Providing Secrecy with Lattice Codes

Recent results have shown that lattice codes can be used to construct good channel codes, source codes and physical layer network codes for Gaussian channels. On the other hand, for Gaussian channels with secrecy constraints, efforts to date rely on random codes. In this work, we provide a tool to bridge these two areas so that the secrecy rate can be computed when lattice codes are used. In particular, we address the problem of bounding equivocation rates under nonlinear modulus operation that is present in lattice encoders/decoders. The technique is then demonstrated in two Gaussian channel examples: (1) a Gaussian wiretap channel with a cooperative jammer, and (2) a multi-hop line network from a source to a destination with untrusted intermediate relay nodes from whom the information needs to be kept secret. In both cases, lattice codes are used to facilitate cooperative jamming. In the second case, interestingly, we demonstrate that a non-vanishing positive secrecy rate is achievable regardless of the number of hops.

💡 Research Summary

The paper bridges the gap between the powerful structured lattice codes that have been shown to achieve capacity‑approaching performance on Gaussian channels and the traditional random‑coding approaches that dominate information‑theoretic secrecy analyses. The authors develop a new analytical tool that quantifies how much information is lost when lattice points are reduced modulo a coarse lattice—a non‑linear operation that is intrinsic to most lattice encoders and decoders. Their central result (Theorem 1) proves that for an N‑dimensional lattice, the modulo operation discards at most one bit per dimension, i.e., at most N bits in total. This bound is derived by partitioning the lattice into 2ᴺ cosets of the sub‑lattice 2Λ and showing that the residue class together with a small integer index T (1 ≤ T ≤ 2ᴺ) uniquely determines the original sum of two lattice points. A supporting lemma shows that when one of the summed points is uniformly distributed over the fundamental region, the sum is statistically independent of the other point—exactly the property needed for cooperative jamming.

Armed with this result, the authors analyze two secrecy scenarios.

1. Gaussian Wiretap Channel with a Cooperative Jammer.
   The legitimate transmitter (S) and the jammer (CJ) both use the same nested lattice code (Λ, Λ₁) and add independent dithers known to all legitimate parties. The legitimate receiver (D) can remove the dithers and decode the lattice point reliably, while the eavesdropper (E) observes the sum of the two lattice points plus Gaussian noise. By Theorem 1, E’s observation can be reduced to at most N extra bits of uncertainty (the index T). Consequently the equivocation satisfies H(W | E) ≥ n