers, all interested in a common message, whereas group 2 is the set of eavesdroppers. We establish new lower and upper bounds on the secure degrees of freedom (d.o.f.). Our lower bound is based on the recently proposed real interference alignment scheme. The upper bound provides the first known example which illustrates that the pairwise upper bound used in earlier works is not tight. The second scenario we study is the compound private broadcast channel. Each group is interested in a message that must be pro- tected from the other group. Upper and lower bounds on the d.o.f. are developed by extending the results on the compound wiretap channel. Index Terms—Common message, compound channels, deter- ministic approximation, information theoretic security, interfer- ence alignment, multi-level coding, private broadcast channels, wiretap channel.