Secure Transmission with Multiple Antennas: The MISOME Wiretap Channel (0708.4219v1)

Abstract: The role of multiple antennas for secure communication is investigated within the framework of Wyner's wiretap channel. We characterize the secrecy capacity in terms of generalized eigenvalues when the sender and eavesdropper have multiple antennas, the intended receiver has a single antenna, and the channel matrices are fixed and known to all the terminals, and show that a beamforming strategy is capacity-achieving. In addition, we show that in the high signal-to-noise (SNR) ratio regime the penalty for not knowing eavesdropper's channel is small--a simple ``secure space-time code'' that can be thought of as masked beamforming and radiates power isotropically attains near-optimal performance. In the limit of large number of antennas, we obtain a realization-independent characterization of the secrecy capacity as a function of the number $\beta$: the number of eavesdropper antennas per sender antenna. We show that the eavesdropper is comparatively ineffective when $\beta<1$, but that for $\beta\ge2$ the eavesdropper can drive the secrecy capacity to zero, thereby blocking secure communication to the intended receiver. Extensions to ergodic fading channels are also provided.

• The paper establishes that the secrecy capacity can be characterized via generalized eigenvalue analysis and optimal beamforming strategies in MISOME channels.
• It derives a tight secrecy capacity upper bound and demonstrates that a near-optimal masked beamforming scheme achieves strong performance in high SNR regimes.
• The study further analyzes scaling laws for large systems and extends its results to fast fading channels, offering valuable insights for secure wireless network design.

Secure Transmission with Multiple Antennas: The MISOME Wiretap Channel

The paper "Secure Transmission with Multiple Antennas: The MISOME Wiretap Channel" by Ashish Khisti and Gregory W. Wornell explores the critical investigation of utilizing multiple antennas for secure communication within the framework of Wyner's wiretap channel. This paper provides a nuanced characterization of the secrecy capacity in scenarios where both the sender and eavesdropper are equipped with multiple antennas, while the intended receiver has a single antenna; herein referred to as the MISOME (multi-input, single-output, multi-eavesdropper) wiretap channel.

Summary of Contributions and Key Results

The primary contributions and key results of this paper include:

1. Secrecy Capacity Characterization: The paper establishes that in the context where both channel matrices (sender to receiver and sender to eavesdropper) are fixed and known to all terminals, the secrecy capacity can be expressed in terms of the generalized eigenvalues. Specifically, it shows that a beamforming strategy is capacity-achieving. In cases of high signal-to-noise ratio (SNR), even when the eavesdropper's channel is unknown, the performance degradation is marginal.
2. Secrecy Capacity Upper Bound: An upper bound on the secrecy capacity is established through a genie-aided analysis. This upper bound is shown to be achievable, thus confirming its tightness. The resulting secrecy capacity is given by:

$$C(P) = \left\{\log \lambda_{\text{max}} \left( \Sigma + P \mathbf{h}_r \mathbf{h}_r^\dagger, \Sigma + P \mathbf{H}_e^\dagger \mathbf{H}_e \right) \right\}^+,$$

where $\lambda_{\text{max}}$ represents the largest generalized eigenvalue.

1. High SNR Regime Analysis: In high SNR conditions, the secrecy capacity either reaches a finite limit when the channel to the receiver is within the span of the eavesdropper's channel or scales logarithmically with SNR otherwise. The paper provides the high SNR asymptotic characterization:

$$\lim_{P \to \infty} C(P) = \{\log \max \left( \lambda_{\text{max}} \left( \mathbf{h}_r \mathbf{h}_r^\dagger, \mathbf{H}_e^\dagger \mathbf{H}_e \right) \right) \}^+ \quad \text{if } \mathbf{H}_e^\perp \mathbf{h}_r = \mathbf{0},$$

otherwise,

$$\lim_{P \to \infty} \left[ C(P) - \log P \right] = \log \|\mathbf{H}_e^\perp \mathbf{h}_r \|^2.$$

1. Masked Beamforming Scheme: A suboptimal, simpler scheme termed masked beamforming also achieves near-optimal performance in the high SNR regime. This scheme does not require knowledge of the eavesdropper's channel matrix and radiates power isotropically. The achievable rate using masked beamforming is given by:

$$R_{MB}(P) = \left\{\log \lambda_{\text{max}} \left( \frac{P}{t} \mathbf{h}_r \mathbf{h}_r^\dagger, \Sigma + \frac{P}{t} \mathbf{H}_e^\dagger \mathbf{H}_e \right) + \log \left( 1 + \frac{t}{P \|\mathbf{h}_r\|^2} \right) \right\}^+.$$

1. Scaling Laws for Large Systems: The paper also investigates the scaling behavior of secrecy capacity in the large system limit with a fixed SNR. In particular, the asymptotic scaling laws reveal that the eavesdropper's capability to block secure communication significantly hinges on the antenna ratio $\beta = e/t$. When $\beta < 1$, the eavesdropper is ineffective, but for $\beta \geq 2$, the secrecy capacity approaches zero.
2. Fading Channels: Extensions to fast fading channels where the channel states vary over time are also analyzed. Upper and lower bounds on the secrecy capacity are derived, demonstrating that the asymptotic behaviors observed in fixed channels persist in the ergodic fading scenarios.

Practical and Theoretical Implications

The implications of these findings span both practical and theoretical realms. Practically, the results guide the design of secure communication systems in wireless networks, emphasizing the deployment of multiple antennas and strategic beamforming. Theoretically, the paper contributes to the understanding of physical layer security, particularly in determining the limits of secure communication in multi-antenna systems.

Future Directions

The paper opens several avenues for future research. This includes tighter bounds for fast fading channels, extending the analysis to more general MIMOME channels (multi-input, multi-output, multi-eavesdropper), and exploring physical layer security mechanisms in conjunction with application-layer cryptographic methods for enhanced security in wireless communication systems.

In summary, the paper provides a comprehensive analysis of the MISOME wiretap channel, including novel results on its secrecy capacity, effective high-SNR strategies, and practical coding schemes. These contributions are essential for advancing secure communication methodologies in multi-antenna wireless networks.