Exercises

$C_s = [R_B - R_E]^+ = \max(R_B - R_E, 0)$ where $R_B, R_E$ are Bob's and Eve's achievable rates under a common signaling scheme.

If Bob's rate exceeds Eve's, positive secrecy is achievable by transmitting $R_B$ bits with $R_E$ bits of confusion — Eve can't decode anything. If Bob's rate ≤ Eve's, no positive secrecy. $\blacksquare$

$R_B \approx \log_2 \text{SNR}_{B}$, $R_E \approx \log_2 \text{SNR}_{E}$. Difference: $\log_2(\text{SNR}_{B}/\text{SNR}_{E})$.

With RIS: $\text{SNR}_{B} = N^2 \text{SNR}_{\text{base}}$, $\text{SNR}_{E} \sim \text{SNR}_{\text{base}}$. Ratio = $N^2$, $\log_2(N^2) = 2 \log_2 N$ bits secrecy. $\blacksquare$

$2 \log_2 N = 2 \log_2 256 = 16$ bits/s/Hz.

This is the upper-bound scaling. Real gains depend on Eve's specific position; 10-14 bits/s/Hz is typical with optimal tuning. Still a dramatic boost over no-RIS. $\blacksquare$

Comm-optimal RIS: all $N$ elements focus coherently on Bob. Secrecy-optimal RIS: some elements focus on Bob; others null at Eve. Dividing elements between these two tasks reduces Bob's coherent combining slightly.

Comm-optimal Bob rate: $\log_2(1 + N^2 \text{SNR})$. Secrecy-optimal Bob rate: $\log_2(1 + (N/2)^2 \text{SNR}) = \log_2(1 + N^2/4 \text{SNR})$ (roughly half elements dedicated to Bob under optimal split). Difference: $-2$ bits/s/Hz.

Paying 2 bits/s/Hz of Bob's comm rate to buy $\sim 14$ bits/s/Hz of secrecy rate. Favorable tradeoff. $\blacksquare$

Maximize $|\mathbf{h}_B^H \mathbf{v}|^2 / |\mathbf{h}_E^H \mathbf{v}|^2$ under $\|\mathbf{v}\| = 1$.

$\mathbf{v}^{H} \mathbf{H}_B \mathbf{v} / \mathbf{v}^{H} \mathbf{H}_E \mathbf{v}$ with $\mathbf{H}_B = \mathbf{h}_B \mathbf{h}_B^H$, $\mathbf{H}_E = \mathbf{h}_E \mathbf{h}_E^H$. Maximum via generalized eigenvalue: $\mathbf{v}^\star$ is the dominant generalized eigenvector of $(\mathbf{H}_B, \mathbf{H}_E)$.

Add a small $\rho \mathbf{I}$ to $\mathbf{H}_E$ to avoid singularity; otherwise closed-form. $\blacksquare$

AN transmitted along directions orthogonal to Bob's effective channel ($\mathbf{h}_B^H \mathbf{A} = \mathbf{0}$).

Bob sees no AN (orthogonal). Eve (at random direction) sees AN with non-zero projection on her channel → jamming.

AN is computed from $\mathbf{h}_B$ only. Eve's channel is irrelevant to the jamming design. The statistical assumption is just that Eve's channel is "generic" — not aligned with Bob's. $\blacksquare$

Data: $(1-0.4)P_t = 0.6P_t$. AN: $0.4 P_t$.

$R_B \propto \log_2(1 + 0.6 \text{SNR})$ vs. pure-comm $\log_2(1 + \text{SNR})$. At high SNR: difference $\approx \log_2(0.6) = -0.74$ bits/s/Hz.

Eve's SNR reduced by $\sim 10\times$ via AN in her null space. Eve's rate drops by $\log_2(10) \approx 3.3$ bits/s/Hz. Net secrecy change: +$3.3 - 0.74 = +2.6$ bits/s/Hz (over no-AN). Favorable tradeoff. $\blacksquare$

Guarantees: deterministic; secrecy achieved for ANY Eve in $\mathcal{U}_E$. Conservative: lower nominal secrecy. Use case: high-stakes applications (military, gov).

Guarantees: expected value or high-probability; possibly poor on specific Eve realizations. Less conservative: higher nominal secrecy. Use case: commercial deployments with accepted small leakage probability.

Depends on adversary model and consequences of leakage. For 5G/6G commercial: stochastic is common. For classified: worst-case is standard. $\blacksquare$

$R_s \sim \log_2(\text{SNR}_{B} / \text{SNR}_{E}) \sim 2 \log_2 N$. Linear (in log) growth in $N$.

Comm rate: $\log_2(1 + N^2 \text{SNR}) \approx 2 \log_2 N$ also at high SNR. Interestingly, secrecy and comm have the same scaling in $N$ — but the additive constants differ.

Realistic Eve CSI uncertainty reduces achievable secrecy; robust designs have slower scaling (logarithmic in inverse uncertainty radius). $\blacksquare$

Fix $(\mathbf{v}, \boldsymbol{\Phi})$: compute $\text{SNR}_{E}^{\text{wc}} = \max_{\mathbf{h}_E \in \mathcal{U}_E} |\mathbf{h}_E^H \mathbf{v}|^2$.

Quadratic in $\mathbf{h}_E$; constraint $\|\mathbf{h}_E - \bar{\mathbf{h}}_E\| \leq \rho$ is quadratic. By S-procedure, this has a dual SDP form.

Outer max (over $\mathbf{v}, \boldsymbol{\Phi}$) combines with inner max. Using S-procedure and Lagrangian duality, express the full problem as a convex SDP in lifted variables.

The full SDP has $O((N_t + N)^2)$ variables; solvable in reasonable time for $N_t \leq 32, N \leq 128$. For larger $N$, use manifold methods on the passive side with SDR on the active side. $\blacksquare$

Provides guarantees under specific attack model (passive, bounded Eve).

Authentication, replay, MITM attacks operate above PHY. PL security doesn't address them.

Use PL security + cryptography + physical security. PL security is one layer in defense-in-depth. Don't rely on it alone for mission-critical systems. $\blacksquare$

AN is direction-agnostic: jams the null space uniformly (including where Eve isn't).

RIS can focus jamming in Eve's specific direction while leaving other directions clean.

Multiple users are served simultaneously; Eve is in one specific direction. AN would jam all non-Bob directions (including other users); RIS jamming is directional. Other users are served without AN interference.

Up to $\sim 5$-$10$ dB additional secrecy rate in multi-user scenarios via RIS-directional jamming. $\blacksquare$

Bob at 5 m, Eve could be at 1-10 m. Worst: Eve at 1 m, very close to router. Bob's SNR lower than Eve's. $R_s^{\text{wc}} = 0$ typically.

Robust optimization: SDP maximizes worst-case secrecy. Typical result: $R_s^{\text{wc}} \sim 3$-$5$ bits/s/Hz positive. Guaranteed for any Eve in the uncertainty set.

Sufficient for small-payload secure messaging (control signals, key exchange). Bulk data still needs encryption. $\blacksquare$

Perfect CSI: optimal nulling at Eve's position. Secrecy very high.

CSI is delayed; Eve may move away from the null. Secrecy reduces based on how far she moves relative to the nulling beamwidth.

Use worst-case over Eve's movement region (e.g., within mobility radius over coherence time). Secrecy guarantee degrades gracefully.

Update RIS phases at a rate faster than Eve's mobility. $\sim 100$-$1000$ updates/sec is sufficient for most mobility. Or use AN-based approach which doesn't track Eve. $\blacksquare$

Eve likely outside the glass wall (5-30 m away). Bob (legitimate meeting attendee) inside.

Place RIS inside the conference room, concealed from outside. Focus on attendee direction; null toward glass walls.

Robust SDP over Eve's uncertainty set (outside walls). AN fraction $\sim 40\%$. Per-attendee key establishment via RIS-secured channel.

RIS provides PHY-layer secrecy for key establishment. Subsequent bulk data encrypted with the established key.

Worst-case secrecy rate: $\sim 5$ bits/s/Hz for any Eve outside walls. Sufficient for key exchange and control signaling. $\blacksquare$