Properties of LDGM-LDPC codes with applications to secrecy coding.

摘要

The ensemble of low-density generator-matrix/low-density parity-check (LDGM-LDPC) codes has been proposed in literature. In this thesis, an irregular LDGM-LDPC code is studied as a sub-code of an LDPC code with some randomly punctured output-bits. It is shown that the LDGM-LDPC codes achieve rates arbitrarily close to the channel-capacity of the binary-input symmetric-output memoryless (BISOM) channel with a finite lower-bound on the complexity. The measure of complexity is the average-degree (per information-bit) of the check-nodes for the factor-graph of the code. A lower-bound on the average degree of the check-nodes of the irregular LDGM-LDPC codes is obtained. The bound does not depend on the decoder used at the receiver. The stability condition for decoding the irregular LDGM-LDPC codes over the binary-erasure channel (BEC) under iterative-decoding with message-passing is described. The LDGM-LDPC codes are capacity achieving with bounded complexity and possess natural binning/nesting structure. These codes are applied to secrecy coding. The problem of secrecy coding for the type-II binary symmetric memoryless wiretap channel is studied. In this model, the main channel is binary-input and noiseless and the eavesdropper channel is binary-symmetric memoryless. A coding strategy based on secure nested codes is proposed. A capacity achieving length-n code for the eavesdropper channel bins the space {0, 1}n into co-sets which are used for secret messaging. The resulting co-set scheme achieves secrecy capacity of the type-II binary symmetric memoryless channel. As an example, the ensemble of capacity-achieving regular low-density generator-matrix/low-density parity-check (LDGM-LDPC) codes is studied as a basis for binning. The previous result is generalized to the case of a noisy main-channel. The problem of secrecy-coding for a specific type-I wiretap channel is studied. In the type-I wiretap channel under consideration, the main channel is a binary-input symmetric-output memoryless (BISOM) channel and the eavesdropper channel is a binary-symmetric channel (BSC). A secure-nested-code that achieves perfect-secrecy for the above type-I channel is proposed. The secure-nested-code is based on a nested regular LDGM-LDPC code construction.