{"ID":2856035,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.10901","arxiv_id":"2510.10901","title":"A Symmetric-Key Cryptosystem Based on the Burnside Ring of a Compact Lie Group","abstract":"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.","short_abstract":"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 Burns...","url_abs":"https://arxiv.org/abs/2510.10901","url_pdf":"https://arxiv.org/pdf/2510.10901v1","authors":"[\"Ziad Ghanem\"]","published":"2025-10-13T01:57:22Z","proceeding":"cs.CR","tasks":"[\"cs.CR\",\"math.RA\"]","methods":"[]","has_code":false}