The Secrecy Capacity of the MIMO Wiretap Channel (0710.1920v1)

Abstract: We consider the MIMO wiretap channel, that is a MIMO broadcast channel where the transmitter sends some confidential information to one user which is a legitimate receiver, while the other user is an eavesdropper. Perfect secrecy is achieved when the the transmitter and the legitimate receiver can communicate at some positive rate, while insuring that the eavesdropper gets zero bits of information. In this paper, we compute the perfect secrecy capacity of the multiple antenna MIMO broadcast channel, where the number of antennas is arbitrary for both the transmitter and the two receivers.

• The paper derives the analytical secrecy capacity for MIMO wiretap channels by computing the rate difference between the legitimate receiver and eavesdropper.
• It extends Wyner’s classic model to non-degraded, multiple-antenna settings using advanced mutual information techniques and linear algebra.
• The findings provide a framework for designing secure multi-antenna wireless systems and pave the way for future research on practical implementations.

The Secrecy Capacity of the MIMO Wiretap Channel

This paper, authored by Frédérique Oggier and Babak Hassibi, investigates the secrecy capacity of the Multiple-Input Multiple-Output (MIMO) wiretap channel. In essence, the MIMO wiretap channel involves a transmitter (Alice) with multiple antennas, attempting to securely communicate with a legitimate receiver (Bob) who also has multiple antennas, while preventing an eavesdropper (Eve) from gaining any information.

Introduction

The paper addresses the challenge of ensuring security in wireless communication, leveraging the concept of information-theoretic security. This approach ensures that even with unlimited computational power, the eavesdropper cannot extract any useful information. The foundational model for this analysis is the wiretap channel, introduced by Wyner. The paper extends this concept to the MIMO setting, aiming to compute the perfect secrecy capacity—the maximum possible transmission rate to Bob while ensuring zero information leakage to Eve.

Perfect Secrecy Capacity

The crux of the paper lies in computing the secrecy capacity for the MIMO wiretap channel. The authors define perfect secrecy capacity as the rate at which Alice can transmit to Bob without Eve obtaining any information. This is mathematically formalized using mutual information, where perfect secrecy implies the mutual information between Eve's received signal and the transmitted message is zero.

Main Contributions

1. Analytical Derivation of Secrecy Capacity: The paper derives the secrecy capacity of the MIMO wiretap channel. This involves complex mathematical formulations, leveraging concepts from linear algebra and information theory.
2. Non-Degraded Channels: Unlike Wyner's assumption of degraded channels, the authors consider the more general and challenging scenario where the broadcast MIMO channel is not degraded. This is a significant step as it broadens the applicability of the results.
3. Proof Technique: The paper introduces a novel proof technique for the converse, applicable even when the channel is not degraded. This technique includes deriving upper bounds on mutual information and optimizing over input distributions and noise correlations.

Numerical and Analytical Results

The primary result of the paper is formalized in the theorem that states the secrecy capacity of the MIMO wiretap channel. Mathematically, it is given by:

$$C_S = \max_{K_X \succeq \mathbf{0}} \left( \log \det (I + H_M K_X H_M^*) - \log \det (I + H_E K_X H_E^*) \right),$$

where $K_X$ is the covariance matrix of the transmitted signal subject to a power constraint, $H_M$ and $H_E$ are the channel matrices for Bob and Eve, respectively.

The authors' derivation confirms that the secrecy capacity is indeed the difference between the capacities of the channel to Bob and the channel to Eve, extending Wyner's results to the MIMO scenario without the degraded assumption.

Implications

Theoretical Implications:

• The results bridge the gap between single antenna wiretap channels and their MIMO counterparts, providing a comprehensive understanding of secrecy in the presence of multiple antennas.
• It also opens avenues for further studies into more complex scenarios, such as fading conditions and time-varying channels.

Practical Implications:

• The findings could significantly impact the design of secure wireless communication systems.
• In practice, systems can now be designed with multiple antennas and still ensure secure communication using the principles derived in this paper.

Future Directions

The paper suggests several future research directions including:

• Applying the derived secrecy capacity to practical MIMO systems and validating assumptions through experimental setups.
• Extending the model to cases with imperfect channel state information at the transmitter.
• Investigating the impact of correlated eavesdropper channels or cooperative eavesdroppers on secrecy capacity.

Conclusion

In summary, this paper makes a substantial contribution to the field of secure wireless communications, providing a rigorous analytical determination of the secrecy capacity for the MIMO wiretap channel. By extending Wyner's classic model to more realistic settings with non-degraded channels and multiple antennas, it sets the foundation for advanced research and real-world applications in wireless security.