equation-asymmetry-information-security

name	equation-asymmetry-information-security
description	Equation Asymmetry Degree (EAD) framework for unifying secrecy and covertness in information-theoretic security. EAD = 1 - r/n governs both equivocation and detection error probability. Applies to MIMO wiretap, secure network coding, FRFT multi-angle transmission, traffic steganography, post-quantum security. Use when analyzing information-theoretic security, secrecy capacity, covertness, or designing secure communication protocols.
metadata	{"arxiv_id":"2606.10374","published":"2026-06-09","authors":"Wang Hao, Zhang Kuang","tags":["information-theory","security","secrecy","covertness","equation-asymmetry","post-quantum"]}

Equation Asymmetry Degree (EAD) Framework

Core Concept

The Equation Asymmetry Degree EAD = Φ = 1 - r/n where:

• n = signal embedding dimension
• r = effective rank of adversary's observation matrix

This single parameter simultaneously governs:

1. Secrecy — measured by equivocation H(X|Y)
2. Covertness — measured by detection error probability P_e

Key Theorems

Theorem 1 (Finite Fields): Equivocation lower bound H(X|Y) ≥ Φ·log|F| with exact probabilistic conditions on F_q.

Theorem 2 (Secrecy Capacity): Complete achievability and converse proofs for secrecy capacity C_s in terms of Φ.

Theorem 5' (Continuous Gaussian): High-SNR secrecy capacity asymptotics + 2-Wasserstein distance covertness condition W_2 ≤ ε·Φ.

Theorem 6 (Monotonicity): Both secrecy capacity and detection error probability are monotone functions of Φ (Pearson correlation 0.997 in experiments).

Theorem 7 (EAD-SDoF Equivalence): Φ = SDoF/n where SDoF is the secure degrees of freedom.

Theorem 8 (Strong Converse): Strong converse theorem for secrecy capacity on finite fields.

Theorem 9 (Post-Quantum Security): Post-quantum security follows from information-theoretic hardness of underdetermined linear systems (Ax = b where A is m×n with m < n).

Unified Form

Seven existing security schemes unified under common form y = Ax + e:

1. Matrix embedding
2. MIMO wiretap channels
3. Secure network coding
4. FRFT multi-angle transmission
5. Traffic steganography
6. Group-key secure summation
7. MDS secure summation

Usage Patterns

Pattern 1: Security Scheme Analysis Given a security protocol → compute EAD = 1 - r/n → bound equivocation and covertness → compare with other schemes on same Φ scale.

Pattern 2: Post-Quantum Assessment Evaluate whether a scheme's security reduces to underdetermined linear system solving → if yes, Theorem 9 applies → information-theoretic post-quantum security guaranteed.

Pattern 3: Design Optimization Maximize Φ = 1 - r/n by increasing embedding dimension n or reducing adversary's effective rank r through signal design.

Activation Keywords

• equation asymmetry
• EAD framework
• information-theoretic security
• secrecy capacity
• covertness
• secure network coding
• MIMO wiretap
• post-quantum security linear system
• 方程不对称度
• 信息论安全
• 保密容量