Complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices. In this work, we study such lattices and their reduction algorithms. Firstly, when the lattice is spanned over a two dimensional basis, we show that the algebraic variant of Gauss's algorithm returns a basis that corresponds to the successive minima of the lattice if the chosen ring is Euclidean. Secondly, we extend the celebrated Lenstra-Lenstra-Lovász (LLL) reduction from over real bases to over complex bases. Properties and implementations of the algorithm are examined. In particular, satisfying Lovász's condition requires the ring to be Euclidean. Lastly, we numerically show the time-advantage of using algebraic LLL by considering lattice bases generated from wireless communications and cryptography.

中文翻译：

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虚二次域上的格约化

复数基，连同由虚二次整数环定义的直和，产生代数格。在这项工作中，我们研究了这样的格和它们的约简算法。首先，当晶格跨越二维基时，我们表明如果选择的环是欧几里得的，高斯算法的代数变体返回对应于晶格的连续最小值的基。其次，我们将著名的 Lenstra-Lenstra-Lovász (LLL) 约简从超实数基数扩展到超复数基数。检查了算法的特性和实现。特别是，满足 Lovász 条件需要环是欧几里得的。最后，我们通过考虑无线通信和密码学生成的格基，在数值上展示了使用代数 LLL 的时间优势。