Local Information-Theoretic Security via Euclidean Geometry

TL;DR

This work develops Local Information-Theoretic Security (LITS) by applying Euclidean Information Theory to study secrecy around a fixed operating point in discrete memoryless wiretap channels. It recasts the nonconvex Secret Information Coupling (SIC) problem into a local quadratic program via second-order approximations, and shows that optimal rate/leakage multipliers can be obtained by solving a linear program whose constraints are governed by generalized eigenvalues of channel-derived matrices. A key contribution is the secret local contraction coefficient, defined as the largest generalized eigenvalue of the pencil (V,Λ), which quantifies the local leakage efficiency and connects to the SIC solution. The framework is validated on general multi-mode channels and the binary symmetric wiretap channel (BSWC), revealing distinct operational regimes and offering analytical design principles for secure transmission under finite-resource constraints.

Abstract

This paper introduces a methodology based on Euclidean information theory to investigate local properties of secure communication over discrete memoryless wiretap channels. We formulate a constrained optimization problem that maximizes a legitimate user's information rate while imposing explicit upper bounds on both the information leakage to an eavesdropper and the informational cost of encoding the secret message. By leveraging local geometric approximations, this inherently non-convex problem is transformed into a tractable quadratic programming structure. It is demonstrated that the optimal Lagrange multipliers governing this approximated problem can be found by solving a linear program. The constraints of this linear program are derived from Karush-Kuhn-Tucker conditions and are expressed in terms of the generalized eigenvalues of channel-derived matrices. This framework facilitates the derivation of an analytical formula for an approximate local secrecy capacity. Furthermore, we define and analyze a new class of secret local contraction coefficients. These coefficients, characterized as the largest generalized eigenvalues of a matrix pencil, quantify the maximum achievable ratio of approximate utility to approximate leakage, thus measuring the intrinsic local leakage efficiency of the channel. We establish bounds connecting these local coefficients to their global counterparts defined over true mutual information measures. The efficacy of the proposed framework is demonstrated through detailed analysis and numerical illustrations for both general multi-mode channels and the canonical binary symmetric wiretap channel.

Figures (12)

• Figure 1: IB Comparison for a BSC($p_{\mathrm{bob}}=0.1$) with uniform $P_X$ and binary $U$. Solid blue represents the true IB curve (Blahut-Arimoto solution) and red circles dashed the analytical EIT-IB.
• Figure 2: The LP solution for optimal multipliers for a numerically generated channel with $|\mathcal{X}|=5$. The optimal vertex is found at the boundary of the feasible region defined by the four eigenmode constraint lines.
• Figure 3: Normalized approximate local secrecy capacity as a function of the constraint ratio.
• Figure 4: $C_{\mathrm{SIC}}$ as a function of Eve's channel quality for different output quantization levels $|\mathcal{Z}|$, with $|X|=8$, Bob's $E_{b}/N_{0}=8.0 \text{dB}, R=0.5, \Theta=0.1$
• Figure 5: Comparison of the EIT-approximated local secrecy capacity (dashed red line) for $q_{\mathrm{eve}}= 0.45, R=1.0, \Theta/R=0.085$, with the true secrecy capacity (solid blue line) for the BSWC.

Theorems & Definitions (49)

• Remark 1
• Lemma 1
• proof
• Definition 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof