代数学与欧几里德格：最优格约化及超越
Algebraic and Euclidean Lattices: Optimal Lattice Reduction and Beyond
摘要：
我们引入了一个框架，推广格约简算法的模格，以实际和有效地解决$γ$ -Hermite Module-SVP问题在任意分圆域。其核心思想是利用子域的结构来设计一个双重递归的归约策略：既在模块的秩中递归，也在我们正在研究的领域中递归。此外，我们展示了如何利用固有的辛几何存在于塔的领域，以提供一个显着的速度减少秩二模块。秩上的递归策略也可以应用于欧氏格的约简，并且我们可以在几乎与矩阵乘法相同的时间内渐近地进行约简。作为副产品的设计，这些快速减少，我们也推广到所有分圆领域，并提供了许多以前的数论算法的加速。量化，我们表明，一个模块的秩2分圆域的程度$n$可以减少近似因子$2^{\tilde{O}(n)}$内的时间$\tilde{O}(n^2B)$，其中$B$是位长度的条目。对于足够大的$B$，这种复杂性缩小为$\tilde{O}(n^{\log_2 3}B)$。这最后一个结果是特别引人注目的，因为它低于由使用所谓的势的LLL算法的经典分析给出的$n^2B$交换的估计。
英文摘要：
We introduce a framework generalizing lattice reduction algorithms to module lattices in order to practically and efficiently solve the $γ$-Hermite Module-SVP problem over arbitrary cyclotomic fields. The core idea is to exploit the structure of the subfields for designing a doubly-recursive strategy of reduction: both recursive in the rank of the module and in the field we are working in. Besides, we demonstrate how to leverage the inherent symplectic geometry existing in the tower of fields to provide a significant speed-up of the reduction for rank two modules. The recursive strategy over the rank can also be applied to the reduction of Euclidean lattices, and we can perform a reduction in asymptotically almost the same time as matrix multiplication. As a byproduct of the design of these fast reductions, we also generalize to all cyclotomic fields and provide speedups for many previous number theoretical algorithms. Quantitatively, we show that a module of rank 2 over a cyclotomic field of degree $n$ can be heuristically reduced within approximation factor $2^{\tilde{O}(n)}$ in time $\tilde{O}(n^2B)$, where $B$ is the bitlength of the entries. For $B$ large enough, this complexity shrinks to $\tilde{O}(n^{\log_2 3}B)$. This last result is particularly striking as it goes below the estimate of $n^2B$ swaps given by the classical analysis of the LLL algorithm using the so-called potential.
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