Maher Mamah | Searcharxiv

arXiv2026

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

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.

quant-ph↗

arXiv2025

Enhanced Algorithms for the Representation of integers by Binary Quadratic forms: Reduction to Subset Sum

In this paper, we present efficient algorithms for solving the Diophantine equation $f(x, y) = m$ for an arbitrary definite binary quadratic form $f$, given the factorization of $m$. While Cornacchia's algorithm to solve $x^2 + dy^2 = m$ is efficient in many cases, its runtime becomes exponentially large when $m$ is highly composite and encounters subtleties when generalized to arbitrary forms $f$. To address these issues, we give a reduction from our problem to an instance of the Subset sum, a weakly NP complete problem, allowing for more efficient solutions. Leveraging this approach, we develop deterministic algorithms that adapt to different cases based on $\mathrm{disc}(f)$ and $ m $. In particular, when $|\mathrm{disc}(f)| = \mathrm{polylog}(m) $, we provide a polynomial time solution that remains efficient regardless of the structure of $ m $. For more general cases, we present an algorithm that improves upon Cornacchia's method, achieving a quadratic speedup. Recently, the problem of representing integers by a form $ f $ found important applications in elliptic curves and isogeny based cryptography, where these algorithms are central to solving norm form equations.

math.NT↗