Secrecy Energy Efficiency in Wireless Networks

• Secrecy Energy Efficiency (SEE) is a metric that measures secure data transmission per unit of energy, balancing secrecy throughput with power consumption.
• SEE is expressed as ratios like bits/Joule or bps/Hz/J, reflecting variations in models from circuit power to propulsion energy in UAV and RIS settings.
• Optimizing SEE involves fractional programming, convex approximations, and robust techniques to manage the trade-off between secrecy performance and energy use.

Searching arXiv for recent and foundational work on secrecy energy efficiency to ground the article in published papers. arxiv_search.query({"^^^^^^^^3search_query3 energy efficiency\" OR ^^^^^^^^3all:\3 energy efficiency\"","start":^^^^3search_query3^^^^,"max_results":^^^^3all:\3search_query3 I found recent and earlier arXiv papers directly on secrecy/secure energy efficiency, including RIS-, UAV-, SWIPT-, and cognitive-radio settings. I’ll use the provided paper set and cite the most relevant arXiv IDs in a neutral encyclopedia-style synthesis. Secrecy energy efficiency (SEE) is a physical-layer security metric that quantifies secure information delivery per unit energy expenditure. Across the literature, it is typically defined as a ratio between a secrecy-throughput quantity—such as secrecy rate, secrecy sum-rate, average secrecy capacity, or outage-weighted secure throughput—and a power-consumption model that may include transmit power, circuit power, RIS hardware power, or UAV propulsion energy. The same ratio appears in normalized forms such as bits/Joule, bps/Hz/J, or nats/s/Hz/Joule, depending on whether bandwidth normalization is retained explicitly. By ^^^^3 OR all:\3search_query3all:\3^^^^7, SEE had already emerged in cooperative relaying, full-duplex jamming, UAV-enabled networking, and cognitive radio, and subsequent work extended it to SWIPT, RIS/IRS, fluid antennas, VLC-RSMA, THz relaying, aerial MEC, and explainable deep-learning-based resource allocation (&&&^^^^3search_query3^^^^&&&, &&&^^^^3all:\3^^^^&&&, &&&^^^^3 OR all:\3^^^^&&&, Ghadi et al., 2024, Fotock et al., 9 Jul 2025).

^^^^3all:\3^^^^. Conceptual definition and metric variants

At its core, SEE measures the tradeoff between confidentiality and energy expenditure. In the fluid-antenna wiretap setting, SEE is defined as the ratio of average secrecy capacity (ASC) to total power consumption, yielding secure bits/s/Hz per Joule (Ghadi et al., 2024). In IRS-assisted MIMOME wiretap channels, the corresponding quantity is written as

PRESERVED_PLACEHOLDER_^^^^3search_query3^^^^

where PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^ is the secrecy rate and PRESERVED_PLACEHOLDER_^^^^3 OR all:\3^^^^ includes amplifier inefficiency and IRS circuit power (Mukherjee et al., 2022). In RIS-aided uplink networks, the numerator is a secrecy sum-rate (SSR), whereas in cooperative TAS/MRC relaying the numerator is a target secrecy rate weighted by the probability of non-outage, $\mathcal R(1-p_{\text{out}})$ (Fotock et al., 9 Jul 2025, &&&^^^^3search_query3^^^^&&&).

This variability is not a contradiction; it reflects different operational regimes. Some formulations optimize an ergodic secrecy quantity, some a deterministic slot-wise secrecy rate, and some an outage- or fairness-aware surrogate. In UAV relaying, SEE may be defined as secrecy throughput divided by propulsion energy consumption, explicitly prioritizing endurance-limited mobility (Xiao et al., 2018). In THz untrusted-UAV relaying, the design target becomes the minimum SEE across users, so that fairness is imposed on the secrecy-per-Joule objective rather than on secrecy rate alone (&&&^^^^3all:\3search_query3^^^^&&&). This suggests that SEE is better understood as a family of secrecy-to-energy ratios than as a single universal formula.

A recurring point in the literature is that SEE is not equivalent to secrecy-rate maximization. Several papers explicitly show that maximizing secrecy rate, SSR, or total secure bits can lead to materially different solutions from maximizing secure bits per Joule (&&&^^^^3 OR all:\3^^^^&&&, &&&^^^^3all:\3 OR all:\3^^^^&&&, &&&^^^^3all:\33^^^^&&&). The distinction is especially sharp when circuit power, RIS hardware power, or propulsion energy dominate the denominator.

^^^^3 OR all:\3^^^^. Mathematical structure and power-consumption models

The secrecy term usually takes a wiretap form. Representative expressions include the instantaneous secrecy capacity

$$\mathcal C_s(\gamma_B,\gamma_E)=\max\left\{\log_2(1+\gamma_B)-\log_2(1+\gamma_E),0\right\}$$

in fluid-antenna systems (Ghadi et al., 2024), the MIMO wiretap secrecy rate

$$C(\mathbf X,\boldsymbol\theta)=\Big[\ln|\mathbf I+\mathbf H_B\mathbf X\mathbf H_B^H|-\ln|\mathbf I+\mathbf H_E\mathbf X\mathbf H_E^H|\Big]_+$$

in IRS-assisted MIMOME channels (Mukherjee et al., 2022), and secrecy sum-rates built from per-user legitimate and eavesdropper SINRs in RIS-aided cellular uplinks (Fotock et al., 9 Jul 2025). Other settings employ worst-case secrecy rates across multiple eavesdroppers, proportional secrecy-rate constraints across users, or average secrecy rate over a horizon (&&&^^^^3all:\3^^^^7&&&, &&&^^^^3all:\33^^^^&&&, &&&^^^^3all:\3search_query3^^^^&&&).

The denominator is equally model-dependent. In compact terrestrial wiretap models, total consumed power often has the form transmit power scaled by amplifier efficiency plus constant circuit power; for example,

$P_{\mathrm{tot}}=\frac{P}{\alpha}+P_c$

in the fluid-antenna wiretap channel (Ghadi et al., 2024). IRS/RIS formulations add static per-element power and fixed hardware offsets. In active-RIS uplinks, the RIS RF amplification term enters explicitly through a trace expression, and the total power becomes the sum of active RIS power, user transmit powers, and static hardware consumption (Fotock et al., 9 Jul 2025). In OFDM cognitive radio, only CBS power and fixed circuit power appear in the SEE denominator, while secrecy constraints are imposed simultaneously for CU and PU (&&&^^^^3 OR all:\3 OR all:\3^^^^&&&).

In UAV systems the power model changes qualitatively. Several works neglect communication energy relative to propulsion energy and define SEE with a denominator dominated by fixed-wing or rotary-wing flight power (Xiao et al., 2018, &&&^^^^3all:\33^^^^&&&, &&&^^^^3all:\3search_query3^^^^&&&). This modeling choice alters the optimization geometry: the primary energy penalty is then induced by velocity, acceleration, and trajectory, not merely by radiated power. A plausible implication is that “green security” in aerial networks is fundamentally a mobility-control problem rather than only a beamforming problem.

3. Communication architectures in which SEE is studied

SEE has been investigated in a wide range of secure wireless architectures. Cooperative relaying and jamming are among the earliest. In TAS/MRC cooperative systems, SEE is optimized under secrecy outage probability constraints while comparing CSI-aided decode-and-forward and artificial-noise relaying (&&&^^^^3search_query3^^^^&&&). In MIMOME wiretap channels with a full-duplex receiver transmitting artificial noise while receiving information, the question is whether the secrecy gain of full-duplex jamming survives once self-interference cancellation power and residual self-interference are accounted for (&&&^^^^3all:\3^^^^&&&).

Cognitive radio and SWIPT introduce additional coupling between secrecy, interference management, and energy transfer. In MISO underlay cognitive radio with an energy receiver acting as a potential passive eavesdropper, SEE is optimized subject to secrecy-rate, harvested-energy, interference-leakage, and transmit-power constraints (&&&^^^^3 OR all:\3^^^^&&&). MISOME-SWIPT formulations further combine beamforming, artificial noise, robust eavesdropper and EHN uncertainty handling, and proportional secrecy-rate fairness (&&&^^^^3all:\3^^^^7&&&, &&&3^^^^3search_query3^^^^&&&). SWIPT-in-DAS extends the concept to distributed antenna systems with power-splitting receivers at both legitimate users and eavesdroppers, and even defines an outage probability of SEE when CSI is unavailable (&&&3^^^^3all:\3^^^^&&&).

RIS/IRS-assisted systems form another major class. SEE has been studied in IRS-assisted MIMOME wiretap channels (Mukherjee et al., 2022), RIS-aided uplink cellular networks comparing active and nearly-passive RIS (Fotock et al., 9 Jul 2025), IRS-assisted VLC MISO networks with RSMA (Guo et al., 2024), and RIS-aided aerial MEC offloading with UAV trajectory and task partitioning (Abdalla et al., 16 May 2025). These works treat RIS/IRS as a lever for improving secrecy rate, but also emphasize its hardware power cost.

Recent work expands SEE into specialized architectures: planar fluid antenna systems with copula-modeled correlated ports (Ghadi et al., 2024), THz UAV relaying with an untrusted relay and destination-assisted cooperative jamming (&&&^^^^3all:\3search_query3^^^^&&&), finite- versus infinite-horizon reinforcement learning for energy-harvesting secure transmission with full-duplex destination jamming (&&&^^^^3all:\3 OR all:\3^^^^&&&), and ambient backscatter multi-user NOMA with explainable deep learning and SHAP (Alam et al., 25 Nov 2025). In ISAC-MIMO, SEE is embedded in a hierarchical Stackelberg–GNE–Bayesian framework that jointly allocates data, artificial noise, sensing power, and geometry-aware cooperative jamming (&&&4^^^^3search_query3^^^^&&&).

4. Optimization and analytical methodologies

Because SEE is almost always fractional and non-convex, the dominant algorithmic pattern is fractional programming combined with local convexification. Dinkelbach’s method appears repeatedly in cognitive radio, OFDM, UAV relaying, THz relaying, and MIMOME/IRS settings (&&&^^^^3 OR all:\3^^^^&&&, &&&^^^^3 OR all:\3 OR all:\3^^^^&&&, Xiao et al., 2018, &&&^^^^3all:\33^^^^&&&, &&&^^^^3all:\3search_query3^^^^&&&, Mukherjee et al., 2022). The typical transformation replaces

$\max \frac{f(x)}{g(x)}$

with iterative solution of

$\max \; f(x)-\lambda g(x),$

updating $\lambda$ until the residual vanishes. This converts the ratio structure into a sequence of subtractive problems.

The remaining non-convexities are then handled by difference-of-concave decompositions, successive convex approximation (SCA), sequential fractional programming, or semidefinite relaxation. Robust formulations under bounded CSI uncertainty rely heavily on the PRESERVED_PLACEHOLDER_^^^^3all:\3search_query3^^^^-procedure and LMI reformulations, especially in MISOME-SWIPT and imperfect-CSI cognitive radio (&&&^^^^3all:\3^^^^7&&&, &&&3^^^^3search_query3^^^^&&&, Zhang et al., 2019). IRS-assisted MIMOME optimization uses a penalty dual decomposition based alternating gradient projection (PDDAPG) method and shows linear scaling with IRS size in practical regimes (Mukherjee et al., 2022). Active/nearly-passive RIS optimization alternates over transmit powers, RIS coefficients, and receive filters, with LMMSE combiners and convergence to KKT points under both perfect and statistical CSI (Fotock et al., 9 Jul 2025).

Stochastic and learning-based methods appear when channel evolution, combinatorial action spaces, or horizon effects become central. FHJPA and IHJPA solve finite- and infinite-horizon SEE optimization in energy-harvesting secure transmission via backward induction and policy iteration, respectively (&&&^^^^3all:\3 OR all:\3^^^^&&&). DS-PPO is used for SEE maximization in IRS-assisted VLC-RSMA with mixed continuous-discrete variables (Guo et al., 2024), while DDPG is used in RIS-aided aerial MEC to jointly optimize trajectory, scheduling, offloading, and RIS phase shifts (Abdalla et al., 16 May 2025). In ambient backscatter NOMA, closed-form structure is retained for one- and two-BD cases, but particle swarm optimization and FNN-based predictors are introduced for larger numbers of backscatter devices, with SHAP used to interpret the learned policy (Alam et al., 25 Nov 2025).

Analytical SEE evaluation is also an active line. In fluid antennas, copula theory and Gauss-Laguerre quadrature yield compact SEE expressions under arbitrary correlated fading and correlated Rayleigh fading (Ghadi et al., 2024). In SWIPT-in-DAS, closed-form outage probability expressions are derived using Erlang statistics, and the outage of SEE itself becomes the object of analysis (&&&3^^^^3all:\3^^^^&&&).

5. Recurrent trade-offs and empirical regularities

Several empirical regularities recur across otherwise different models. First, SEE commonly increases with transmit power only up to a point, after which it saturates or decreases because power consumption grows faster than secrecy throughput. This pattern is reported for fluid antennas, IRS-assisted MIMOME, SWIPT-in-DAS, OFDM cognitive radio, RIS-aided cellular uplinks, and other settings (Ghadi et al., 2024, Mukherjee et al., 2022, &&&3^^^^3all:\3^^^^&&&, &&&^^^^3 OR all:\3 OR all:\3^^^^&&&, Fotock et al., 9 Jul 2025). A common misconception is therefore that more transmit power necessarily improves secure energy efficiency; the literature consistently rejects that view.

Second, additional hardware resources are not uniformly beneficial. Larger IRS/RIS sizes may improve secrecy rate but can hurt SEE when per-element power is significant, so an optimal surface size can exist (Mukherjee et al., 2022). Active RIS can outperform nearly-passive RIS in rate, yet achieve worse SEE once amplifier static power and noise amplification are included (Fotock et al., 9 Jul 2025). More antennas can likewise increase circuit power enough that SEE becomes non-monotonic in antenna count (&&&^^^^3search_query3^^^^&&&). Full-duplex jamming improves secrecy rate, but only yields marginal SEE gains under many conditions because SIC cost and residual self-interference offset the benefit (&&&^^^^3all:\3^^^^&&&).

Third, mobility introduces a specific secrecy–energy geometry. In UAV relaying and multi-UAV cooperative jamming, trajectories that improve legitimate links or jamming effectiveness may simultaneously incur large propulsion costs. Reported optimal behaviors include collecting data near the source, forwarding near the destination, avoiding the eavesdropper during transmission, and using smoother or figure-eight-like paths when the time horizon permits (Xiao et al., 2018, &&&^^^^3all:\33^^^^&&&). In THz untrusted-UAV relaying, the best minimum SEE emerges from jointly balancing user scheduling, BS jamming power, relay power, and UAV trajectory under molecular absorption and propulsion limits (&&&^^^^3all:\3search_query3^^^^&&&).

Fourth, secrecy constraints tied to fairness, harvesting, or eavesdropper models materially change the SEE optimum. Proportional secrecy-rate constraints, minimum harvested-energy constraints, and eavesdropper energy-harvesting caps all reduce feasible SEE in exchange for fairness or security assurances (&&&^^^^3all:\3^^^^7&&&, &&&3^^^^3all:\3^^^^&&&). This suggests that SEE is highly sensitive to constraint semantics, not only to channel conditions.

6. Relation to adjacent metrics, modeling caveats, and open directions

SEE is closely related to secrecy rate, secrecy outage probability, energy efficiency, and secrecy throughput, but it is not reducible to any of them. In cooperative outage-based formulations, SEE is defined using secrecy outage probability explicitly (&&&^^^^3search_query3^^^^&&&). In finite-horizon energy-harvesting systems, maximizing average SEE can conflict with maximizing expected total transmitted secure bits; the greedy policy may achieve lower SEE yet higher secure throughput in energy-plentiful regimes (&&&^^^^3all:\3 OR all:\3^^^^&&&). In ISAC, SEE is combined with secrecy-deficit and sensing-entropy penalties rather than treated as an isolated communications metric (&&&4^^^^3search_query3^^^^&&&).

The literature also reveals several modeling caveats. Some works include only radiated and circuit power; others include RIS static power; UAV formulations often exclude communication circuitry in favor of propulsion-only denominators (Xiao et al., 2018, &&&^^^^3all:\33^^^^&&&). Some treat harvested energy at legitimate or illegitimate nodes as a constraint rather than a subtraction in operator power consumption (&&&^^^^3all:\3^^^^7&&&, &&&3^^^^3all:\3^^^^&&&). Full-duplex models may or may not account for SIC-related static power and residual self-interference (&&&^^^^3all:\3^^^^&&&). As a result, SEE values are comparable only within a shared power model.

Open directions are implied by the surveyed formulations. One is robustness: papers already address perfect, statistical, and bounded-uncertainty CSI, but the gap between these assumptions remains substantial in practice (Fotock et al., 9 Jul 2025, Zhang et al., 2019). Another is explainability: SHAP-based interpretation of FNN predictors in ambient backscatter NOMA indicates that the dominant composite channel components control learned SEE decisions in a way consistent with the analytical model (Alam et al., 25 Nov 2025). A further direction is cross-layer coupling. The aerial MEC and finite-horizon RL works indicate that task partitioning, queueing horizon, battery dynamics, scheduling, and mobility can all enter the secrecy-per-Joule objective directly (Abdalla et al., 16 May 2025, &&&^^^^3all:\3 OR all:\3^^^^&&&). This suggests that SEE is evolving from a beamforming metric into a general systems metric for secure, energy-aware wireless design.