الملخص
We study vector subset sum over $\mathbb{F}_3^n$: given $m$ random vectors from $\mathbb{F}_3^n$, find a nonempty subset that sums to zero; the smaller $m$, the more difficult it is to find such a subset. Chen, Liu, and Zhandry (EUROCRYPT'22) introduced an efficient quantum algorithm that solves this problem when $m\approx n^2/2$, where a naive classical algorithm would require exponential time. Subsequently, Kothari, O'Donnell, and Wu (STOC'2026) gave an efficient classical algorithm that only requires $m \approx n^2/3$ vectors, thus removing the hope for an exponential quantum advantage in this parameter regime. Using the framework of Chen, Liu, and Zhandry, we give quantum algorithms that require much fewer input vectors, renewing the possibility of an exponential quantum speedup: for any fixed $ε>0$, our quantum algorithm solves $\mathbb{F}_3$-subset sum in polynomial time with $m=ε\cdot n^2$ vectors. More generally, we establish a full sample--time tradeoff that interpolates between exponential and polynomial runtime. The main ingredient is a deterministic classical algorithm for the binary-error Learning-with-Errors problem, which is of independent cryptographic interest. For this, we rigorously establish a sample--time tradeoff that was predicted by earlier algebraic heuristics. For vector subset sums over larger fields, we also significantly improve classical algorithms in Kothari, O'Donnell, and Wu (STOC'2026).
الكلمات المفتاحية
الموضوع
بيانات النشر
- المجلة
- غير متاح
- وصول مفتوح
- وصول مفتوح أخضر
اقتبس هذه المقالة
APA 7
Kothari, R., Metger, T., O'Donnell, R., Shutty, N., & Wu, K. (2026). Exponential quantum speedup for $\mathbb{F}_3^n$-Subset-Sum? Or, rigorous classical algorithms for Binary-Error LWE. https://omanscience.com/ar/articles/exponential-quantum-speedup-for-mathbb-f-3-n-subset-sum-or-rigorous-classical-algorithms-for-binary-error-lwe
MLA 9
Kothari, Robin, et al. "Exponential quantum speedup for $\mathbb{F}_3^n$-Subset-Sum? Or, rigorous classical algorithms for Binary-Error LWE." https://omanscience.com/ar/articles/exponential-quantum-speedup-for-mathbb-f-3-n-subset-sum-or-rigorous-classical-algorithms-for-binary-error-lwe.
شيكاغو (المؤلف–التاريخ)
Kothari, Robin, Tony Metger, Ryan O'Donnell, Noah Shutty, and Kewen Wu. 2026. "Exponential quantum speedup for $\mathbb{F}_3^n$-Subset-Sum? Or, rigorous classical algorithms for Binary-Error LWE." https://omanscience.com/ar/articles/exponential-quantum-speedup-for-mathbb-f-3-n-subset-sum-or-rigorous-classical-algorithms-for-binary-error-lwe.
هارفارد
Kothari, R., Metger, T., O'Donnell, R., Shutty, N. and Wu, K. (2026) 'Exponential quantum speedup for $\mathbb{F}_3^n$-Subset-Sum? Or, rigorous classical algorithms for Binary-Error LWE', Available at: https://omanscience.com/ar/articles/exponential-quantum-speedup-for-mathbb-f-3-n-subset-sum-or-rigorous-classical-algorithms-for-binary-error-lwe.
فانكوفر
Kothari R, Metger T, O'Donnell R, Shutty N, Wu K. Exponential quantum speedup for $\mathbb{F}_3^n$-Subset-Sum? Or, rigorous classical algorithms for Binary-Error LWE. https://omanscience.com/ar/articles/exponential-quantum-speedup-for-mathbb-f-3-n-subset-sum-or-rigorous-classical-algorithms-for-binary-error-lwe
IEEE
R. Kothari, T. Metger, R. O'Donnell, N. Shutty, and K. Wu, "Exponential quantum speedup for $\mathbb{F}_3^n$-Subset-Sum? Or, rigorous classical algorithms for Binary-Error LWE," https://omanscience.com/ar/articles/exponential-quantum-speedup-for-mathbb-f-3-n-subset-sum-or-rigorous-classical-algorithms-for-binary-error-lwe.