We propose a new coding scheme using only one lattice that achieves the 1/2 log(1 + SNR) capacity of the additive white Gaussian noise (AWGN) channel with lattice decoding, which is provable for signal-to-noise ratio SNR > e at present. The scheme applies a discrete Gaussian distribution over an AWGN-good lattice, but otherwise does not require a shaping lattice or dither. Thus, it significantly simplifies the default lattice coding scheme of Erez and Zamir which involves a quantization good lattice as well as an AWGN-good lattice. Using the flatness factor, we show that the error probability of the proposed scheme under minimum mean-square error lattice decoding is almost the same as that of Erez and Zamir, for any rate up to the AWGN channel capacity. We introduce the notion of good constellations, which carry almost the same mutual information as that of continuous Gaussian inputs. We also address the implementation of Gaussian shaping for the proposed lattice Gaussian coding scheme.

We consider the problem of secure distributed matrix computation (SDMC), where a user queries a function of data matrices generated at distributed source nodes. We assume the availability of $N$ honest but curious computation servers, which are connected to the sources, the user, and each other through orthogonal and reliable communication links. Our goal is to minimize the amount of data that must be transmitted from the sources to the servers, called the upload cost , while guaranteeing that no $T$ colluding servers can learn any information about the source matrices, and the user cannot learn any information beyond the computation result. We first focus on secure distributed matrix multiplication (SDMM), considering two matrices, and propose a novel polynomial coding scheme using the properties of finite field discrete Fourier transform, which achieves an upload cost significantly lower than the existing results in the literature. We then generalize the proposed scheme to include straggler mitigation, and to the multiplication of multiple matrices while keeping the input matrices, the intermediate computation results, as well as the final result secure against any $T$ colluding servers. We also consider a special case, called computation with own data, where the data matrices used for computation belong to the user. In this case, we drop the security requirement against the user, and show that the proposed scheme achieves the minimal upload cost. We then propose methods for performing other common matrix computations securely on distributed servers, including changing the parameters of secret sharing, matrix transpose, matrix exponentiation, solving a linear system, and matrix inversion, which are then used to show how arbitrary matrix polynomials can be computed securely on distributed servers using the proposed procedure.

The use of Lenstra-Lenstra-Lovasz (LLL) lattice reduction significantly improves the performance of zero-forcing (ZF) and successive interference cancellation (SIC) decoders in multi-input multi-output (MIMO) communications. Capitalizing on the observation that the decision region of SIC is determined by the Gram-Schmidt vectors rather than the basis itself, we propose the use of effective LLL reduction in SIC decoding, where size reduction is only performed for pairs of consecutive basis vectors. We establish the theoretic upper bound O(n³ log n) on the average complexity of effective LLL reduction for the i.i.d. Gaussian model of MIMO fading channels, which is an order lower than previously thought. Moreover, an effectively LLL-reduced basis can easily be transformed into the standard LLL-reduced basis for the purpose of ZF decoding.

Vector perturbation (VP) precoding is a promising technique for multiuser communication systems operating in the downlink. In this work, we introduce a hybrid framework to improve the performance of lattice reduction (LR) aided precoding in VP. First, we perform a simple precoding using zero forcing (ZF) or successive interference cancellation (SIC) based on a reduced lattice basis. Since the signal space after LR-ZF or LR-SIC precoding can be shown to be bounded to a small range, then along with sufficient orthogonality of the lattice basis guaranteed by LR, they collectively pave the way for the subsequent application of an approximate message passing (AMP) algorithm, which further boosts the performance of any suboptimal precoder. Our work shows that the AMP algorithm can be beneficial for a lattice decoding problem whose data symbols lie in integers $\mathbb{Z}$ and entries of the lattice basis may bot be i.i.d. Gaussian. Numerical results confirm the low-complexity AMP algorithm can improve the symbol error rate (SER) performance of LR aided precoding significantly. Lastly, the hybrid scheme is also proved effective when solving the data detection problem of massive MIMO systems without using LR.

Sampling from the lattice Gaussian distribution has emerged as an important problem in coding, decoding, and cryptography. In this paper, the classic Metropolis-Hastings (MH) algorithm in Markov chain Monte Carlo methods is adopted for lattice Gaussian sampling. Two MH-based algorithms are proposed, which overcome the limitation of Klein's algorithm. The first one, referred to as the independent Metropolis-Hastings-Klein (MHK) algorithm, establishes a Markov chain via an independent proposal distribution. We show that the Markov chain arising from this independent MHK algorithm is uniformly ergodic, namely, it converges to the stationary distribution exponentially fast regardless of the initial state. Moreover, the rate of convergence is analyzed in terms of the theta series, leading to predictable mixing time. A symmetric Metropolis-Klein algorithm is also proposed, which is proven to be geometrically ergodic.

Sampling from the lattice Gaussian distribution plays an important role in various research fields. In this paper, the Markov chain Monte Carlo (MCMC)-based sampling technique is advanced in several fronts. First, the spectral gap for the independent Metropolis-Hastings-Klein (MHK) algorithm is derived, which is then extended to Peikert's algorithm and rejection sampling; we show that independent MHK exhibits faster convergence. Then, the performance of bounded distance decoding (BDD) using MCMC is analyzed, revealing a flexible trade-off between the decoding radius and complexity. MCMC is further applied to trapdoor sampling, again offering a trade-off between security and complexity. Finally, the independent multiple-try Metropolis-Klein (MTMK) algorithm is proposed to enhance the convergence rate. The proposed algorithms allow parallel implementation, which is beneficial for practical applications.