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arXiv · 2602.02263

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

Maher Mamah·Jake Doliskani·David Jao
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Abstract

In this paper, we study the problem of sampling random supersingular elliptic curves with unknown endomorphism rings. This problem has recently gained considerable attention as many isogeny-based cryptographic protocols require such ``secure'' curves for instantation, while existing methods achieve this only in a trusted-setup setting. We present the first provable quantum polynomial-time algorithms for sampling such curves with high probability, one of which is based on an algorithm of Booher et. al. One variant runs heuristically in $\tilde{O}(\log^{4} p)$ quantum gate complexity, and in $\tilde{O}(\log^{13} p)$ under the Generalized Riemann Hypothesis, and outputs a curve that is provably secure assuming average-case hardness of the endomorphism ring problem. Another variant samples uniform $\mathcal{O}$-oriented curves with unknown endomorphism rings, for any imaginary quadratic order $\mathcal O$, with security based on the hardness of Vectorization problem. When accompanied by an interactive quantum computation verification protocol our algorithms provide a secure instantiation of the CGL hash function and related primitives. Our analysis relies on a new spectral delocalization result for supersingular $\ell$-isogeny graphs: we prove the Quantum Unique Ergodicity conjecture and provide numerical evidence for complete eigenvector delocalization. We also 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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Maher Mamah, Jake Doliskani, David Jao. 2026-02-02. On the Spectral theory of Isogeny Graphs and Quantum Sampling of Secure Supersingular Elliptic curves. https://arxiv.org/abs/2602.02263

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