Isogeny Graphs in Superposition and Quantum Onion Routing

2025-10-01

AI Summary

This paper addresses the security challenges of anonymous communication in the quantum era by proposing the first quantum-resistant Quantum Onion Routing (QOR) framework. To counter quantum threats to classical symmetric encryption, QOR innovatively integrates abelian ideal class group actions for hierarchical encryption, leverages distance-regular graphs and their Bose–Mesner algebras to ensure operation commutativity, and designs a non-local key exchange mechanism intrinsically realized via continuous-time quantum walks (CTQWs). Compared to classical onion routing, QOR achieves both post-quantum security and link anonymity. The work presents two concrete implementations: a generic quantum oracle-based construction and a CTQW-native protocol, both validated via Qiskit-based prototyping. Its core contribution lies in the first application of distance-regular graph theory and quantum walks to anonymous routing—establishing a provably secure, scalable, and quantum-era anonymous communication paradigm.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

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.

Problem

• Developing quantum onion routing using symmetric encryption
• Establishing secure session keys via quantum-resistant key exchange
• Implementing quantum routing via class group actions and walks

Innovation

• Symmetric quantum onion routing using class group actions
• Non-local key exchange resilient to quantum attacks
• Implementation via quantum oracles and continuous-time quantum walks