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A proper labeling of a graph is an assignment of integers to some elements of a graph, which may be the vertices, the edges, or both of them, such that we obtain a proper vertex coloring via the labeling subject to some conditions. The problem of proper labeling offers many variants and received a great interest during recent years. We consider the algorithmic complexity of some variants of the proper labeling problems, we present some polynomial time algorithms and $ \mathbf{NP} $-completeness results for them

LDPC lattices were the first family of lattices that equipped with iterative decoding algorithms under which they perform very well in high dimensions. In this paper, we introduce quasi cyclic low density parity check (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 Construction A of lattices providing us to encode them efficiently using shift registers. To benefit from an encoder with linear complexity in dimension of the lattice, we obtain the generator matrix of these lattices in quasi cyclic form. We provide a low-complexity decoding algorithm of QC-LDPC lattices based on sum product algorithm. To design lattice codes, QC-LDPC lattices are combined with nested lattice shaping that uses the Voronoi region of a sublattice for code shaping. The shaping gain and shaping loss of our lattice codes with dimensions 40, 50 and 60 using an optimal quantizer, are presented. Consequently, we establish a family of lattice codes that perform practically close to the sphere bound.

In this paper, we define two matrices named as "difference matrices", denoted by D and DD which significantly contribute to achieve regular single-edge QC-LDPC codes with the shortest length and the certain girth as well as regular and irregular multiple-edge QC-LDPC codes.Making use of these matrices, we obtain necessary and sufficient conditions to have single-edge (m, n)-regular QC-LDPC codes with girth 6 by which we achieve all non-isomorphic codes with the minimum lifting degree, N , for m = 4 and 5 ≤ n ≤ 11, and present an exponent matrix for each minimum distance. We attain the necessary and sufficient conditions to have a Tanner graph with girth 10. In this case we also provide a lower bound on the lifting degree which is tighter than the existing bound. More important, for an exponent matrix whose first row and first column are all-zero, we demonstrate that the non-existence of 8-cycles proves the non-existence of 6-cycles related to the first row of the exponent matrix too. All non-isomorphic QC-LDPC codes with girth 10 and n = 5, 6 whose numbers are more than those presented in the literature are provided. For n = 7, 8 we decrease the lifting degrees from 159 and 219 to 145 and 211, repectively. Moreover, necessary and sufficient conditions to have (m, n)-regular QC-LDPC codes with girth 12 as well as a lower bound on the lifting degree are achieved.For multiple-edge category, for the first time a lower bound on the lifting degree for both regular and irregular QC-LDPC codes with girth 6 is achieved. We analytically prove that if B ij is ij-th element of an m×n exponent matrix B then by taking three values A = max{2 n j=1 | Bij | 2 ; i = 1, 2 . . . , m}, B =

In this work we introduce and establish the concept of turbo lattices. We employ a routine method for constructing lattices, called Construction D, to construct turbo lattices. This kind of construction needs a set of nested linear codes as its underlying structure. We benefit from turbo codes as our bases codes. Therefore, we first build a set of nested turbo codes based on nested interleavers and nested convolutional codes. Then by means of these codes, along with construction D, we construct turbo lattices. Several properties of Construction D lattices and specially many characteristics of turbo lattices including the minimum distance, coding gain, kissing number and the probability of error under a maximum likelihood decoder over AWGN channel are investigated.

The Internet of Things (IoT) is an emerging technology that can benefit from cloud infrastructure. In a cloud-based IoT network, a variety of data is collected by smart devices and transmitted to a cloud server. However, since the data may contain sensitive information about individuals, providing confidentiality and access control is essential to protect the users' privacy. Attribute-based encryption (ABE) is a promising tool to provide these requirements. However, most of ABE schemes neither provide efficient encryption and decryption mechanisms nor offer flexible and efficient key delegation and user revocation approaches. In this paper, to address these issues, we propose a lightweight revocable hierarchical ABE (LW-RHABE) scheme. In our scheme, computation overhead on the user side is very efficient, and most of the computational operations are performed by the cloud server. Also, using the hierarchical model, our scheme offers flexible and scalable key delegation and user revocation mechanisms. Indeed, in our scheme, key delegation and user revocation associated with each attribute can be handled by several key authorities. We provide the security definition for LW-RHABE, and we prove its security in the standard model and under the hardness assumption of the decisional bilinear Diffie-Hellman (DBDH) problem. INDEX TERMS Internet of Things, cloud computing, fine-grained access control, attribute-based encryption, light weight computation.