Performance Limits of Lattice Reduction over Imaginary Quadratic Fields with Applications to Compute-and-Forward

Abstract

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-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, 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. Index Termslattice reduction, Gauss's algorithm, LLL, compute-and-forward. Initially coding lattices from ConstructionA over rational integers Z or Gaussian integers Z[i] are the main enabler in showing the achievable information rate in C&F. Recently, however, the Construction A lattices have been extended to over the ring of Eisenstein integers Z[ω] [4], [5] and other rings of imaginary quadratic integers [6], This work was presented in part at the IEEE Information Theory Workshop 2018, Guangzhou, China, and submitted in part to the IEEE Information Theory Workshop 2019, Visby, Gotland, Sweden. S. Lyu is with the College

On the Optimality of Gauss’s Algorithm over Euclidean Imaginary Quadratic Fields

“…Napias first generalised the LLL algorithm to lattices spanned over imaginary quadratic rings and certain quaternion fields [6]; later Fieker, Pohst and Stehle investigated more fundamental properties of algebraic lattices [7], [8]. Recently Kim and Lee proposed an efficient LLL algorithm over bi-quadratic field whose quantization step requires a Euclidean domain [4], while our work on LLL over imaginary quadratic fields showed that a Euclidean domain is needed to make the algorithm convergent [9].…”

Optimal Space-Time Block Code Designs Based on Irreducible Polynomials of Degree Two

“…which are considered in most communication problems, is that there are some recently researches that consider as base field a general ring of algebraic integer, for example [23] and [24]. Moreover, the lattice reduction has also been generalized to a ring of imaginary quadratic integers [25] and only the rings from Q( √ −d), where d takes the values 1, 2, 3, 7 and 11 can be used to define Lovász condition [26].…”