On the Spectral theory of Isogeny Graphs and Quantum Sampling of Hard Supersingular Elliptic curves

• URL: http://arxiv.org/abs/2602.02263v1
• Date: Mon, 02 Feb 2026 16:04:54 GMT
• Title: On the Spectral theory of Isogeny Graphs and Quantum Sampling of Hard Supersingular Elliptic curves
• Authors: David Jao, Maher Mamah,
• Abstract summary: We present the first provable quantum-time algorithm that samples a random hard supersingular elliptic curve with high probability.<n>Our algorithm gives a secure instantiation of the hash function and other cryptographic primitives.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper we study the problem of sampling random supersingular elliptic curves with unknown endomorphism rings. This task has recently attracted significant attention, as the secure instantiation of many isogeny-based cryptographic protocols relies on the ability to sample such ``hard'' curves. Existing approaches, however, achieve this only in a trusted-setup setting. We present the first provable quantum polynomial-time algorithm that samples a random hard supersingular elliptic curve with high probability.Our algorithm runs heuristically in $\tilde{O}\!\left(\log^{4}p\right)$ quantum gate complexity and in $\tilde{O}\!\left(\log^{13} p\right)$ under the Generalized Riemann Hypothesis. As a consequence, our algorithm gives a secure instantiation of the CGL hash function and other cryptographic primitives. Our analysis relies on a new spectral delocalization result for supersingular $\ell$-isogeny graphs: we prove the Quantum Unique Ergodicity conjecture, and we further provide numerical evidence for complete eigenvector delocalization; this theoretical result may be of independent interest. Along the way, we prove a stronger $\varepsilon$-separation property for eigenvalues of isogeny graphs than that predicted in the quantum money protocol of Kane, Sharif, and Silverberg, thereby removing a key heuristic assumption in their construction.

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