Onion routing provides anonymity by layering encryption so that no relay can link sender to destination. A quantum analogue faces a core obstacle: layered quantum encryption generally requires symmetric encryption schemes, whereas classically one would rely on public-key encryption. We propose a symmetric-encryption-based quantum onion routing (QOR) scheme by instantiating each layer with the abelian ideal class group action from the Theory of Complex Multiplication. Session keys are established locally via a Diffie-Hellman key exchange between neighbors in the chain of communication. Furthermore, we propose a novel ''non-local'' key exchange between the sender and receiver. The underlying problem remains hard even for quantum adversaries and underpins the security of current post-quantum schemes. We connect our construction to isogeny graphs and their association schemes, using the Bose-Mesner algebra to formalize commutativity and guide implementation. We give two implementation paths: (i) a universal quantum oracle evaluating the class group action with polynomially many quantum resources, and (ii) an intrinsically quantum approach via continuous-time quantum walks (CTQWs), outlined here and developed in a companion paper. A small Qiskit example illustrates the mechanics (by design, not the efficiency) of the QOR.

翻译：洋葱路由通过分层加密提供匿名性，使得任何中继节点都无法将发送者与目的地关联。其量子类比面临一个核心障碍：分层量子加密通常需要对称加密方案，而经典方案则可依赖公钥加密。我们提出一种基于对称加密的量子洋葱路由方案，其每层加密通过复乘理论中的阿贝尔理想类群作用实现。会话密钥通过通信链中相邻节点间的Diffie-Hellman密钥交换在本地建立。此外，我们提出一种发送者与接收者间新颖的"非局域"密钥交换方案。该方案的基础问题即使对量子敌手而言仍保持困难性，并为当前后量子密码方案提供安全支撑。我们将该构造与同源图及其结合方案相关联，利用Bose-Mesner代数形式化交换性并指导实现。我们提供两条实现路径：（i）使用多项式量子资源评估类群作用的通用量子预言机；（ii）通过连续时间量子行走的本征量子方法（本文概述其框架，具体内容在姊妹篇中展开）。文中通过一个小型Qiskit示例说明量子洋葱路由的运作机制（该示例旨在展示原理，而非体现效率）。