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

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.