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起来。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（QOR）方案，通过用复乘理论中的阿贝尔理想类群作用实例化每一层来实现。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，我们提出了一种新颖的发送者与接收者间"非局域"密钥交换方法。即使对于量子敌手，该底层问题仍然保持困难性，并为当前后量子方案的安全性提供支撑。我们将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。我们给出两条实现路径：（i）使用多项式量级量子资源评估类群作用的通用量子预言机，以及（ii）通过连续时间量子行走（CTQW）的固有量子方法（本文概述其框架，并在姊妹篇中详细展开）。一个简短的Qiskit示例展示了QOR的运行机制（设计上侧重原理性而非效率）。