改进的子集和的经典与量子算法
Improved Classical and Quantum Algorithms for Subset-Sum
摘要：
我们提出了新的经典和量子算法解决随机子集和的情况。首先，我们改进了Becker-Coron-Joux算法（EUROCHEMPT 2011），从$\tilde{\mathcal{O}}(2^{0.291 n})$降到$\tilde{\mathcal{O}}(2^{0.283 n})$，使用更一般的表示，值为$\{-1,0,1,2\}$。接下来，我们在几个方向上改进了这个问题的量子算法。通过结合Howgrave-Graham-Joux算法（EUROCHEMICAL PT 2010）和量子搜索，我们设计了一种具有渐近成本$\tilde{\mathcal{O}}(2^{0.236 n})$的算法，其成本低于基于Bernstein，Jeffery，Lange和Meurer提出的相同经典算法的量子行走（PQCHEMICAL PTO 2013）。该算法的优点是使用具有量子随机访问的\emph{classical}存储器，而以前已知的算法使用量子行走框架，并且需要具有量子随机访问的\emph{quantum}存储器。我们还提出了新的子集和量子行走，比Helm和May（TQC 2018）给出的$\tilde{\mathcal{O}}(2^{0.226 n})$之前的最佳时间复杂度更好。我们结合我们的新技术，以达到一个时间$\tilde{\mathcal{O}}(2^{0.216 n})$。这个时间取决于Helm和May对量子行走更新的启发式，这也是以前的算法所需要的。我们展示了如何部分克服这种启发式，我们得到了一个算法与量子时间$\tilde{\mathcal{O}}(2^{0.218 n})$只需要标准的经典子集和算法。
英文摘要：
We present new classical and quantum algorithms for solving random subset-sum instances. First, we improve over the Becker-Coron-Joux algorithm (EUROCRYPT 2011) from $\tilde{\mathcal{O}}(2^{0.291 n})$ downto $\tilde{\mathcal{O}}(2^{0.283 n})$, using more general representations with values in $\{-1,0,1,2\}$. Next, we improve the state of the art of quantum algorithms for this problem in several directions. By combining the Howgrave-Graham-Joux algorithm (EUROCRYPT 2010) and quantum search, we devise an algorithm with asymptotic cost $\tilde{\mathcal{O}}(2^{0.236 n})$, lower than the cost of the quantum walk based on the same classical algorithm proposed by Bernstein, Jeffery, Lange and Meurer (PQCRYPTO 2013). This algorithm has the advantage of using \emph{classical} memory with quantum random access, while the previously known algorithms used the quantum walk framework, and required \emph{quantum} memory with quantum random access. We also propose new quantum walks for subset-sum, performing better than the previous best time complexity of $\tilde{\mathcal{O}}(2^{0.226 n})$ given by Helm and May (TQC 2018). We combine our new techniques to reach a time $\tilde{\mathcal{O}}(2^{0.216 n})$. This time is dependent on a heuristic on quantum walk updates, formalized by Helm and May, that is also required by the previous algorithms. We show how to partially overcome this heuristic, and we obtain an algorithm with quantum time $\tilde{\mathcal{O}}(2^{0.218 n})$ requiring only the standard classical subset-sum heuristics.
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