Ferkans — Interactive Telecom Tutor

Secrecy with Multiple Antennas

The scalar wiretap channel requires the eavesdropper to have a worse channel than the legitimate receiver. In practice, we cannot always guarantee this. But with multiple antennas, the transmitter gains a powerful tool: it can steer the signal toward the intended receiver and away from the eavesdropper, even when the eavesdropper's scalar channel quality is comparable.

This section develops the secrecy capacity of the MIMO wiretap channel and introduces the technique of artificial noise injection — one of the most practical ideas in physical-layer security.

Definition:
The MIMO Wiretap Channel

The MIMO wiretap channel consists of:

Transmitter with $n_t$ antennas
Legitimate receiver (Bob) with $n_r$ antennas: $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}_B$
Eavesdropper (Eve) with $n_e$ antennas: $\mathbf{z} = \mathbf{H}_{E}\mathbf{x} + \mathbf{w}_E$

where $\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$ is Bob's channel, $\mathbf{H}_{E} \in \mathbb{C}^{n_e \times n_t}$ is Eve's channel, $\mathbf{w}_B \sim \mathcal{CN}(\mathbf{0}, \mathbf{I}_{n_r})$ and $\mathbf{w}_E \sim \mathcal{CN}(\mathbf{0}, \mathbf{I}_{n_e})$ are noise vectors, and $\mathbf{x}$ satisfies a power constraint $\text{tr}(\boldsymbol{\Sigma}_{X}) \leq P$ where $\boldsymbol{\Sigma}_{X} = \mathbb{E}[\mathbf{x}\mathbf{x}^H]$ .

Theorem: Secrecy Capacity of the MIMO Wiretap Channel

The secrecy capacity of the MIMO Gaussian wiretap channel is $C_s = \max_{\boldsymbol{\Sigma}_{X} \succeq 0,\, \text{tr}(\boldsymbol{\Sigma}_{X}) \leq P} \bigl[\log\det(\mathbf{I} + \mathbf{H}\boldsymbol{\Sigma}_{X}\mathbf{H}^{H}) - \log\det(\mathbf{I} + \mathbf{H}_{E}\boldsymbol{\Sigma}_{X}\mathbf{H}_{E}^{H})\bigr].$

This is a difference of two concave functions in $\boldsymbol{\Sigma}_{X}$ , making the optimization non-convex in general. However, the problem can be solved via a convex reformulation using the MIMO broadcast channel duality.

The secrecy capacity has the same form as the scalar case — mutual information to Bob minus mutual information to Eve — but now the optimization is over the input covariance matrix $\boldsymbol{\Sigma}_{X}$ rather than a scalar input distribution. The transmitter can choose the spatial direction and power allocation to maximize the gap between what Bob and Eve receive.

When the transmitter has many more antennas than Eve ( $n_t \gg n_e$ ), the secrecy capacity approaches the non-secrecy capacity — the eavesdropper can be effectively shut out by beamforming in the null space of $\mathbf{H}_{E}$ .

Proof

Achievability

The achievability follows from the general wiretap coding theorem applied to the MIMO channel. The Gaussian input distribution $\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, \boldsymbol{\Sigma}_{X})$ achieves $R_s = \log\det(\mathbf{I} + \mathbf{H}\boldsymbol{\Sigma}_{X}\mathbf{H}^{H}) - \log\det(\mathbf{I} + \mathbf{H}_{E}\boldsymbol{\Sigma}_{X}\mathbf{H}_{E}^{H})$ for any $\boldsymbol{\Sigma}_{X} \succeq 0$ with $\text{tr}(\boldsymbol{\Sigma}_{X}) \leq P$ .

Converse

The converse uses the entropy-power inequality and the maximum-entropy property of the Gaussian distribution to show that Gaussian input is optimal. The proof parallels the MIMO capacity converse (Chapter 10) with the additional secrecy constraint handled by the equivocation chain.

Optimization

The optimization over $\boldsymbol{\Sigma}_{X}$ is a difference-of-concave-functions problem. It can be solved using:

Generalized eigenvalue decomposition of $(\mathbf{H}^{H}\mathbf{H}, \mathbf{H}_{E}^{H}\mathbf{H}_{E})$
Convex-concave procedure (iterative linearization)
MIMO BC duality (Khisti and Wornell, 2010) $\blacksquare$

,

Definition:
Artificial Noise Injection

Artificial noise (AN) injection is a practical technique for enhancing secrecy in MIMO systems. The transmitter splits its signal into two components:

$\mathbf{x} = \mathbf{v}_s s + \mathbf{V}_{\text{AN}} \mathbf{n}_{\text{AN}}$

where:

$\mathbf{v}_s$ is the beamforming vector for the secret message $s$
$\mathbf{V}_{\text{AN}}$ spans the null space of Bob's channel: $\mathbf{H}\mathbf{V}_{\text{AN}} = \mathbf{0}$
$\mathbf{n}_{\text{AN}} \sim \mathcal{CN}(\mathbf{0}, \sigma_{\text{AN}}^2 \mathbf{I})$ is the artificial noise

Bob sees only $\mathbf{H}\mathbf{v}_s s + \mathbf{w}_B$ (the AN is nulled out by design). Eve sees $\mathbf{H}_{E}\mathbf{v}_s s + \mathbf{H}_{E}\mathbf{V}_{\text{AN}}\mathbf{n}_{\text{AN}} + \mathbf{w}_E$ (the AN degrades her channel).

The power is split: fraction $\alpha$ for the message, fraction $(1-\alpha)$ for artificial noise, with $\alpha$ optimized for maximum secrecy rate.

Artificial noise requires $n_t > n_r$ (more transmit antennas than Bob's receive antennas) so that the null space of $\mathbf{H}$ is non-trivial. In massive MIMO ( $n_t \gg n_r$ ), the null space is large, and most of the transmit power can be allocated to AN while still serving Bob effectively.

Artificial noise

Randomly generated noise transmitted by Alice in the null space of the legitimate channel, designed to degrade the eavesdropper's reception without affecting the legitimate receiver. A key technique in MIMO physical-layer security.

Example: MISO Wiretap Channel with Artificial Noise

Alice has $n_t = 4$ antennas, Bob has $n_r = 1$ antenna, Eve has $n_e = 1$ antenna. The channels are $\mathbf{h}_B^H \in \mathbb{C}^{1 \times 4}$ (Bob) and $\mathbf{h}_E^H \in \mathbb{C}^{1 \times 4}$ (Eve), with power $P = 10$ .

(a) Design the artificial noise scheme.

(b) Compute the secrecy rate as a function of the power split $\alpha$ .

Solution

Part (a): AN design

Beamforming vector: $\mathbf{v}_s = \mathbf{h}_B / \|\mathbf{h}_B\|$ (matched filter to Bob).

AN subspace: Let $\mathbf{V}_{\text{AN}} \in \mathbb{C}^{4 \times 3}$ be an orthonormal basis for the null space of $\mathbf{h}_B^H$ , i.e., $\mathbf{h}_B^H \mathbf{V}_{\text{AN}} = \mathbf{0}$ .

Transmitted signal: $\mathbf{x} = \sqrt{\alpha P} \mathbf{v}_s s + \sqrt{\frac{(1-\alpha)P}{3}} \mathbf{V}_{\text{AN}} \mathbf{n}_{\text{AN}}$ where $s$ has unit power and $\mathbf{n}_{\text{AN}} \sim \mathcal{CN}(\mathbf{0}, \mathbf{I}_3)$ .

Part (b): Secrecy rate

Bob's SNR: $\text{SNR}_B = \alpha P \|\mathbf{h}_B\|^2$ (AN is nulled out).

Eve's SINR: $\text{SINR}_E = \frac{\alpha P |\mathbf{h}_E^H \mathbf{v}_s|^2}{1 + \frac{(1-\alpha)P}{3}\|\mathbf{h}_E^H \mathbf{V}_{\text{AN}}\|^2}.$

Secrecy rate: $R_s(\alpha) = \left[\log(1 + \text{SNR}_B) - \log(1 + \text{SINR}_E)\right]^+.$

The optimal $\alpha^*$ maximizes $R_s(\alpha)$ . Typically, $\alpha^* \in [0.3, 0.7]$ — a significant fraction of power goes to artificial noise.

Artificial Noise: Power Split vs. Secrecy Rate

Explore how the secrecy rate depends on the fraction of power allocated to the message ( $\alpha$ ) vs. artificial noise ( $1-\alpha$ ). The optimal power split balances Bob's SNR against Eve's interference level.

Parameters

n_t

(transmit antennas)4

P

(dB)15

Total transmit power in dB

Eve angle (degrees)45

Angular separation between Bob and Eve

MIMO Secrecy Capacity vs. Number of Antennas

See how the secrecy capacity grows with the number of transmit antennas. With many antennas, the transmitter can effectively null the eavesdropper, and the secrecy capacity approaches the non-secrecy capacity.

Parameters

\text{SNR}

(dB)15

n_e

(Eve antennas)1

Common Mistake: Assuming Perfect Knowledge of Eve's Channel

Mistake:

Designing a physical-layer security scheme assuming the transmitter knows $\mathbf{H}_{E}$ (Eve's channel matrix) perfectly.

Correction:

In practice, the transmitter rarely knows Eve's channel. The artificial noise technique is attractive precisely because it works without ECSI (Eavesdropper's CSI): the AN is transmitted isotropically in Bob's null space, degrading Eve regardless of her location. Schemes that require ECSI are theoretically interesting but practically unrealistic except in controlled environments.

Comparison of Secrecy Techniques

Technique	Requires ECSI?	Requires key?	Security level	Practical maturity
Wiretap coding	No (statistical model)	No	Information-theoretic	Low (research)
Artificial noise	No	No	Information-theoretic	Medium (prototypes)
Secret key from channel	No	Generated	Information-theoretic	Medium (Wi-Fi demos)
AES encryption	No	Yes (pre-shared)	Computational	High (deployed)
Quantum key distribution	No	Generated	Information-theoretic	Low-Medium (point-to-point)

Theorem: Secrecy Degrees of Freedom

For the MIMO wiretap channel with $n_t$ transmit, $n_r$ receive, and $n_e$ eavesdropper antennas, the secrecy degrees of freedom (secure spatial multiplexing gain) is $d_s = [d_{\text{main}} - d_{\text{eve}}]^+ = [\min(n_t, n_r) - \min(n_t, n_e)]^+$ when the channel matrices are generic (full rank with probability 1).

In particular, when $n_t > n_r + n_e$ , $d_s = n_r$ — the eavesdropper has no effect on the degrees of freedom.

The secrecy DoF tells us how many independent secret streams can be transmitted simultaneously. When the transmitter has enough antennas to null both Bob's interference and Eve's reception, the secrecy DoF equals the non-secrecy DoF. This is the regime where massive MIMO makes physical-layer security essentially "free" — the extra antennas can simultaneously serve Bob and confuse Eve.

Proof

Upper bound

The secrecy capacity is bounded by $C_s \leq \log\det(\mathbf{I} + \text{SNR}\mathbf{H}\mathbf{H}^{H}) - \log\det(\mathbf{I} + \text{SNR}\mathbf{H}_{E}\mathbf{H}_{E}^{H})$ at the best covariance. At high SNR, the first term scales as $d_{\text{main}}\log\text{SNR}$ and the second as $d_{\text{eve}}\log\text{SNR}$ .

Achievability

Use GSVD-based precoding: the generalized SVD of $(\mathbf{H}, \mathbf{H}_{E})$ decomposes the channel into parallel sub-channels where some are visible only to Bob, some only to Eve, and some to both. The secret streams are sent on the sub-channels where Bob has an advantage.

Massive MIMO regime

When $n_t > n_r + n_e$ , the null space of $\mathbf{H}_{E}$ has dimension at least $n_t - n_e > n_r$ . We can find $n_r$ directions that reach Bob but are invisible to Eve, achieving $d_s = n_r$ . $\blacksquare$

Massive MIMO Makes Secrecy Free

The secrecy DoF result has a striking practical implication: in massive MIMO systems with $n_t \gg n_r + n_e$ , the secrecy penalty vanishes. The transmitter has so many degrees of freedom that it can simultaneously serve the legitimate receiver at full rate and completely null the eavesdropper.

This is not just a theoretical observation — it underlies the security advantage of 5G massive MIMO. With 64 or 128 transmit antennas serving single-antenna users, the excess spatial dimensions can be used to inject artificial noise at no cost to the legitimate user's rate. Physical-layer security becomes a "bonus feature" of the antenna array, not an additional system component.

Historical Note: Artificial Noise: Goel and Negi (2008)

2008

The artificial noise technique was introduced by Goel and Negi in 2008, building on the MIMO wiretap channel work of Khisti and Wornell. The idea is elegant in its simplicity: instead of trying to optimize the signal for secrecy (which requires knowledge of Eve's channel), simply transmit noise in Bob's null space. This degrades Eve's channel without affecting Bob, achieving a positive secrecy rate even without knowing $\mathbf{H}_{E}$ .

The technique has become the most widely cited result in physical-layer security, with thousands of follow-up works extending it to relay networks, cooperative jamming, and intelligent reflecting surfaces.

Quick Check

In a massive MIMO system with $n_t = 64$ , $n_r = 1$ , $n_e = 4$ , what are the secrecy degrees of freedom?

$d_s = 1$ — since $\min(64, 1) - \min(64, 4) = 1 - 4 < 0$ , we get $d_s = [1 - 4]^+ = 0$

$d_s = 1$ — the transmitter has enough antennas to null Eve while serving Bob at full rate

$d_s = 60$ — the null space of Eve has dimension 60

Correction:

d_s = 1

— the transmitter has enough antennas to null Eve while serving Bob at full rate

With $n_t = 64 > n_r + n_e = 5$ , the transmitter can beamform to Bob (achieving 1 DoF) while transmitting artificial noise in Eve's direction. The secrecy DoF equals the non-secrecy DoF: $d_s = n_r = 1$ .

Key Takeaway

The MIMO wiretap channel secrecy capacity is the maximum difference between the log-determinants of the legitimate and eavesdropper channels, optimized over the input covariance. Artificial noise injection provides a practical, ECSI-free approach: transmit noise in Bob's null space to degrade Eve without affecting Bob. In the massive MIMO regime ( $n_t \gg n_r + n_e$ ), secrecy comes at zero rate cost — the secrecy DoF equals the non-secrecy DoF.

Artificial Noise Injection in MIMO Wiretap

Visualizes how a multi-antenna transmitter beamforms the signal to the legitimate receiver while injecting artificial noise in the null space to degrade the eavesdropper's channel.

Broadcast Channel with Confidential Messages

Secrecy with Multiple Antennas

Definition: The MIMO Wiretap Channel

Theorem: Secrecy Capacity of the MIMO Wiretap Channel

Achievability

Converse

Optimization

Definition: Artificial Noise Injection

Artificial noise

Example: MISO Wiretap Channel with Artificial Noise

Part (a): AN design

Part (b): Secrecy rate

Artificial Noise: Power Split vs. Secrecy Rate

Parameters

MIMO Secrecy Capacity vs. Number of Antennas

Parameters

Common Mistake: Assuming Perfect Knowledge of Eve's Channel

Comparison of Secrecy Techniques

Theorem: Secrecy Degrees of Freedom

Upper bound

Achievability

Massive MIMO regime

Massive MIMO Makes Secrecy Free

Historical Note: Artificial Noise: Goel and Negi (2008)

Quick Check

Key Takeaway

Artificial Noise Injection in MIMO Wiretap

Definition:
The MIMO Wiretap Channel

Definition:
Artificial Noise Injection