A Symmetric-Key Cryptosystem Based on the Burnside Ring of a Compact Lie Group
Ziad Ghanem
TL;DR
The paper addresses the vulnerability of classical linear ciphers by building a symmetric-key cryptosystem that operates in the Burnside ring $A(G)$ of a compact Lie group, with a concrete focus on $G=O(2)$. Messages are encoded as finitely supported elements of $A(G)$ and encrypted via Burnside multiplication by an involutory key $k$ derived from a selected set of irreducible representations, enabling encryption without compromising the infinite-dimensional ambient space. For $G=O(2)$, encryption preserves plaintext support within a finite generating set, and security analyses show that finite observations restrict the action to a finite-rank submodule, while the key remains information-theoretically non-identifiable from such data. However, the scheme is not IND-CPA secure; a one-query chosen-plaintext distinguisher based on dihedral probes demonstrates deterministic insecurity. This work blends algebraic topology, representation theory, and cryptography, introducing a novel algebraic primitive that illustrates how equivariant structures and the Burnside ring can inform cryptographic design and security limitations.
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.