There is a concept in graph theory known as a chord which has not been considered before in relation to trapping sets of Tanner graphs. A chord of a cycle is an edge outside the cycle which connects two vertices of that cycle. It is proved that short cycles with a chord are the root of several trapping sets and eliminating them increases the minimum distance$d_{\min }$of a code. We provide new analytic lower bounds on$d_{\min }$of LDPC codes with girths 6 and 8 and column weight$\gamma $in which the short cycles are all chordless. We prove, analytically, that$d_{\min }\geq 2\gamma $for girth 6 and$d_{\min }\geq \frac {3(\gamma -1)^{2}}{\gamma \ln \gamma -\gamma +1}$for girth 8. Comparing these bounds with the existing bound$\gamma +1$for girth-6 LDPC codes shows the positive and significant influence of eliminating these cycles. A method to construct protograph-based LDPC codes with different girths and free of short cycles with a chord is given which is applicable to any type of protographs, simple and multi-edge, regular and irregular. The conditions to remove small trapping sets from the Tanner graph of a multi-edge QC-LDPC code are given. Numerical results indicate that the application of our method to QC-LDPC codes improves existing results.

With the rapid development of cloud computing and Internet of Things (IoT), smart health (s-health) is anticipated to enhance healthcare quality significantly. However, data integrity, user anonymity, and authentication concerns have not been adequately addressed in s-health. Remote data integrity checking (RDIC) and digital signature schemes have great potential to address these requirements. Nevertheless, the direct adoption of these schemes suffers from two flaws. Firstly, they incur prohibitively high computation and communication overhead. Secondly, they leak sensitive health information about patients and do not provide complete anonymity. To address these issues, we introduce$\mathbf {A^{3}B}$-$\mathbf {RDV}$, an aggregate anonymous attribute-based remote data verification scheme. In$\mathbf {A^{3}B}$-$\mathbf {RDV}$, the integrity of an arbitrary number of cloud data files can be verified at once without downloading the whole data, thereby saving communication and computation resources. Moreover, in$\mathbf {A^{3}B}$-$\mathbf {RDV}$, data owners can be authenticated by performing highly efficient operations. Also,$\mathbf {A^{3}B}$-$\mathbf {RDV}$provides complete anonymity and supports dishonest-user traceability. We provide security definitions for$\mathbf {A^{3}B}$-$\mathbf {RDV}$and prove its security under the hardness assumption of the bilinear Diffie-Hellman (BDH) problem. Performance comparisons and experimental results indicate that$\mathbf {A^{3}B}$-$\mathbf {RDV}$is more efficient and expressive than state-of-the-art approaches.

LDPC codes based on multi-edge protographs potentially have larger minimum distances compared to their counterparts, single-edge protographs. However, considering different features of their Tanner graph, such as short cycles, girth and other graphical structures, is harder than for Tanner graphs from single-edge protographs. Here, we provide a novel approach to construct the parity-check matrix of an LDPC code which is based on trades obtained from block designs. We employ our method to construct multi-edge quasi-cyclic (QC) LDPC codes.We use those trade-based matrices to define base matrices of multi-edge protographs. The construction of exponent matrices corresponding to these base matrices has less complexity than the ones proposed in the literature. We prove that these base matrices result in QC-LDPC codes with smaller lower bounds on the lifting degree than existing ones.

We consider multiple-edge QC-LDPC codes with a base matrix of large size. We propose a new method, the degree reduction method, to obtain exponent matrices of these codes which considerably reduces the complexity of the search algorithm. We also provide a necessary and sufficient condition to avoid 4-cycles from occurrence in the Tanner graph of codes obtained using our method. Then, we apply our method to quasi-cyclic protograph-based Raptor-Like LDPC (QC-PBRL-LDPC) codes whose base matrices are multiple-edge. Numerical results show that as a consequence of this study we can obtain the minimum lifting degree of QC-PBRL-LDPC codes with girth at least 6. Thus, the lengths of the obtained codes are much smaller than those of their counterpart short-length codes in the literature.

Block-fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. Frequency-hopping schemes and orthogonal frequency division multiplexing (OFDM) can conveniently be modelled as BF channels. Applying lattices in this type of channel entails dividing a lattice point into multiple blocks such that fading is constant within a block but changes, independently, across blocks. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. Recently, the original binary Construction A for lattices, due to Forney, has been generalized to a lattice construction from totally real and complex multiplication (CM) fields. This generalized algebraic Construction A of lattices provides signal space diversity, intrinsically, which is the main requirement for the signal sets designed for fading channels. In this paper, we construct full-diversity algebraic lattices for BF channels using Construction A over totally real number fields. We propose two new decoding methods for these family of lattices which have complexity that grows linearly in the dimension of the lattice. The first decoder is proposed for full-diversity algebraic LDPC lattices which are generalized Construction A lattices with a binary LDPC code as underlying code. This decoding method contains iterative and non-iterative phases. In order to implement the iterative phase of our decoding algorithm, we propose the definition of a parity-check matrix and Tanner graph for full-diversity algebraic Construction A lattices. We also prove that using an underlying LDPC code that achieves the outage probability limit over one-block-fading channel, the constructed algebraic LDPC lattices together with the proposed decoding method admit diversity order $n$ over an $n$ -block-fading channel. Then, we modify the proposed algorithm by removing its iterative phase which enables full-diversity practical decoding of all generalized Construction A lattices without any assumption about their underlying code. In contrast with the known results on AWGN channels in which non-binary Construction A lattices always outperform the binary ones, we provide some instances showing that algebraic Construction A lattices obtained from binary codes outperform the ones based on non-binary codes in block fading channels. Since available lattice construction methods from totally real and complex multiplication (CM) fields do not provide diversity in the binary case, we generalize algebraic Construction A lattices over a wider family of number fields namely monogenic number fields.

We propose a new nonlinear Rao-Nam like symmetric key encryption scheme. In our design, we employ a specific type of coded modulation schemes namely quasi-cyclic low-density parity-check (QC-LDPC) lattice-codes which have low-complexity encoding and decoding algorithms. Due to the application of coded modulation schemes in our design, the proposed scheme performs encryption, encoding and modulation simultaneously. Therefore, we regard the proposed scheme as a joint cryptosystem. The proposed joint cryptosystem withstands all variants of chosen plaintext attacks applied on Rao-Nam like cryptosystems due to its nonlinearity. Moreover, some conditions implying the uniformity of the ciphertexts distribution are introduced through our analysis. Our scheme is efficient and admits small key size. These features are obtained due to several reasons including the quasi-cyclic form of the generator and the parity-check matrices of QC-LDPC lattice-codes, and the simple hardware structure for generating the permutation matrix, the intentional error vector and the nonlinear functions used in our design. The QC-LDPC lattice-codes facilitate high-rate transmission which is suitable for bandlimited AWGN channels. Our simulations indicate that QC-LDPC lattice-codes outperform the error performance of high-order coded modulation schemes based on QAM modulations. Hence, our scheme provides secure, reliable and efficient data transmission in bandlimited AWGN channels.

Neural network decoding algorithms are recently introduced by Nachmani et al. to decode high-density parity-check (HDPC) codes. In contrast with iterative decoding algorithms such as sum-product or min-sum algorithms in which the weight of each edge is set to 1, in the neural network decoding algorithms, the weight of every edge depends on its impact in the transmitted codeword. In this paper, we provide a novel feed-forward neural network lattice decoding algorithm suitable to decode lattices constructed based on Construction A, whose underlying codes have HDPC matrices. We first establish the concept of feed-forward neural network for HDPC codes and improve their decoding algorithms compared to Nachmani et al. We then apply our proposed decoder for a Construction A lattice with HDPC underlying code, for which the well-known iterative decoding algorithms show poor performances. The main advantage of our proposed algorithm is that instead of assigning and training weights for all edges, which turns out to be time-consuming especially for high-density parity-check matrices, we concentrate on edges which are present in most of 4-cycles and removing them gives a girth -6 Tanner graph. This approach, by slight modifications using updated LLRs instead of initial ones, simultaneously accelerates the training process and improves the error performance of our proposed decoding algorithm.

In this paper, we give lower bounds on the size of (a, b) elementary trapping sets (ETSs) of variable-regular LDPC codes with any girth, g, and irregular ones with girth 8, where a is the size and b is the number of degree-one check nodes. Our proposed lower bounds are analytical, applicable to all values of g and b, based on graph theories and tighter than the existing ones in the literature. Our results mostly depend on the girth. We also propose results, which are independent of the girth and rely on the variables a, b, y, and the column weight value. We obtain the tightest lower bounds on the size of ETSs of variable-regular LDPC codes with girth eight. These results provide us with a chance to present a method to achieve the minimum size of ETSs of irregular LDPC codes with girth, especially those whose column weight values are a subset of (2, 3, 4, 5, 61 and fulfill the inequality (b/a) <; 1. Moreover, we present a range of numerical results about (a, b) ETSs with girths 8 and 10 to compare the tightness between our proposed lower bounds and the existing bounds in the literature.

Low density parity check (LDPC) lattices were the first family of lattices equipped with iterative decoding algorithms. We introduce quasi-cyclic LDPC (QC LDPC) lattices as a special case of LDPC lattices with one binary QC-LDPC code as their underlying code. These lattices are obtained from the Construction A of lattices providing us to encode them efficiently using shift registers. To benefit from an encoder with linear complexity in the lattice dimension, we obtain the generator matrix of these lattices in quasi-cyclic form. We generalize the proposed quasi-cyclic form of the generator matrix for other Construction A lattices, namely the LDA lattices, with a non-binary QC-LDPC code as their underlying code. We provide a low-complexity decoding algorithm of QC LDPC-lattices based on the sum product algorithm. To design lattice codes, QC LDPC-lattices are combined with the nested lattice shaping that uses the Voronoi region of a sublattice for shaping. The shaping gain and the shaping loss of our lattice codes with dimensions 40, 50, and 60 using an optimal quantizer, are presented. The guidelines for applying efficient shaping methods, like hypercube shaping, for QC LDPC-lattices are also given. Consequently, we establish a family of lattice codes that perform practically close to the sphere bound.

Low-density parity-check (LDPC) lattices were the first family of lattices to show efficient decoding in high dimensions. We consider a case of these lattices with one binary LDPC code as an underlying code. We employ encoding and decoding of the LDPC lattices in a cooperative transmission framework. We establish two efficient shaping based on hypercube and Voronoi shaping, to obtain LDPC lattice codes. Then, we propose the implementation of block Markov encoding for one-way and two-way relay networks using LDPC lattice codes. An efficient method is also required for decomposing full-rate codebook into lower rate codebooks. We apply different decomposition schemes for one-way and two-way relay channels, which are the altered versions of the decomposition methods of low density lattice code (LDLC) lattices. Due to the lower complexity of the decoding for LDPC lattices comparing with LDLCs, the complexity of our schemes is significantly lower than the ones proposed for LDLCs. The efficiency of the proposed schemes is presented using simulation results that indicate the outperforming behavior of LDLCs over LDPC lattice codes in the same dimensions. However, having lower decoding complexity enables us to increase the dimension of the lattice to compensate the existing gap between the performance of the LDPC lattice codes and the LDLCs.

In this paper, we propose new constructions for regular girth-8 quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices (CPM). The constructions assume symmetries in the structure of the parity-check matrix and employ a greedy exhaustive search algorithm to find the permutation shifts of the CPMs. As a result of symmetries, the new codes have a more compact representation compared with their counterparts. In majority of cases, also, they achieve the girth 8 at a shorter block length for the same degree distribution (code rate). Deterministic (explicit) constructions are also presented to expand the proposed parity-check matrices to larger block lengths and higher rates. The proposed long high-rate codes are often substantially shorter than regular girth-8 QC-LDPC codes of similar rate in the literature. Simulation results demonstrate that the proposed symmetric codes have competitive performance in comparison with similar existing QC-LDPC codes that lack symmetry.

LDPC lattices were the first family of lattices which have an efficient decoding algorithm in high dimensions over an AWGN channel. When we consider Construction D' of lattices with one binary LDPC code as its underlying code, 1-level LDPC lattices are obtained. Block fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. In this type of channel, a lattice point is divided into multiple blocks such that fading is constant within a block but changes, independently, across blocks. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. In this paper we construct full diversity 1-level LDPC lattices for block fading channels. We propose a new iterative decoding method for these family of lattices which has complexity that grows linearly in the dimension of lattice.

In this paper, we study the cycle structure of quasi-cyclic (QC) low-density parity-check (LDPC) codes with the goal of obtaining the shortest code with a given degree distribution and girth. We focus on QC-LDPC codes, whose Tanner graphs are cyclic liftings of fully connected base graphs of size 3 × n, n ≥ 4, and obtain minimal lifting degrees that result in girths 6 and 8. This is performed through an efficient exhaustive search, and as a result, we also find all the possible non-isomorphic codes with the same minimum block length, girth, and degree distribution. The exhaustive search, which is ordinarily a formidable task, is made possible by pruning the search space of many codes that are isomorphic to those previously examined in the search process. Many of the pruning techniques proposed in this paper are also applicable to QC-LDPC codes with base graphs other than the 3 × n fully connected ones discussed here, as well as to codes with a larger girth. To further demonstrate the effectiveness of the pruning techniques, we use them to search for QC-LDPC codes with girths 10 and 12, and find a number of such codes that have a shorter block length compared with the best known similar codes in the literature. In addition, motivated by the exhaustive search results, we tighten the lower bound on the block length of QC-LDPC codes of girth 6 constructed from fully connected 3 × n base graphs, and construct codes that achieve the lower bound for an arbitrary value of n ≥ 4.

In this paper we present Grobner bases for lattices given in a general form, including integer and non-integer lattices. Grdot{o}bner bases for binary linear codes were introduced by Borges-Quintana et al. . We extend their work to non-binary group block codes. Then, given a lattice Λ and its associated label code L, which is a group code, we define an ideal for L. A Grobner basis is assigned to Λ as the Grobner basis of its label code L. Since the associated label code for integer and non-integer lattices are group codes, the assigned Grobner bases can be obtained for both cases. Using this Grobner basis an algebraic decoding algorithm is introduced. We provide an example of the decoding method for a lower dimension lattice. We explain that the complexity of this decoding method depends on the division algorithm and show this decoding method has polynomial time complexity. Experiments for some versions of root lattices (E_7 and E_8) show that for low SNR the performance of these lattices is near to the lower bounds given in .

Let p be a prime number. We propose a new method for solving sparse linear systems of equations modulo p when the coefficient matrix has column degree at most 2. This algorithm is based on a well-known decoding algorithm for Low-Density Parity-Check (LDPC) codes called bit-flipping (BF) algorithm. We modify and extend this hard-decision decoding algorithm. The complexity of this algorithm is linear in terms of the number of columns n and the number of nonzero coefficients u of the matrix. We give a detailed small example, and report on computational results for larger systems.

Low-density parity-check codes (LDPC) can have an impressive performance under iterative decoding algorithms. In this paper we introduce a method to construct high coding gain lattices with low decoding complexity based on LDPC codes. To construct such lattices we apply Construction D', due to Bos, Conway, and Sloane, to a set of parity checks defining a family of nested LDPC codes. For the decoding algorithm, we generalize the application of max-sum algorithm to the Tanner graph of lattices. Bounds on the decoding complexity are derived and our analysis shows that using LDPC codes results in low decoding complexity for the proposed lattices. The progressive edge growth (PEG) algorithm is then extended to construct a class of nested regular LDPC codes which are in turn used to generate low density parity check lattices. Using this approach, a class of two-level lattices is constructed. The performance of this class improves when the dimension increases and is within 3 dB of the Shannon limit for error probabilities of about 10-6. This is while the decoding complexity is still quite manageable even for dimensions of a few thousands