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.