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.

We study the relation between near resolvable designs and ${\mathbb{F}_q}$ -linear codes over $\mathbb{F}_q^b$. We use the incidence matrix of a near resolvable design to construct a parity-check matrix of an ${\mathbb{F}_q}$-linear code. We show an equivalence between the construction of a new class of near resolvable designs NRB(rb + 1, r), that we call r-complete, and the well-known ${\mathbb{F}_q}$-linear codes over $\mathbb{F}_q^b$ with length n and dimension n – r which have the following good properties: (i) they are maximum distance separable (MDS), (ii) they are low-density, and (iii) they reach the maximum length of any MDS lowest density code.

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.

Let [ n { = {1, …, n } and let Ω n be the set of all mappings from [ n { to itself. Let f be a random uniform element of Ω n and let T( f ) and B( f ) denote, respectively, the least common multiple and the product of the length of the cycles of f . Harris proved in 1973 that T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for random uniform mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k ( r ) ≥ 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x k + a over the integers modulo p , with p ≡ 1 (mod k). We exhibit and discuss our numerical results on this heuristic.

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.

In this paper, we describe an image encryption scheme based on the multiple-parameter fractional number-theoretic transform (MFrNTT). In order to define the MFrNTT used in our proposal, we introduce a systematic procedure to construct a number-theoretic transform eigenbasis from chosen parameters. Our results indicate that the proposed scheme can resist to cryptographical attacks usually considered in the related literature; at the same time, our method is very competitive in terms of encryption/decryption speed when compared with other existing approaches. We also demonstrate that our scheme can be easily adjusted to deal with different types of images and that it involves a larger number of free parameters than other NTT-based image encryption techniques.

The qtx(q+1)tordered orthogonal arrays (OOAs) of strength t over the alphabet Fq were constructed using linear feedback shift register sequences (LFSRs) defined by primitive polynomials in Fq[x]. In this paper, we extend this result to all polynomials in Fq[x] which satisfy some fairly simple restrictions, i.e., the restrictions that are automatically satisfied by primitive polynomials. While these restrictions sometimes reduce the number of columns produced from (q + 1)t to a smaller multiple oft, in many cases, we still obtain the maximum number of columns in the constructed OOA when using non-primitive polynomials. For 2 ≤ q ≤ 9 and small t, we generate OOAs in this manner for all permissible polynomials of degree t in Fq[x] and compare the results to the ones produced in [2], [16], and [17] showing how close the arrays are to being “full” orthogonal arrays. Unusually for the finite fields, our arrays based on the non-primitive irreducible and even reducible polynomials are closer to the orthogonal arrays than those built from the primitive polynomials.

Let f be a uniformly random element of the set of all mappings from [n] = {1, ..., n} to itself. Let T(f) and B(f) denote, respectively, the least common multiple and the product of the lengths of the cycles of f. Harris proved in 1973 that log T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for uniform random mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k(r) >= 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x^k + a over the integers modulo p, where k divides p - 1. We exhibit and discuss our numerical results on this heuristic.

A generating matrix is a matrix such that, when multiplied by an eigenvector of a discrete transform, a new eigenvector is obtained. In this paper, we introduce a family of generating matrices of DFT eigenvectors. We demonstrate that, if a specific initial set of eigenvectors is chosen, using the referred family of matrices, a Hermite-Gaussian-like DFT eigenbasis is obtained. Such an eigenbasis is then employed to define a discrete fractional Fourier transform which numerically approximates the corresponding continuous transform.

Shamir's Secret Sharing Scheme is well established and widely used. It allows a so-called Dealer to split and share a secret k among n Participants such that at least t shares are needed to reconstruct k, where 0 <; t ≤ n. Nothing about the secret can be learned from less than t shares. To split secret k, the Dealer generates a polynomial f, whose independent term is k and the coefficients are randomly selected using a uniform distribution. A share is a pair (x, f(x)) where x is also chosen randomly using a uniform distribution. This scheme is useful, for example, to distribute cryptographic keys among different cloud providers and to create multi-factor authentication. The security of Shamir's Secret Sharing Scheme is usually analyzed using a threat model where the Dealer is trusted to split and share secrets as described above. In this paper, we demonstrate that there exists a different threat model where a malicious Dealer can compute shares such that a subset of less than t shares is allowed to reconstruct the secret. We refer to such subsets as hidden sets. We formally define hidden sets and prove lower bounds on the number of possible hidden sets for polynomials of degree t - 1. Yet, we show how to detect hidden sets given a set of n shares and describe how to create hidden sets while sharing a secret using a modification of Shamir's scheme.

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.

We present a new construction of ordered orthogonal arrays (OOAs) of strength t with (q + 1)t columns over a finite field Fqusing linear feedback shift register sequences (LFSRs). OOAs are naturally related to (t, m, s)-nets, linear codes, and MDS codes. Our construction selects suitable columns from the array formed by all subintervals of length (qt-1)/(q-1) of an LFSR sequence generated by a primitive polynomial of degree t over Fq. We prove properties about the relative positions of runs in an LFSR, which guarantee that the constructed OOA has strength t. The set of parameters of our OOAs are the same as the ones given by Rosenbloom and Tsfasman (1997) and Skriganov (2002), but the constructed arrays are different. We experimentally verify that our OOAs are stronger than the Rosenbloom-Tsfasman-Skriganov OOAs in the sense that ours are “closer” to being a “full” orthogonal array. We also discuss how our OOA construction relates to previous techniques to build OOAs from a set of linearly independent vectors over Fq, as well as to hypergraph homomorphisms.

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.

We describe the functional graph of the multiplication-by-$n$ map in a cycle group and use this to obtain the structure of the functional graph associated with a Rédei function over a nonbinary finite field $\mathbb{F}_q$. In particular, we obtain two descriptions of the tree attached to the cyclic nodes in these graphs and provide period and preperiod estimates for Rédei functions. We also extend characterizations of Rédei permutations by describing their decomposition into disjoint cycles. Finally, we obtain some results on the length of the cycles related to Rédei permutations and we give an algorithm to construct Rédei permutations with prescribed length cycles in a geometric progression.

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 .

The concepts of ambiguity and deficiency for a bijection on a finite Abelian group were recently introduced. In this paper, we present some further fundamental results on the ambiguity and deficiency of functions; in particular, we note that they are invariant under the well-known Carlet-Charpin-Zinoviev-equivalence, we obtain upper and lower bounds on the ambiguity and deficiency of differentially k-uniform functions, and we give a lower bound on the nonlinearity of functions that achieve the lower bound of ambiguity and deficiency. In addition, we provide an explicit formula in terms of the ranks of matrices on the ambiguity and deficiency of a Dembowski-Ostrom (DO) polynomial, and using this technique, we find exact values for known cases of DO permutations with few terms. We also derive exact values for the ambiguities and deficiencies of DO permutations obtained from trace functions. The key relationship between the above polynomials is that they all have linearized difference map.

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.

The concepts of ambiguity and deficiency for a given bijection on a finite Abelian group were recently introduced [13]. In this work we investigate the ambiguity and deficiency of some well-known polynomials which satisfy Dn(x+y, xy) = xn+ynfor every x, y ϵ Fqand n ϵ N, as well as linearized polynomials and Dembowski-Ostrom polynomials (DO polynomials). For some specific values of n (related to q) these polynomials generate permutations on Fq. We derive explicitly the ambiguity and deficiency of some of them. Numerical results on the ambiguity and deficiency of the others are also provided. Some of these polynomials are almost perfect nonlinear (APN) functions.

We introduce the concepts of weighted ambiguity and deficiency for a mapping between two finite Abelian groups of the same size. Then, we study the optimum lower bounds of these measures for permutations of an Abelian group. A construction of permutations, by modifying some permutation functions over finite fields, is given. Their ambiguity and deficiency is investigated; most of these functions are APN permutations. We show that, when they are not optimal, the Möbius function in the multiplicative group of \BBFqis closer to being optimal in ambiguity than the inverse function in the additive group of \BBFq. We note that the inverse function over \BBF28is used in AES. Finally, we conclude that a twisted permutation polynomial of a finite field is again closer to being optimal in ambiguity than the APN function employed in the SAFER cryptosystem.

Quickselect with median-of-3 is largely used in practice and its behavior is fairly well understood. However, the following natural adaptive variant, which we call proportion-from-3 , had not been previously analyzed: “choose as pivot the smallest of the sample if the relative rank of the sought element is below 1/3, the largest if the relative rank is above 2/3, and the median if the relative rank is between 1/3 and 2/3.” We first analyze the average number of comparisons made when using proportion-from-2 and then for proportion-from-3. We also analyze ν-find, a generalization of proportion-from-3 with interval breakpoints at ν and 1-ν. We show that there exists an optimal value of ν and we also provide the range of values of ν where ν-find outperforms median-of-3. Then, we consider the average total cost of these strategies, which takes into account the cost of both comparisons and exchanges. Our results strongly suggest that a suitable implementation of ν-find could be the method of choice in a practical setting. We also study the behavior of proportion-from- s with s >3 and in particular we show that proportion-from- s -like strategies are optimal when s →∞.

In this paper, we present a new method for multiplying polynomials in Chebyshev form. Our approach has two steps. First, the well-known Karatsuba's algorithm is applied to polynomials constructed by using Chebyshev coefficients. Then, from the obtained result, extra arithmetic operations are used to write the final result in Chebyshev form. The proposed algorithm has a quadratic computational complexity. We also compare our method to other approaches.

In this paper, we define and study error locating arrays (ELAs), which can be used in software testing for locating faulty interactions among parameters or components in a system. We give constructions of ELAs for arbitrary strength t, based on covering arrays. We show that the number of tests given by ELAs grows as $O(\log k)$, where k is the number of parameters/components in the system, assuming other quantities (the number g of values per parameter, the strength t of faulty interactions, and the number d of faulty interactions) are bounded by a constant. We then give a series of results for the case of pairwise interactions ($t=2$). We study the computational complexity of deciding whether a graph describing the faulty pairwise interactions is “locatable.” We characterize the locatable graphs for the binary case ($g=2$). We design and analyze efficient algorithms that locate errors under certain assumptions on the structure of the faulty pairwise interactions. Under the assumption of known “safe values,” our algorithm performs a number of tests that is polynomial in $\log k$ and d, where k is the number of parameters in the system and d is an upper bound on the number of faulty pairwise interactions. For the binary alphabet case, we provide an algorithm that does not require safe values and runs in expected polynomial time in $\log k$ whenever $d\in O(\log\log k)$.

In this paper, a new definition of Chebyshev polynomials over prime finite fields is introduced. Our approach uses a finite field trigonometry and reveals some aspects concerning the security of a recently proposed public-key encryption algorithm based on those polynomials. Particularly, we show that recovering the corresponding plaintext from a given ciphertext involves the discrete logarithm problem.

In this paper, we extend previously known results on the complexities of normal elements. Using algorithms that exhaustively test field elements, we are able to provide the distribution of the complexity of normal elements for binary fields with degree extensions up to 39. We also provide current results on the smallest known complexity for the remaining degree extensions up to 512 by using a combination of constructive theorems and known exact values. We give an algorithm to exhaustively search field elements by using Gray codes, which allows us to reuse previous computations. We compare this with a standard method. We analyze this algorithm and show both experimentally and asymptotically that the Gray code optimization gives substantial savings. The total computation of the distribution of the complexity of normal elements for degrees up to 39 in our experiments allows us to draw several conjectures. In particular, our data provides remarkable evidence for the conjecture that the complexity of normal elements follows a normal distribution. Finally, we conjecture that there is no linear bound on the minimum complexity with respect to the degree of the extension.

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

The orthogonal matching pursuit (OMP) algorithm is an adaptive nonlinear algorithm for signal decomposition using an overcomplete dictionary. A tree-search based orthogonal matching pursuit (TB-OMP) has been proposed (Cotter et al. (2001)). Although the TB-OMP algorithm improves the approximation performance, its computation time requirement increases exponentially making the algorithm impractical for certain applications. In this paper, we propose the flexible tree-search based orthogonal matching pursuit (FTB-OMP). The algorithm provides design parameters that give flexibility to establish a tradeoff between approximation performance and experimental time complexity. Sparse signal representations are frequently required in problems related to signal processing and communication areas. The proposed FTB-OMP algorithm is a promising solution for such problems.

In communications system design, it is frequently assumed that source symbols are equiprobable. However, in real life applications, this is not the case since most sources produce Gaussian samples. In this paper, we introduce a Karhunen Loeve transform (KLT) based integer to integer transform, I/sub 2/I KLT, over GF(q) that forces the symbols to uniform distribution. This transform can be used as an interface between sources with different distributions and communication systems designed according to uniform distributions.

In order to exploit word-oriented operations for linear-feedback shift registers (LFSRs), Tsaban and Vishne [2002] introduced the notion of linear transformation shift registers (TSRs). An implementation of their primitive TSR generating algorithm shows that the LFSR are paired for all transformations. We prove that the characteristic polynomials of a pair of LFSRs are either both irreducible or both reducible for all transformations. This allows some time improvement when finding primitive TSRs. The authors give a full enumeration of all primitive TSRs with transformations of order 8 and LFSRs of order 3, 4, 5, and 6.