Lattice Reduction over Imaginary Quadratic Fields with an Application to Compute-and-Forward

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. First, 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 in polynomial time if the chosen ring is Euclidean. Second, we extend the celebrated Lenstra-Lenstra-Lov\'asz (LLL) reduction from over real bases to over complex bases. Properties and implementations of the algorithm are examined. In particular, satisfying Lov\'asz's condition requires the ring to be Euclidean. Lastly, as an application, we use the algebraic algorithms to find the network coding matrices 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.Comment: submitted to IEEE Transactions on Information Theor