一种改进的用于三元组格筛法的量子算法
An Improved Quantum Algorithm for 3-Tuple Lattice Sieving
AI总结：
研究针对高维格中最短向量问题的三元组筛法，利用预处理关联中心点聚焦搜索，改进量子时间复杂度，提升至\(2^{0.2846d}\)，并在总内存限制为\(2^{0.1887d}\)时成为已知最快的SVP量子算法。
摘要：
高维格中最短向量问题（SVP）的假设难度是后量子密码学的基石之一。已知对SVP最快的启发式攻击是通过筛法。k元组筛法是一种迭代方法，每次迭代处理大量固定范数的格向量，通过对k个输入向量求和与求差生成相同数量但范数稍小的格向量，多次迭代可产生短格向量。最快的攻击是k = 2时，但增大k可减少攻击所需内存。本文利用预处理步骤将给定格向量与附近“中心点”关联以聚焦搜索，将三元组筛法的量子时间复杂度从\(2^{0.3098d}\)提升到\(2^{0.2846d}\)。该算法使用\(2^{0.1887d}\)经典比特和QCRAM比特以及\(2^{o(d)}\)量子比特，是已知总内存限制为\(2^{0.1887d}\)时最快的SVP量子算法。
英文摘要：
The assumed hardness of the Shortest Vector Problem in high-dimensional lattices is one of the cornerstones of post-quantum cryptography. The fastest known heuristic attacks on SVP are via so-called sieving methods. While these still take exponential time in the dimension $d$, they are significantly faster than non-heuristic approaches and their heuristic assumptions are verified by extensive experiments. $k$-Tuple sieving is an iterative method where each iteration takes as input a large number of lattice vectors of a certain norm, and produces an equal number of lattice vectors of slightly smaller norm, by taking sums and differences of $k$ of the input vectors. Iterating these ''sieving steps'' sufficiently many times produces a short lattice vector. The fastest attacks (both classical and quantum) are for $k=2$, but taking larger $k$ reduces the amount of memory required for the attack. In this paper we improve the quantum time complexity of 3-tuple sieving from $2^{0.3098 d}$ to $2^{0.2846 d}$, using a two-level amplitude amplification aided by a preprocessing step that associates the given lattice vectors with nearby ''center points'' to focus the search on the neighborhoods of these center points. Our algorithm uses $2^{0.1887d}$ classical bits and QCRAM bits, and $2^{o(d)}$ qubits. This is the fastest known quantum algorithm for SVP when total memory is limited to $2^{0.1887d}$.
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