Average Fractional Equivocation (AFE)

• Average Fractional Equivocation (AFE) is a performance metric that quantifies the fraction of message uncertainty remaining at an eavesdropper, based on normalized conditional entropy.
• It refines binary secrecy measures by statistically averaging equivocation over channel fading and system dynamics, thus capturing partial secrecy.
• AFE informs system design by optimizing coding rates, diversity configurations, and power allocation to enhance secure wireless communications.

Average Fractional Equivocation (AFE) is a fundamental performance metric in secure communications, quantifying the average proportion of message uncertainty that remains at an eavesdropper. Conceived to provide finer granularity than binary secrecy criteria such as outage probability, AFE serves as an asymptotic lower bound for decoding error probability and reflects partial secrecy, especially relevant in scenarios where perfect secrecy is unattainable due to channel, system, or coding limitations. It is rigorously defined as the expected value of fractional equivocation—the conditional entropy of the message given the eavesdropper’s observation, normalized by the message entropy—averaged over the stochastic variations of the channel or other system parameters.

1. Definition and Mathematical Formulation

AFE is mathematically expressed as

$\text{AFE} = \mathbb{E}\left[ \Lambda \right]$

where the fractional equivocation $\Lambda$ for a given realization is

$\Lambda = \frac{H(M|Z)}{H(M)}$

with %%%%1%%%% denoting the entropy of the message and $H(M|Z)$ the conditional entropy given the eavesdropper’s observation $Z$. In practice, for wireless channels under fading, $\Lambda$ becomes a random variable due to channel fluctuations, and AFE quantifies its expectation over the fading statistics (Osorio et al., 2019, Mora et al., 15 Sep 2025). In the wiretap channel regime, AFE can also be formulated in terms of achievable secrecy and information rates: $\text{AFE} = \frac{R_e}{R_1}$ where $R_e$ is the equivocation (or secrecy) rate and $R_1$ is the private message rate (Ekrem et al., 2010, Marina et al., 2011). For finite-blocklength codes, AFE is assessed as the average normalized equivocation per symbol (Pfister et al., 2017):

Symbol	Mathematical Expression	Interpretation
$\Lambda$	$\frac{H(M|Z)}{H(M)}$	Fraction of message uncertainty
$\bar{\Lambda}$ (AFE)	$\mathbb{E}[\Lambda]$	Average over channel realizations
AFE (Rate)	$R_e / R_1$	Fraction of private message concealed

2. Relationship to Partial Secrecy and System Performance Metrics

AFE is central in the analysis of partial secrecy regimes, providing a more nuanced and actionable view compared to secrecy outage metrics, which are inherently binary. AFE quantifies the degree to which the secrecy criterion is statistically met rather than simply triggered. This is especially pertinent for practical wireless systems under quasi-static fading, cooperative relaying, or diversity-enhanced architectures (Osorio et al., 2019, Mora et al., 15 Sep 2025). Unlike generalized secrecy outage probability (GSOP), which examines probability of strictly zero information leakage, AFE reflects expected uncertainty remaining over the operational distribution of the channel characteristics.

• Asymptotic Error Probability Lower Bound: High AFE values (close to one) indicate robust protection—large residual uncertainty at the eavesdropper—whereas low values denote increased information leakage.
• Integral Formulation: In fading channels, AFE is obtained via integrals conditioned on the instantaneous signal-to-noise ratio (SNR) ratios, e.g.

$$\bar{\Lambda} = 1 - \frac{1}{\ln(2^{R_s})} \int_{1}^{2^{R_s}} \frac{1}{z} F_\Phi(z) dz$$

with $F_\Phi(z)$ being the cumulative distribution function of a random variable representing SNR ratios (Mora et al., 15 Sep 2025).

3. Information-Theoretic Foundations and Coding Schemes

Information-theoretic secrecy frameworks, such as the single-letter Csiszár–Körner region for wiretap channels, naturally embed AFE within their characterizations. Optimal secrecy codes seek to maximize equivocation rate $R_e$ subject to constraints imposed by channel capacities and the achievable information rates. In the Gaussian MIMO wiretap channel, equivocation is characterized via log-determinant formulas: $$R_e \leq \frac{1}{2} \log \frac{\left| \mathbf{S}_Y + \mathbf{K} \right|}{\left| \mathbf{S}_Y \right|} - \frac{1}{2} \log \frac{\left| \mathbf{S}_Z + \mathbf{K} \right|}{\left| \mathbf{S}_Z \right|}$$ allowing direct evaluation of AFE as $R_e/R_1$ (Ekrem et al., 2010).

Advanced coding techniques, such as stochastic superposition coding with layered binning, facilitate separate control over equivocation of semantic and observed source components (Kozlov et al., 15 Sep 2025). In practical quantum wiretap scenarios, non-uniformity in auxiliary randomness degrades AFE through additional penalty terms related to Rényi entropy (Hayashi, 2012). Privacy amplification, message splitting, and optimal power allocation further influence achievable AFE in both broadcast and cooperative networks (Hayashi et al., 2011, Marina et al., 2011, Osorio et al., 2019).

4. Evaluation in Classical and Modern Fading Models

AFE is computationally tractable for a broad class of fading channel models, including the generalized multicluster fluctuating two-ray (MFTR) distribution. In such settings, exact closed-form or efficient high-SNR approximations are available, and complexity remains constant regardless of diversity order (Mora et al., 15 Sep 2025). Diversity, measured by the number of antennas per node (maximal ratio combining, MRC), directly enhances AFE, reflecting improved confusion of the eavesdropper as the legitimate receiver’s channel improves. Monte Carlo simulations extensively validate analytic and approximate AFE formulas and reveal its sensitivity to power allocation, secrecy rate, and fading severity.

5. Connections to Rate-Distortion Theory and Alternative Metrics

Recent work interprets equivocation—and thus AFE—as a special case within distortion-based secrecy characterizations, particularly under log-loss distortion functions (Cuff, 2013). This perspective integrates AFE into rate-distortion-equivocation tradeoff regions and suggests the design space includes more refined secrecy metrics sensitive to both fidelity and secrecy requirements. For semantic communication, AFE of the semantic source component is measured as $A_s/H(S)$, with $H(S)$ the semantic entropy and $A_s$ the equivocation at the eavesdropper (Kozlov et al., 15 Sep 2025). Distortion-induced compression inherently boosts equivocation, with explicit formulas linking AFE to achievable rate-distortion pairs.

6. Finite Blocklength Considerations and Code Optimization

For wiretap codes of finite blocklength, average equivocation is estimated via simulation—using techniques like Monte Carlo sampling of erasure positions in coset-based codes (Pfister et al., 2017). The normalized equivocation per symbol serves as AFE, and the “achievability gap” quantifies the deviation from ideal secrecy at finite lengths. Structured codes (e.g., Hamming, simplex) attain higher AFE and narrower gaps under small to moderate blocklength constraints, highlighting the role of code selection in practical secrecy engineering.

7. Robustness, Continuity Bounds, and Limitations

AFE’s robustness to distributional variations is characterized by tight uniform continuity bounds for conditional entropy. If two joint distributions on finite alphabets differ by total variation $\varepsilon$, then

$$|H(X|Y) - H(X'|Y')| \leq \varepsilon \cdot \log(|X| - 1) + h(\varepsilon)$$

where $h(\cdot)$ is the binary entropy function (Alhejji et al., 2019). This enables quantification of sensitivity in AFE when channel or source models are perturbed. Applicability is constrained to finite alphabet systems and relies on entropy invariance properties; extension to infinite-alphabet regimes remains an open issue.

8. Practical System Design and Optimization

AFE provides concrete utility in wireless system design, guiding optimization of coding rates, diversity configurations, and power allocation schemes to balance reliability and secrecy. In relay networks with untrusted relays and destination-based jamming, optimizing the AFE (integral over channel realizations) leads to tangible design criteria (Osorio et al., 2019). Diversity gains at the legitimate receiver consistently improve AFE; analytic tools enable fast evaluation and deployment across a variety of fading environments and operational regimes (Mora et al., 15 Sep 2025). Rate-distortion-equivocation frameworks allow joint design of fidelity and secrecy, with explicit control over semantic uncertainty at adversaries (Kozlov et al., 15 Sep 2025).

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Average Fractional Equivocation (AFE) serves as a rigorous, quantitative tool for analyzing and designing secure communication systems, especially in partial secrecy regimes and under finitary or non-Gaussian channel models. Its analytic foundations, tractable computation, and clear ties to both information-theoretic and practical metrics position it as a preferred measure in contemporary physical-layer security analysis.