Ferkans — Interactive Telecom Tutor

The Joint Security Optimization

Section 15.1 defined the secrecy capacity $C_s$ as a function of $(\mathbf{v}, \boldsymbol{\Phi})$ . Section 15.2 formalizes the optimization problem and presents practical algorithms. Structurally similar to Chapter 5's joint comm optimization, but the objective is the difference of two log-SINRs, not a sum — a new optimization flavor.

Definition:
Joint RIS Secrecy Rate Maximization

The secrecy rate maximization is

$\boxed{\;\max_{\mathbf{v}, \boldsymbol{\Phi}}\; R_s(\mathbf{v}, \boldsymbol{\Phi}) = [\log_2(1 + \text{SNR}_{B}) - \log_2(1 + \text{SNR}_{E})]^+\;}$

subject to power constraints and $|\phi_n| = 1$ .

Without the $[\cdot]^+$ : the objective is a difference of log- SINRs. At high SNR, this reduces to $\log_2(\text{SNR}_{B}/\text{SNR}_{E})$ . The design goal: maximize the channel contrast between Bob and Eve — not just Bob's SINR.

,

Theorem: High-SNR Secrecy Approximation

Under high-SNR regime, the secrecy-rate optimization simplifies to

$\max_{\mathbf{v}, \boldsymbol{\Phi}} \frac{|\mathbf{h}_B^H(\boldsymbol{\Phi}) \mathbf{v}|^2}{|\mathbf{h}_E^H(\boldsymbol{\Phi}) \mathbf{v}|^2}$

(a ratio, not a difference). This is a generalized eigenvalue problem in $\mathbf{v}$ for fixed $\boldsymbol{\Phi}$ ; becomes a harder non-convex problem in $\boldsymbol{\Phi}$ .

At high SNR, $\log(1 + \text{SNR}) \approx \log(\text{SNR})$ . Secrecy rate reduces to $\log(\text{SNR}_{B}/\text{SNR}_{E}) = \log|\mathbf{h}_B^H \mathbf{v}|^2 - \log|\mathbf{h}_E^H \mathbf{v}|^2$ . The optimization aims to maximize the ratio of signal powers at Bob vs. Eve — a geometric interpretation.

Proof

High-SNR expansion

$\log(1 + x) = \log x + O(1/x)$ at large $x$ . $R_s \approx \log \text{SNR}_{B} - \log \text{SNR}_{E} = \log(\text{SNR}_{B}/\text{SNR}_{E})$ .

Ratio

$\text{SNR}_{B}/\text{SNR}_{E} = \sigma^2_{E} |\mathbf{h}_B^H \mathbf{v}|^2 / (\sigma^2 |\mathbf{h}_E^H \mathbf{v}|^2)$ . Under equal noise powers: maximize $|\mathbf{h}_B^H \mathbf{v}|^2 / |\mathbf{h}_E^H \mathbf{v}|^2$ .

Generalized eigenvalue

For fixed $\boldsymbol{\Phi}$ , the optimal $\mathbf{v}$ is the dominant generalized eigenvector of $(\mathbf{h}_B \mathbf{h}_B^H, \mathbf{h}_E \mathbf{h}_E^H)$ . Closed form. Chained with AO on $\boldsymbol{\Phi}$ . $\blacksquare$

AO for RIS Secrecy Rate

Complexity:

O(T \cdot (N_t^{3} + C_{\text{passive}}))

; T ~ 15 iterations

Input:

\mathbf{h}_{B,d}, \mathbf{h}_{B,2}, \mathbf{h}_{E,d}, \mathbf{h}_{E,2}, \mathbf{H}_1

, power

P_t

.

Output:

(\mathbf{v}^\star, \boldsymbol{\Phi}^\star)

maximizing

R_s

.

1. Initialize

\boldsymbol{\Phi}^{(0)}

(e.g., matched to Bob).

2. Repeat

t = 0, 1, \ldots

:

3.

\quad

Compute

\mathbf{h}_B, \mathbf{h}_E

given

\boldsymbol{\Phi}^{(t)}

.

4.

\quad

Active update: solve generalized eigenvalue problem for

\mathbf{v}^{(t+1)}

(dominant vector of

(\mathbf{h}_B\mathbf{h}_B^H, \mathbf{h}_E\mathbf{h}_E^H + \rho \mathbf{I})

).

5.

\quad

Passive update: solve

\max_{|\phi_n|=1} \log(1 + \text{SNR}_{B}) - \log(1 + \text{SNR}_{E})

via SDR or manifold method. Can include artificial-noise term.

6.

\quad

Check convergence.

7. return

(\mathbf{v}, \boldsymbol{\Phi})

.

The active update has a closed-form solution via generalized eigenvalue decomposition (unlike WMMSE for sum rate). The passive update is the harder subproblem; SDR gives tight relaxation for single-Eve scenarios.

Secrecy Rate AO Convergence

Trace the secrecy rate vs. iteration under AO. Compare the RIS-aided secrecy with a no-RIS baseline. The RIS contribution grows over iterations as $\boldsymbol{\Phi}$ learns to null Eve.

Parameters

RIS elements

N

64

BS antennas

N_t

4

Transmit SNR (dB)10

Bob-Eve channel correlation0.2

Example: Expected Secrecy Gain from RIS

$N_t = 4, N = 64, P_t/\sigma^2 = 10\text{ dB}$ . Bob and Eve have independent Rayleigh channels. No-RIS baseline has zero secrecy (Eve randomly better half the time). Estimate the RIS-aided secrecy rate.

Solution

No-RIS

Half the realizations have Eve > Bob: $R_s = 0$ . Half have Bob > Eve: $R_s \sim \log_2(\text{SNR}_{B}/\text{SNR}_{E})$ , small. Expected $R_s \sim 0.5$ - $1$ bit/s/Hz.

With RIS

Optimized $\boldsymbol{\Phi}$ always focuses on Bob (deterministically choose to favor Bob). $\text{SNR}_{B} \sim N^2 \cdot \text{SNR}$ ; $\text{SNR}_{E} \sim 1 \cdot \text{SNR}$ . Ratio $\sim N^2 = 4096$ . $R_s \sim \log_2(4096) = 12$ bits/s/Hz.

Net gain

RIS provides $\sim 10$ - $11$ additional bits/s/Hz of secrecy rate. This is the "secure throughput boost" — directly improving the information that can be transmitted without leakage. $\blacksquare$

Secrecy vs. Communication Tradeoff

Maximizing secrecy rate is not the same as maximizing Bob's rate. For comm: focus only on Bob. For secrecy: focus on Bob and null at Eve.

Consequence: under RIS-aided secrecy optimization, Bob's received rate may be slightly lower than under pure comm optimization, because some DoF are spent nulling Eve.

In practice, the secrecy-optimal $\boldsymbol{\Phi}$ gives Bob $\sim 90$ - $95\%$ of the pure-comm rate, with the remainder spent ensuring Eve's rate stays small. Acceptable tradeoff for security-critical services.

Common Mistake: Check for Positive Secrecy

Mistake:

"Apply the secrecy maximization AO blindly; trust whatever it outputs."

Correction:

If $C_s = 0$ is achievable with positive $R_B - R_E$ at the input, the optimization may find negative-secrecy solutions (Eve stronger than Bob), returning $R_s = 0$ . Always verify:

Is the input channel geometry favorable? (Bob's geometric RIS path exists with adequate $\alpha \beta$ .)
Does the AO converge to positive $R_s$ ? If not, consider alternative: artificial noise (Section 15.3) or different RIS placement.
Under imperfect Eve CSI, apply robust design (Section 15.4) or abandon secrecy and use cryptography instead.

Secrecy Rate Maximization with RIS