Nonhomogeneous LDPC Codes and Their Application to Encrypted Communication

Given a parity check matrix (PCM) H, its associated low-density parity-check (LDPC) code is the solution space of the homogeneous system of linear equations Hx = 0. In this paper, we propose to adopt the solutions to Hx = b (a nonhomogeneous system of linear equations) as the code words, where b is a nonzero vector and is called the code's target syndrome. We use the term "nonhomogeneous LDPC (abbreviated as NH-LDPC) codes" to refer to the codes obtained this way. In particular, we propose a systematic NH-LDPC coding scheme. We show that the PCM needs to possess some properties for the systematic NH-LDPC encoding to work. The decoding of NH-LDPC codes is almost the same as the decoding of homogeneous (i.e. conventional) LDPC codes (i.e. by sum-product algorithm and iterative decoding), only with slight modification. As an example of practical application, we propose to use the NH-LDPC codes in encrypted data transmission, where the target syndrome b is used as the secret key (available only to legitimate users) for encryption and decryption. One problem is that, even without the knowledge of b, an illegitimate user can still demodulate a codeword's systematic part, albeit with a much higher error probability than legitimate ones. To counter this problem, a descrambler is incorporated at the receiver to magnify the effect of the errors from demodulation so that the illegitimate user can not obtain intelligible information content. Simulations show that the proposed joint coding-encryption scheme works well as long as the channel signal-to-noise ratio (SNR) is properly controlled to lie within a specific range.