Isogeny Graphs in Superposition and Quantum Onion Routing

• URL: http://arxiv.org/abs/2510.01464v1
• Date: Wed, 01 Oct 2025 21:15:22 GMT
• Title: Isogeny Graphs in Superposition and Quantum Onion Routing
• Authors: Eleni Agathocleous, Tobias Hartung, Karl Jansen, Lukas Mansour,
• Abstract summary: Onion routing provides anonymity by layering encryption so that no relay can link sender to destination.<n>We propose a symmetric-enion-based quantum onion routing scheme by instantiating each layer with the abelian ideal class group action from the Theory of Complex multiplication.
• Score: 1.6321617803073527
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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