Joint encryption encoding schemes have been released to fulfill both reliability and security desires in a single step. Using Low Density Parity-Check (LDPC) codes in joint encryption encoding schemes, as an alternative to classical linear codes, would shorten the key size as well as improving error correction capability. In this article, a joint encryption encoding scheme using Quasi-Cyclic Low Density Parity-Check (QC-LDPC) codes based on finite geometry is presented. It is observed that our proposed scheme not only outperforms its predecessors in key size and transmission rate, but also remains secure against all known cryptanalyses of code-based secret key cryptosystems. In this paper, we have proposed an idea to make QC-LDPC based cryptosystems secure against reaction attacks. It is subsequently shown that our scheme benefits from low computational complexity. By taking the advantage of QC-LDPC codes based on finite geometry, the key size of our scheme is very close to its target security level. In addition, using the proposed scheme, a wide range of desirable transmission rates are achievable. This variety of codes makes our cryptosystem suitable for a number of different communication and cryptographic standards such as wireless personal area networks (WPAN) and digital video broadcasting (DVB).

In this paper, a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6. For a J ´ L exponent matrix E, the lower bound Q (E) is obtained explicitly such that (J, L) QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size m ³ Q (E). The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6.

An (a; b) elementary trapping set (ETS), where a and b denote the size and the number of unsatis ed check nodes in the ETS, in uences the performance of low-density parity-check (LDPC) codes. The smallest size of an ETS in LDPC codes with column weight 3 and girth 6 is 4. In this paper, we concentrate on a well-known algebraic-based construction of girth-6 QC-LDPC codes based on powers of a primitive element in a  nite  eld Fq. For this structure, we provide the su cient conditions to obtain 3 n submatrices of an exponent matrix in constructing girth-6 QC-LDPC codes whose ETSs have the size of at least 5. For structures on  nite  eld Fq, where q is a power of 2, all non-isomorphic 3  n submatrices of the exponent matrix which yield QC-LDPC codes free of small ETSs are presented.

In [30] a class of quasi-cyclic LDPC codes has been proposed, whose information rate is 1/2. We generalize that construction to arbitrary rates l-1/l and we provide a Grobner basis for their dual codes, in some generic cases.