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Towards the Secrecy Capacity of the Gaussian MIMO Wire-tap Channel: The 2-2-1 Channel
Shabnam Shafiee, Nan Liu, Sennur Ulukus
TL;DR
The paper asks for the secrecy capacity of a 2-2-1 Gaussian MIMO wire-tap channel, extending beyond cases with a single-antenna receiver. It proposes an achievable Gaussian beam-forming scheme and a tight upper bound, showing that beam-forming without information pre-processing is optimal.
Problem
The secrecy capacity of Gaussian MIMO wire-tap channels with multiple-antenna receivers remains unresolved, so the paper studies the 2-2-1 case.
Method
The paper combines a Gaussian beam-forming achievable scheme with a tight upper bound based on giving the eavesdropper’s signal to the receiver and correlating receiver noises.
Results
The achievable scheme and upper bound meet, establishing the secrecy capacity and showing that beam-forming without information pre-processing is optimal.
Takeaways & Limitations
For the 2-2-1 channel, optimal secrecy transmission uses Gaussian signalling through a unit-rank beam-forming covariance without information pre-processing.
Takeaways & Limitations
The analysis assumes an average power constraint and focuses on the 2-2-1 channel, with full-rank legitimate-channel matrices treated as the nontrivial case.
Abstract
from arXiv · showhide
We find the secrecy capacity of the 2-2-1 Gaussian MIMO wire-tap channel, which consists of a transmitter and a receiver with two antennas each, and an eavesdropper with a single antenna. We determine the secrecy capacity of this channel by proposing an achievable scheme and then developing a tight upper bound that meets the proposed achievable secrecy rate. We show that, for this channel, Gaussian signalling in the form of beam-forming is optimal, and no pre-processing of information is necessary.
1 Introduction
The paper addresses the difficulty of finding secrecy capacity for non-degraded Gaussian MIMO wire-tap channels by studying a tractable case with multiple antennas at both transmitter and receiver. It shows that beam-forming with Gaussian signalling is optimal and develops an upper bound for the secrecy capacity.
- Problem: Because Gaussian MIMO wire-tap channels are generally non-degraded, determining secrecy capacity requires identifying the optimal auxiliary-variable distribution representing information pre-processing.The auxiliary random variable is interpreted as performing pre-processing on the information.
- Contribution: The paper advances beyond prior cases with single-antenna receivers by considering a MIMO channel with multiple antennas at both transmitter and receiver.The general problem is described as intractable, motivating focus on a simple special case.
- Method: The optimal Gaussian signalling scheme has a unit-rank transmit covariance matrix, making beam-forming optimal.The beam-forming direction is chosen to be as orthogonal as possible to the eavesdropper’s direction and as close as possible to the receiver’s two directions.
- Method: The secrecy-capacity upper bound is obtained by giving the eavesdropper’s signal to the receiver.The secrecy capacity of this enhanced channel provides an upper bound to the original channel.
2 System Model
The system model defines the 2-2-1 Gaussian MIMO wire-tap channel under an average power constraint and characterizes its secrecy capacity. The analysis reduces rank-deficient cases to a known 2-1-1 channel and focuses on full-rank H, excluding degraded channels where Gaussian signalling is already optimal without preprocessing.
- System model: The 2-2-1 channel consists of transmitted signal x, legitimate-user output y, and eavesdropper output z, with independent Gaussian noises.The legitimate-user noise has zero mean and identity covariance, while the eavesdropper noise has zero mean and unit variance.
- System model: The transmitted signal satisfies an average power constraint, and secrecy capacity C(P) measures reliable communication while leaving the eavesdropper essentially no better informed.C(P) is defined as the maximum number of bits correctly transmitted to the intended receiver under the secrecy condition.
- Full-rank assumption: When H is not full-rank, singular-value decomposition and channel rotation reduce the system to a 2-1-1 channel with known secrecy capacity.Accordingly, the paper assumes H is full-rank and invertible without loss of generality.
- Degraded versus non-degraded channels: For degraded channels, z is a noisy version of y, so no information preprocessing is necessary and Gaussian signalling is optimal.The paper therefore concentrates on the more difficult full-rank case satisfying the stated non-degraded condition.
3 An Achievable Scheme
The section presents an achievable secrecy scheme based on choosing u = x and restricting the input to Gaussian signals with covariance S under a power constraint. The resulting achievable secrecy rate is given by (8), with zero rate when its maximum is negative.
- Achievable Scheme: Choosing u = x and Gaussian input covariance S with tr(S) ≤ P yields an achievable secrecy rate.The construction uses the Markov relation u → x → yz.
- Achievable Scheme: The resulting achievable secrecy rate is expressed in equation (8).The displayed expression includes the term 2 log(1 + g^TSg).
- Achievable Scheme: If the maximum value in (8) is negative, the achieved secrecy rate is zero.Otherwise, the scheme achieves the secrecy rate specified by the maximization in (8).
I + HTHS
The section proves that the optimal transmit covariance S is unit-rank for the 2-2-1 channel, yielding a beam-forming achievable secrecy rate. It identifies the optimal beam direction through a Rayleigh-quotient eigenvector and shows the resulting secrecy rate is strictly positive.
- Unit-rank optimality: The optimal covariance S is unit-rank because H^T H is invertible and satisfies Lemma 1’s conditions.Thus, the optimal signaling strategy reduces to beam-forming.
- Achievable secrecy rate: The corresponding achievable secrecy rate is maximized over a unit-norm beam vector q.Writing S in unit-rank form converts the optimization into a Rayleigh quotient.
- Beam direction: The optimal beam q_a is the unit-norm eigenvector associated with the largest eigenvalue of B^-1/2 A B^-1/2.This eigenvector provides the maximizing direction for the Rayleigh-quotient expression.
- Positivity: The achievable secrecy rate is strictly positive because H is full rank and H g⊥ ≠ 0 for a unit-norm vector orthogonal to g.Choosing S = P g⊥g⊥^T produces a strictly positive achievable rate, and the optimized rate is at least as large.
4 A Tight Upper Bound
The section constructs a tight secrecy-capacity upper bound by selecting a noise-correlation vector a∗ that makes the bound tractable and unit-rank optimal. The resulting upper bound is 1/2 log λ1, matching the achievable lower bound.
- Upper-bound construction: Theorem 1 gives an upper bound on the secrecy capacity for any correlation vector a satisfying ||a|| < 1.The vector a represents correlation between the legitimate receiver’s and eavesdropper’s Gaussian noises.
- Upper-bound construction: The selected vector a∗ maximizes the relevant second-order polynomial while satisfying ||a∗|| < 1.The maximizer is obtained by choosing 1/α∗ to maximize θ(α).
- Unit-rank optimality: For a∗, the maximizing covariance matrix S is unit-rank.A(a∗) satisfies the conditions of Lemma 1, so arg max U(S, a∗) is unit-rank.
- Evaluating the bound: 1/2 log λ1 is the resulting maximum value of the upper-bound optimization over S ⪰ 0 with tr(S) ≤ P.The optimization reduces to a Rayleigh quotient whose largest eigenvalue is λ1, with λ1 > 1.
- Tightness: 1/2 log λ1 equals the lower bound on secrecy capacity established by the achievable scheme, making the upper bound tight.The equality establishes the secrecy capacity for this channel.
5 Conclusions
The secrecy capacity of the 2-2-1 Gaussian MIMO wire-tap channel is determined by matching an achievable Gaussian beam-forming scheme with a tight upper bound. The derivation relies on the channel’s 2-2-1-specific unit-rank optimal transmit structure and does not establish the general MIMO case.
- 5 Conclusions: The secrecy capacity is determined by solving for the optimum auxiliary-variable and channel-input joint distribution in the Csiszar-Korner formula.A lower bound is obtained by evaluating the formula for a specific choice of auxiliary variable and channel input.
- 5 Conclusions: Gaussian signalling with beam-forming and no information pre-processing achieves the secrecy capacity.The proposed achievable scheme is optimal because a tight upper bound meets its achievable rate.
- 5 Conclusions: The tight upper bound is constructed by giving the eavesdropper’s signal to the legitimate receiver and can be explicitly evaluated and tightened for the 2-2-1 case.The upper-bound construction is defined for a general MIMO wire-tap channel, but its explicit evaluation here uses the 2-2-1 restriction.
- 5 Conclusions: The matching lower and upper bounds have not been established for general MIMO channels because the derivation depends essentially on unit-rank beam-forming transmit matrices.Beam-forming is not likely to remain optimal when the numbers of transmit and receive antennas exceed two.
6 Appendix
The appendix proves Theorem 1 by extending prior lemmas to two legitimate-receiver antennas and derives a secrecy-capacity upper bound using an enhanced degraded channel. The bound is evaluated through correlated noises and an LMMSE-based calculation.
- Proof of Theorem 1: Theorem 1’s proof extends a prior two-part lemma framework to multiple legitimate-receiver antennas, specifically handling two antennas at the legitimate receiver.The proof uses the generalized version of [10, Lemma 1] and extends [10, Lemma 2].
- Upper bound: The upper bound enhances the channel by giving the legitimate receiver access to the eavesdropper’s signal, yielding a more-capable degraded channel with secrecy-capacity formula (78).Because the legitimate receiver is more capable in the enhanced channel, its secrecy capacity upper-bounds that of the original channel.
- Upper bound: The noise correlation parameter a must satisfy ||a|| < 1; it leaves the original secrecy capacity unchanged but affects the upper bound, which remains valid for every such a.The appendix introduces correlation between n_y and n_z and evaluates the resulting conditional mutual information using Gaussian-noise properties and an LMMSE estimator.
- Upper bound: The resulting secrecy-capacity upper bound is expressed in terms of U(S, a) for any a with ||a|| < 1.The appendix concludes the upper-bound derivation after evaluating the relevant covariance and determinant expressions.