Equation Asymmetry: An Algebraic Framework for Unifying Secrecy and Covertness in Information-Theoretic Security

2026-06-08

AI Summary

This work aims to unify the two fundamental objectives of information-theoretic security—secrecy and covertness—by introducing a single metric termed “Equation Asymmetry Degree” (EAD). EAD simultaneously characterizes secrecy and covertness performance through the relationship between the signal embedding dimension and the effective rank of the adversary’s observation matrix. Leveraging information-theoretic tools across finite and Gaussian continuous fields, matrix rank analysis, Wasserstein distance, differential entropy, and MDS coding, the study derives secrecy capacity, a strong converse theorem, and covertness conditions. It establishes an equivalence between EAD and secure degrees of freedom, revealing an intrinsic connection between secrecy capacity and covertness performance. The universality of EAD is validated across seven existing schemes, with experiments demonstrating a correlation of up to 0.997 between EAD and practical security metrics.

Abstract

This paper studies the algebraic structure underlying a broad class of information-theoretic security problems. We define the equation asymmetry degree (EAD) as $Φ= (n - r)/n$, where $n$ is the signal embedding dimension and $r$ is the effective rank of the adversary's observation matrix. This single parameter is shown to simultaneously govern both secrecy (measured by equivocation $H(M|Y_E)$) and covertness (measured by detection error probability $P_e$). On finite fields $\mathbb{F}_q$, we establish the equivocation lower bound $H(M|Y_E) = \min(k, n - r_E) \log q$ with exact probabilistic conditions (Theorem~1), the secrecy capacity $C_s = (n - r_E) \log q$ with complete achievability and converse proofs (Theorem~2), and a strong converse (Theorem~8). In the continuous Gaussian regime, we derive a differential-entropy equivocation bound (Lemma~1), the high-SNR secrecy capacity asymptotics (Lemma~2), and a 2-Wasserstein distance covertness condition $W_2 \approx \sqrt{r_W} \cdot P / (2Nσ) \to 0$ (Theorem~5'). The EAD-SDoF equivalence $d_s = n \cdot Φ$ is established (Theorem~7). Both $η_s$ and $η_c$ are shown to be monotone functions of $Φ$ (Theorem~6), with a Pearson correlation of $0.997$ in continuous-domain experiments. Seven existing security schemes -- matrix embedding, MIMO wiretap, secure network coding, FRFT multi-angle transmission, traffic steganography, group-key secure summation, and MDS secure summation -- are unified under the common form $C_s = (n - r) \log q$. Post-quantum security follows from the information-theoretic hardness of underdetermined linear systems (Theorem~9). All numerical experiments are reproducible with open-source code.

Problem

• information-theoretic security
• secrecy
• covertness
• algebraic framework
• equation asymmetry

Innovation

• Equation Asymmetry
• Information-Theoretic Security
• Secrecy and Covertness Unification
• Secrecy Capacity
• Post-Quantum Security