Construction of Codes for Wiretap Channel and Secret Key Agreement from Correlated Source Outputs by Using Sparse Matrices

The aim of this paper is to prove coding theorems for the wiretap channel coding problem and secret key agreement problem based on the the notion of a hash property for an ensemble of functions. These theorems imply that codes using sparse matrices can achieve the optimal rate. Furthermore, fixed-rate universal coding theorems for a wiretap channel and a secret key agreement are also proved.

💡 Research Summary

The paper addresses two fundamental problems in information‑theoretic security—wiretap channel coding and secret‑key agreement from correlated sources—by introducing a unified coding framework based on a “hash property” of function ensembles. The authors define the hash property as a statistical uniformity condition: for a randomly chosen function from the ensemble, the probability that any two distinct inputs collide is essentially the same for all input pairs. This property guarantees both strong randomness extraction and compression, which are essential for achieving secrecy capacity and optimal key rates.

To construct practical codes that satisfy the hash property, the authors focus on linear transformations defined by sparse matrices. A sparse matrix is a binary matrix in which the number of non‑zero entries per row (or column) grows only logarithmically with the block length n, i.e., O(log n). Such matrices can be stored and multiplied with vectors in O(n log n) time, dramatically reducing the computational and memory overhead compared with dense random matrices traditionally used in information‑theoretic security proofs.

For the wiretap channel, the model consists of a legitimate sender (Alice) who wishes to transmit a secret message M of length k bits over a noisy main channel to a legitimate receiver (Bob), while an eavesdropper (Eve) observes a degraded version of the transmission through a second channel. The coding scheme proceeds as follows: Alice selects a sparse matrix A∈𝔽₂^{k×n} from a publicly known ensemble, computes the codeword Xⁿ = f_A(M, R) = A·