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.

翻译：本文提出了一种基于欧几里得信息论的方法，用于研究离散无记忆窃听信道安全通信的局部特性。我们构建了一个约束优化问题，在明确限制信息向窃听者泄露的上界以及秘密消息编码信息成本的同时，最大化合法用户的信息速率。通过利用局部几何近似，这一本质非凸问题被转化为可处理的二次规划结构。研究证明，控制该近似问题的最优拉格朗日乘子可通过求解线性规划获得。该线性规划的约束条件源自Karush-Kuhn-Tucker条件，并以信道导出矩阵的广义特征值形式表示。该框架有助于推导近似局部保密容量的解析公式。此外，我们定义并分析了一类新的秘密局部收缩系数。这些系数被表征为矩阵束的最大广义特征值，用于量化近似效用与近似泄露的最大可实现比率，从而衡量信道固有的局部泄露效率。我们建立了将这些局部系数与其在真实互信息度量上定义的全局对应物相联系的界限。通过对通用多模信道和典型二进制对称窃听信道进行详细分析和数值演示，验证了所提框架的有效性。