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arXiv:1512.07290 · IEEE Information Theory Workshop (ITW)

Secure Degrees of Freedom for the MIMO Wire-Tap Channel With a Multi-Antenna Cooperative Jammer

Mohamed Nafea, Aylin Yener

Published Nov 1, 2014• 39 citations
arXivHTMLPDF

Abstract

A multiple antenna Gaussian wiretap channel with a multiantenna cooperative jammer (CJ) is considered and the secure degrees of freedom (s.d.o.f.), with N antennas at the sender, receiver, and eavesdropper, is derived for all possible values of the number of antennas at the cooperative jammer, K. In particular, the upper and lower bounds for the s.d.o.f. are provided for different ranges of K and shown to coincide. Gaussian signaling both for transmission and jamming is shown to be sufficient to achieve the s.d.o.f. of the channel, when the s.d.o.f. is integer-valued. By contrast, when the channel has a non-integer s.d.o.f., structured signaling and joint signal space and signal scale alignment are employed to achieve the s.d.o.f.

DOI
10.1109/itw.2014.6970907

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Secure degrees of freedom of N ×N ×M wiretap channel with a K-antenna cooperative jammer

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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.

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A multiple antenna Gaussian wiretap channel with a multiantenna cooperative jammer (CJ) is considered and the secure degrees of freedom (s.d.o.f.), with N antennas at the sender, receiver, and eavesdropper, is derived for all possible values of the number of antennas at the cooperative jammer, K. In particular, the upper and lower bounds for the s.d.o.f. are provided for different ranges of K and shown to coincide. Gaussian signaling both for transmission and jamming is shown to be sufficient to achieve the s.d.o.f. of the channel, when the s.d.o.f. is integer-valued. By contrast, when the channel has a non-integer s.d.o.f., structured signaling and joint signal space and signal scale alignment are employed to achieve the s.d.o.f.
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.
In this work, multiple antenna Gaussian wiretap channels with a multiantenna cooperative jammer are considered. In particular, the focus is on identifying the spatial resources needed at the cooperative jammer in order to achieve a secure degrees of freedom (s.d.o.f.) for the channel in question equal to the degrees of freedom (d.o.f.). In order to accomplish this goal, the cooperative jammer sets out to send correlated jamming signals. Simultaneously, the legitimate transmitter chooses a precoder that perfectly aligns its information signals over the jamming ones at the eavesdropper. Both terminals use structured signaling, i.e., discrete constellations. The proposed approach is shown to achieve full d.o.f. for the channel at hand as long as the cooperative jammer has twice the antennas of the eavesdropper. It is also shown that within 1/2 of the d.o.f. is achievable with one less antenna at the cooperative jammer.
A Gaussian wiretap channel with a helper, i.e., a cooperative jammer, is considered and its secure degrees of freedom (s.d.o.f.) is computed. Previous work showed that the s.d.o.f. is upper bounded by 1 over 2 in this model when all parties are equipped with one antenna each. In this paper, the more challenging scenario where the eavesdropper has multiple antennas is tackled. Relying on structured signaling and cooperative jamming, specifically, by real interference alignment, it is shown that s.d.o.f. of 1 over 2 is achievable irrespective of the number of antennas the eavesdropper may have as long as the cooperative jammer has the same number of antennas as the eavesdropper. The design insight revealed is that the price to pay for the increase in the number of antennas at the eavesdropper is an equivalent increase in the number of antennas at the cooperative jammer in order to maintain the same s.d.o.f.
In this letter, the secure degrees of freedom (SDoF) of 2-user multiple-input multiple-output (MIMO) interference channel with an external N-antenna cooperative jammer (CJ) is considered. In this model, each transmitter intends to send confidential messages to the corresponding receiver and keep secret with unintended receiver. Different from the previous studies on interference channels, our model has an external cooperative jammer, which can improve the SDoF by transmitting interference signals. We derive the converse and give the corresponding achievable schemes to determine the exact SDoF, and study the influence of CJ on the SDoF of this system. We find that adding one CJ antenna has the same effect as adding one transmitting antenna, and has a better effect than adding one receiving antenna. The converse is given by using vector extensions of the role of a helper and secrecy penalty lemmas. For arbitrary values of N and M, we provide different achievable schemes for the corresponding regimes by making use of the symbol extension and spatial alignment techniques.
We investigate the secure degrees of freedom (SDoF) of the multiple-input multiple-output (MIMO) wiretap channel with intersymbol interference (ISI) in the presence of a multi-antenna cooperative jammer. We focus on the practically relevant setting with no channel state information at the transmitters (CSIT). More specifically, the legitimate transmitter and the cooperative jammer only have statistical knowledge of the channel in terms of the effective number of ISI channel taps, i.e., channel impulse response (CIR) lengths, toward the legitimate receiver and the eavesdropper. Our main contribution is to show that in the absence of CSIT, positive SDoF can be achieved by carefully exploiting: 1) the heterogeneity of the CIR lengths toward both receivers and 2) the relative number of antennas at the four terminals. To achieve secrecy, we propose a scheme in which the transmitter strategically sends a mixture of information and artificial noise symbols in a way that exploits the heterogeneity of CIR lengths. Additionally, the cooperative jammer transmits artificial noise symbols in a way that completely masks all the information symbols that are received at the eavesdropping node. Under the proposed scheme, positive SDoF can be achieved, even when the number of antennas at the legitimate receiver is less than the number of antennas at the eavesdropper.
We investigate the secure degrees of freedom (SDoF) of the wiretap and the K user Gaussian broadcast channels with multiple antennas at the transmitter, the legitimate receivers and an unknown number of eavesdroppers each with a number of antennas less than or equal to a known value NE. The channel matrices between the legitimate transmitter and the receivers are available everywhere, while the legitimate pair have no information about the eavesdroppers' channels. We provide the exact sum SDoF for the considered system. A new comprehensive upperbound is deduced and a new achievable scheme based on utilizing jamming is exploited. We prove that cooperative jamming is SDoF optimal even without the eavesdropper CSI available at the transmitters.