Lattice Reduction over Imaginary Quadratic Fields

In this work, we extend the celebrated Lenstra-Lenstra-Lov\'asz (LLL) reduction from over real bases to over complex bases. The complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices. We first analyze properties of these algebraic lattices, and then construct a general algebraic LLL reduction algorithm for them. Properties and implementations of the algorithm are examined. In particular, satisfying Lov\'asz's condition requires the ring to be Euclidean. As an application, we use the algebraic LLL algorithm to find the network coding matrix in compute-and-forward. Such lattice reduction-based approaches have low complexity which is not dictated by the signal-to-noise (SNR) ratio. Moreover, such approaches can not only preserve the degree-of-freedom of computation rates, but ensure the independence in the code space as well.