Classical linear ciphers, such as the Hill cipher, operate on fixed, finite-dimensional modules and are therefore vulnerable to straightforward known-plaintext attacks that recover the key as a fully determined linear operator. We propose a symmetric-key cryptosystem whose linear action takes place instead in the Burnside ring $A(G)$ of a compact Lie group $G$, with emphasis on the case $G=O(2)$. The secret key consists of (i) a compact Lie group $G$; (ii) a secret total ordering of the subgroup orbit-basis of $A(G)$; and (iii) a finite set $S$ of indices of irreducible $G$-representations, whose associated basic degrees define an involutory multiplier $k\in A(G)$. Messages of arbitrary finite length are encoded as finitely supported elements of $A(G)$ and encrypted via the Burnside product with $k$. For $G=O(2)$ we prove that encryption preserves plaintext support among the generators $\{(D_1),\dots,(D_L),(SO(2)),(O(2))\}$, avoiding ciphertext expansion and security leakage. We then analyze security in passive models, showing that any finite set of observations constrains the action only on a finite-rank submodule $W_L\subset A(O(2))$, and we show information-theoretic non-identifiability of the key from such data. Finally, we prove the scheme is \emph{not} IND-CPA secure, by presenting a one-query chosen-plaintext distinguisher based on dihedral probes.

翻译：经典的线性密码（如希尔密码）作用于固定的有限维模，因此容易受到直接已知明文攻击，此类攻击可将密钥恢复为完全确定的线性算子。我们提出了一种对称密钥密码系统，其线性作用发生在紧李群 $G$ 的伯恩赛德环 $A(G)$ 中，重点研究 $G=O(2)$ 的情况。密钥包含：(i) 一个紧李群 $G$；(ii) $A(G)$ 的子群轨道基的一个秘密全序；(iii) 一个有限索引集 $S$，对应于不可约 $G$-表示，其关联的基本次数定义了一个对合乘子 $k\in A(G)$。任意有限长度的消息被编码为 $A(G)$ 的有限支撑元，并通过与 $k$ 的伯恩赛德积进行加密。对于 $G=O(2)$，我们证明了加密在生成元 $\{(D_1),\dots,(D_L),(SO(2)),(O(2))\}$ 中保持了明文的支撑集，避免了密文膨胀和安全泄露。随后，我们在被动模型中分析安全性，表明任何有限观测集仅能将作用约束在有限秩子模 $W_L\subset A(O(2))$ 上，并且我们证明了从此类数据中密钥在信息论上是不可识别的。最后，我们通过提出一种基于二面体探针的单次查询选择明文区分器，证明了该方案 \emph{不} 具有 IND-CPA 安全性。