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.

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