Hyeonhak Kim of Korea University and colleagues have achieved a speedup of O(log⁴p) in attacks against a recent quantum money scheme. Their work targets the instantiation of quantum money using class group actions on elliptic curves, proposed by Montgomery and Sharif in Asiacrypt’24. The researchers used rational points and quadratic twists to create a more efficient verification procedure for the underlying quantum money, offering the first nontrivial algorithmic insight into its structural hardness. This new line of analysis, the team believes, “opens promising directions for future research in elliptic-curve-based quantum cryptography.”

Montgomery-Sharif Quantum Money & Elliptic Curve Encoding

Researchers used rational points and their coordinates to accelerate the evaluation of division polynomials, a key step in breaking the encryption proposed by Montgomery and Sharif at Asiacrypt’24. This algorithmic advancement, while still resulting in exponential time complexity, marks the first nontrivial insight into the structural hardness of this specific instantiation of quantum money. This analysis focuses on a quantum money system built upon class group actions on elliptic curves, a method gaining traction since Wiesner’s initial 1983 theoretical proposal.

The team’s work demonstrates that the previously assumed security of this approach may have relied primarily on the computational difficulty of exhaustive searches, rather than inherent structural properties. By efficiently calculating with rational points on these curves, the researchers created a more streamlined verification procedure for the quantum money itself, effectively reducing the time needed to assess its validity.

The method builds on established mathematical foundations, including Silverman’s work on the arithmetic of elliptic curves and a 1987 paper detailing elliptic curves over finite fields. The implications of this work extend beyond simply accelerating attacks. It provides a new lens through which to evaluate the security of isogeny-based quantum money.

Rational Points Accelerate Cryptanalysis of Isogeny-Based Schemes

The researchers achieved this acceleration by improving the efficiency of evaluating division polynomials, a key component in breaking the cryptographic scheme. This means the process of confirming the validity of the “quantum banknotes” is now faster, a critical factor for any practical implementation of such a system. The team’s work illustrates how a deeper understanding of potential attacks can refine both the security and operational aspects of quantum cryptographic schemes, demonstrating a symbiotic relationship between cryptanalysis and cryptography.

While the current method does not yet provide a practical means to counterfeit banknotes with the parameters currently used, the team estimates the computational resources required for such an attack, providing a benchmark for future security evaluations. The researchers state, suggesting that further exploration of rational points and related techniques could yield even more effective cryptanalytic tools.

Researchers achieved a speedup of O(log⁴p) in attacking this specific scheme, a quantifiable improvement over brute-force methods, by focusing on the efficiency of division polynomial evaluation. This acceleration stems from a novel application of rational point coordinates within the elliptic curve cryptography.

👉 More information
🗞 Cryptanalysis of Isogeny-Based Quantum Money with Rational Points
✍️ Hyeonhak Kim, Donghoe Heo, Seokhie Hong, Sangjin Lee and Suhri Kim
🧠 DOI: https://quantum-journal.org/papers/q-2026-11-10-2263/

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