Almost universal codes for MIMO wiretap channels

Despite several works on secrecy coding for fading and MIMO wiretap channels from an error probability perspective, the construction of information-theoretically secure codes over such channels remains an open problem. In this paper, we consider a fading wiretap channel model where the transmitter has only partial statistical channel state information. Our channel model includes static channels, i.i.d. block fading channels, and ergodic stationary fading with fast decay of large deviations for the eavesdropper's channel. We extend the flatness factor criterion from the Gaussian wiretap channel to fading and MIMO wiretap channels, and establish a simple design criterion where the normalized product distance / minimum determinant of the lattice and its dual should be maximized simultaneously. Moreover, we propose concrete lattice codes satisfying this design criterion, which are built from algebraic number fields with constant root discriminant in the single-antenna case, and from division algebras centered at such number fields in the multiple-antenna case. The proposed lattice codes achieve strong secrecy and semantic security for all rates $R<C_b-C_e-\kappa$, where $C_b$ and $C_e$ are Bob and Eve's channel capacities respectively, and $\kappa$ is an explicit constant gap. Furthermore, these codes are almost universal in the sense that a fixed code is good for secrecy for a wide range of fading models. Finally, we consider a compound wiretap model with a more restricted uncertainty set, and show that rates $R<\bar{C}_b-\bar{C}_e-\kappa$ are achievable, where $\bar{C}_b$ is a lower bound for Bob's capacity and $\bar{C}_e$ is an upper bound for Eve's capacity for all the channels in the set.