Isogeny Graphs in Superposition and Quantum Onion Routing

TL;DR

This work presents a quantum onion routing framework (QOR) that replaces classical public-key assumptions with a layered quantum encryption scheme grounded in the abelian ideal class group action $Cl(\mathcal{O}_{\Delta})$ from the Theory of Complex Multiplication. By leveraging isogeny graphs and their Cayley-graph structure, the authors connect path-finding hardness to robust algebraic tools and provide two concrete implementation paths: a universal quantum oracle with polynomial-time class-group action evaluation and an intrinsically quantum continuous-time quantum walk approach. They detail encryption schemes based on $j$-invariants, formalize the underlying algebra with cyclic association schemes and Bose–Mesner algebra, and describe the isogeny-walk unitary structure that enables quantum superposition of traversal. An explicit five-actor example via a small Qiskit demonstration illustrates mechanics and security considerations, while the discussion addresses parameter choices, subexponential quantum attacks, and practical deployment concerns. Overall, the paper advances a coherent, post-quantum compatible protocol for anonymous quantum communication by unifying number-theoretic isogeny concepts with quantum information primitives.

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.

Figures (7)

• Figure 1: High-Level Overview of the First Part of the Protocol
• Figure 2: Hiding the Quantum Message
• Figure 3: Example Carol's of the "mapper oracle" for a three-qubit index register. In general, this simple mapper oracle implementation requires $2^{\lceil\log_2\Omega\rceil}-1$ controlled shift oracles $U_C$.
• Figure 4: Circuit for repeated squaring for the universal quantum oracle. This circuit applies $g^s$ to the state in the $j$-register depending on the binary decomposition of $s$ stored in the index register.
• Figure 4: The Entire Isogeny Cycle in Superposition

Theorems & Definitions (10)

• Theorem 2.1
• Remark 2.2
• Remark 4.1
• Definition 4.2
• Remark 4.3
• Lemma 4.4
• proof
• Remark 4.5
• Remark 4.6
• Remark 5.1