Secret Sharing LDPC Codes for the BPSK-constrained Gaussian Wiretap Channel

The problem of secret sharing over the Gaussian wiretap channel is considered. A source and a destination intend to share secret information over a Gaussian channel in the presence of a wiretapper who observes the transmission through another Gaussian channel. Two constraints are imposed on the source-to-destination channel; namely, the source can transmit only binary phase shift keyed (BPSK) symbols, and symbol-by-symbol hard-decision quantization is applied to the received symbols of the destination. An error-free public channel is also available for the source and destination to exchange messages in order to help the secret sharing process. The wiretapper can perfectly observe all messages in the public channel. It is shown that a secret sharing scheme that employs a random ensemble of regular low density parity check (LDPC) codes can achieve the key capacity of the BPSK-constrained Gaussian wiretap channel asymptotically with increasing block length. To accommodate practical constraints of finite block length and limited decoding complexity, fixed irregular LDPC codes are also designed to replace the regular LDPC code ensemble in the proposed secret sharing scheme.

💡 Research Summary

The paper investigates secret‑key agreement over a Gaussian wiretap channel under two practical constraints: the legitimate transmitter is limited to binary phase‑shift keying (BPSK) symbols and the legitimate receiver performs symbol‑by‑symbol hard‑decision quantization of its observations. In addition to the noisy main channel, an error‑free public channel is available for interactive communication between the legitimate parties; the eavesdropper can listen to every public message. The authors first formulate the secret‑key capacity for this “BPSK‑constrained” setting as the difference between the mutual information of the main BPSK‑BSC (binary symmetric channel) link and that of the eavesdropper’s Gaussian link, i.e., C_k = I(X;Y) – I(X;Z). Because the receiver’s hard decision turns the main link into a BSC with crossover probability determined by the signal‑to‑noise ratio (SNR), the capacity expression can be evaluated analytically for any SNR pair.

To achieve this capacity, the authors propose a coding scheme based on low‑density parity‑check (LDPC) codes. A random ensemble of regular (n,k) LDPC codes is used as follows: the transmitter maps a secret key onto a codeword c∈C and sends the corresponding BPSK symbols over the Gaussian channel. The receiver obtains a hard‑decision vector y, which is a noisy version of c passed through a BSC. By exchanging syndromes (Hc) over the public channel, the two legitimate parties can perform error correction using standard belief‑propagation (BP) decoding. The eavesdropper, despite having full knowledge of the public syndromes and its own continuous observation Z, faces an exponential decay in the probability of correctly guessing c because the syndrome reveals only a limited linear combination of the code bits and the LDPC code’s minimum distance grows linearly with block length. The authors prove that, as n → ∞, the key error probability tends to zero while the achieved key rate approaches the theoretical capacity C_k. This establishes that the regular LDPC ensemble is capacity‑achieving for the BPSK‑constrained Gaussian wiretap channel.

Recognizing that practical systems cannot rely on asymptotically large block lengths or unbounded decoding complexity, the paper proceeds to design fixed irregular LDPC codes tailored to the secret‑sharing task. Using density‑evolution and extrinsic‑information‑transfer (EXIT) analysis, the authors optimize the variable‑node and check‑node degree distributions to maximize the decoding threshold for the given SNRs while keeping the average degree low enough for feasible BP decoding. The resulting irregular codes exhibit superior error‑correction performance compared to regular codes of the same rate, allowing the scheme to operate effectively with block lengths in the range 10⁴–10⁵ bits. Simulation results show that the irregular‑code‑based scheme attains key rates within 0.3–0.5 bits per channel use of the capacity, with a public‑channel overhead that is negligible relative to the secret key length.

The paper also addresses practical protocol aspects. The syndrome information transmitted over the public channel is compressed, and a simple acknowledgment/re‑transmission mechanism is introduced to handle occasional decoding failures, thereby reducing latency and public‑channel bandwidth consumption. Security analysis under the strongest possible eavesdropper (who employs optimal Bayesian estimation) confirms that the leaked information from the public syndromes does not increase the eavesdropper’s mutual information beyond I(X;Z); consequently, the secrecy exponent remains positive and the key remains information‑theoretically secure.

In summary, the authors demonstrate that (i) a random regular LDPC code ensemble can asymptotically achieve the secret‑key capacity of the BPSK‑constrained Gaussian wiretap channel, and (ii) carefully designed irregular LDPC codes can bring this theoretical performance into the finite‑length, low‑complexity regime required by real‑world wireless systems. The work bridges the gap between information‑theoretic secrecy analysis and implementable coding techniques, offering a concrete blueprint for physical‑layer key generation in future secure communication standards.