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IEEE Transactions on Wireless Communications

Secure Degrees of Freedom of MIMO Rayleigh Block Fading Wiretap Channels With No CSI Anywhere

Ta-Yuan Liu, Pritam Mukherjee, Şennur Ulukuş, Shih‐Chun Lin, Y.-W. Peter Hong

Published Jan 9, 2015• 26 citations
Publisher

Abstract

We consider the block Rayleigh fading multiple-input multiple-output (MIMO) wiretap channel with no prior channel state information (CSI) available at any of the terminals. The channel gains remain constant within a coherence interval of T symbols, and then change to another independent realization in the next coherence interval. The transmitter, the legitimate receiver, and the eavesdropper have nt, nr, and ne antennas, respectively. We determine the exact secure degrees of freedom (s.d.o.f.) of this system when T ≥ 2min(nt,nr). We show that, in this case, the s.d.o.f. is exactly equal to (min(nt,nr)-ne)+(T -min(nt,nr))/T. The first term in this expression can be interpreted as the eavesdropper with ne antennas taking away ne antennas from both the transmitter and the legitimate receiver. The second term can be interpreted as a fraction of the s.d.o.f. being lost due to the lack of CSI at the legitimate receiver. In particular, the fraction loss, min(nt,nr)/T, can be interpreted as the fraction of channel uses dedicated to training the legitimate receiver for it to learn its own CSI. We prove that this s.d.o.f. can be achieved by employing a constant norm channel input, which can be viewed as a generalization of discrete signalling to multiple dimensions.

DOI
10.1109/twc.2015.2390234

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We consider the block Rayleigh fading multiple-input multiple-output (MIMO) wiretap channel with no prior channel state information (CSI) available at any of the terminals. The channel gains remain constant within a coherence interval of T symbols, and then change to another independent realization in the next coherence interval. The transmitter, the legitimate receiver, and the eavesdropper have nt, nr, and ne antennas, respectively. We determine the exact secure degrees of freedom (s.d.o.f.) of this system when T ≥ 2min(nt,nr). We show that, in this case, the s.d.o.f. is exactly equal to (min(nt,nr)-ne)+(T -min(nt,nr))/T. The first term in this expression can be interpreted as the eavesdropper with ne antennas taking away ne antennas from both the transmitter and the legitimate receiver. The second term can be interpreted as a fraction of the s.d.o.f. being lost due to the lack of CSI at the legitimate receiver. In particular, the fraction loss, min(nt,nr)/T, can be interpreted as the fraction of channel uses dedicated to training the legitimate receiver for it to learn its own CSI. We prove that this s.d.o.f. can be achieved by employing a constant norm channel input, which can be viewed as a generalization of discrete signalling to multiple dimensions.
We consider the block Rayleigh fading multiple-input multiple-output (MIMO) wiretap channel with no prior channel state information (CSI) available at any of the terminals. The channel gains remain constant in a coherence time of T symbols, and then change to another independent realization. The transmitter, the legitimate receiver and the eavesdropper have nt, nrand neantennas, respectively. We determine the exact secure degrees of freedom (s.d.o.f.) of this system when T ≥ 2 min(nt, nr). We show that, in this case, the s.d.o.f. is exactly (min(nt, nr) − ne)+(T − min(nt, nr))/T. The first term can be interpreted as the eavesdropper with neantennas taking away neantennas from both the transmitter and the legitimate receiver. The second term can be interpreted as a fraction of s.d.o.f. being lost due to the lack of CSI at the legitimate receiver. In particular, the fraction loss, min(nt, nr)/T, can be interpreted as the fraction of channel uses dedicated to training the legitimate receiver for it to learn its own CSI. We prove that this s.d.o.f. can be achieved by employing a constant norm channel input, which can be viewed as a generalization of discrete signalling to multiple dimensions.
This article considers a two-way multiple-input multiple-output (MIMO) Rayleigh block fading wiretap channel with two full-duplex nodes (Alice and Bob) exchanging messages simultaneously and wiretapped by a $ {N}_{ {e}}$ -antennas eavesdropper (Eve). The channel state remains unchanged over a coherence interval T and is unknown to all terminals, including Alice, Bob, and Eve, at the beginning of the coherence interval. We first show the expression of the optimal signal waveform for Alice and Bob, which maximizes the sum secrecy rate, should be the products of independent random diagonal matrices and isotropically distributed unitary matrices. Then, conditioned on large T and small $ {N}_{ {e}}$ , the upper-bound for the secure degree of freedom (s.d.o.f.) of the two-way wiretap channel is derived, and a constant-norm-signaling (CNS) scheme is presented to achieve this theoretical bound. Finally, we formulate an optimization problem of the s.d.o.f. achieved by the CNS scheme for general cases with arbitrary T and $ {N}_{ {e}}$ , and solve this problem by exploring the monotonicity of the achievable s.d.o.f.. Our result shows that the optimal s.d.o.f. can be independent of coherence time T and becomes the product of the numbers of legitimate nodes's transmitter antennas (NLNTA) for large $ {N}_{ {e}}$ and small NLNTA.
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.
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 (MAC-WT) channel. 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 CSI at the transmitters (CSIT) is available for the eavesdropper's channel. We determine the optimal sum secure degrees of freedom (s.d.o.f.) for each channel model for the regime K ≤ N, and show that in this regime, the MAC-WT channel reduces to the WTH in the absence of eavesdropper CSIT. For the regime N ≤ K ≤ 2N, we obtain the optimal linear s.d.o.f., and show that the MAC-WT channel and the WTH have the same optimal s.d.o.f. when restricted to linear encoding strategies. In the absence of any such restrictions, we provide an upper bound for the sum s.d.o.f. of the MAC-WT channel in the regime N ≤ K ≤ 2N. Our results show that unlike in the single-input single-output case, there is loss of s.d.o.f. for even the WTH due to lack of eavesdropper CSIT when K ≥ N.
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) are considered. In each case, the eavesdropper has K antennas while the remaining terminals have N antennas. 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. The optimal sum secure degrees of freedom (s.d.o.f.) for each channel model is determined for the regime K ≤ N, and we show that in this regime, the MAC-WT channel reduces to the WTH in the absence of eavesdropper CSIT. For the regime N ≤ K ≤ 2N, we obtain the optimal linear s.d.o.f., and show that the MAC-WT channel and the WTH have the same optimal s.d.o.f. when restricted to linear encoding strategies. In the absence of any such restrictions, we provide an upper bound for the sum s.d.o.f. of the MAC-WT chanel in the regime N ≤ K ≤ 2N. Our results show that unlike in the single-input single-output (SISO) case, there is loss of s.d.o.f. for even the WTH due to lack of eavesdropper CSIT, when K ≥ N.