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This paper presents a concrete cryptanalysis of isogeny-based quantum money schemes, leveraging rational points on elliptic curves to achieve a significant speedup over brute-force attacks while providing the first nontrivial algorithmic insight into their structural hardness. The work also derives an efficient verification procedure based on quadratic twists, offering valuable direction for future research in post-quantum cryptographic protocols.
Cryptanalysis of Isogeny-Based Quantum Money with Rational Points
arXiv:2508.00351v2 Announce Type: cross Abstract: Quantum money is a representative application in cryptography that utilizes quantum technology to achieve security that is unattainable with classical systems. Since the theoretical proposal by Wiesner in 1983, the most recent instantiation is the quantum money using class group actions on elliptic curves, proposed by Montgomery and Sharif in Asiacrypt'24. In this work, we propose a concrete cryptanalysis by leveraging the efficiency of evaluating division polynomials with the coordinates of rational points, offering a speedup of $O(\log^4p)$ compared to the brute-force attack. While the resulting attack still runs in exponential time, our method provides the first nontrivial algorithmic insight into their structural hardness. Furthermore, due to the algebraic properties of rational points and quadratic twists, our technique naturally yields a more efficient verification procedure for the underlying quantum money. We believe this new line of analysis opens promising directions for future research in elliptic-curve-based quantum cryptography.
Abstract:Quantum money is a representative application in cryptography that utilizes quantum technology to achieve security that is unattainable with classical systems. Since the theoretical proposal by Wiesner in 1983, the most recent instantiation is the quantum money using class group actions on elliptic curves, proposed by Montgomery and Sharif in Asiacrypt'24. In this work, we propose a concrete cryptanalysis by leveraging the efficiency of evaluating division polynomials with the coordinates of rational points, offering a speedup of $O(\log^4p)$ compared to the brute-force attack. While the resulting attack still runs in exponential time, our method provides the first nontrivial algorithmic insight into their structural hardness. Furthermore, due to the algebraic properties of rational points and quadratic twists, our technique naturally yields a more efficient verification procedure for the underlying quantum money. We believe this new line of analysis opens promising directions for future research in elliptic-curve-based quantum cryptography.
25 pages; accepted for publication in Quantum. Revised proofs and resource estimates
Cryptography and Security (cs.CR); Quantum Physics (quant-ph)
arXiv:2508.00351 [cs.CR]
(or arXiv:2508.00351v2 [cs.CR] for this version)
https://doi.org/10.48550/arXiv.2508.00351 Focus to learn more arXiv-issued DOI via DataCite
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