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In this article, we are going to discuss about how alternant codes over Eisenstein integers are crucial in today’s technologies especially in encoding data and correction of those data. Based on the parity check matrices of alternant codes, the codes are constructed with symbols from finite commutative local rings with unity of Eisenstein integers modulo n. Particularly, the factorization of x n - 1 is over the unit-ring of an appropriate extension of the finite local commutative ring of Eisenstein integers. These codes are built similarly to the BCH codes over the finite integer rings and provide a solid background for the process of encoding the data. This work further proved the applicability of the said algorithm for the purpose of error correction in the discussed alternant codes in order to improve transference effectiveness. Through analyzing the mathematical concepts behind as well as practical applications of alternant codes over Eisenstein integers the possibility of changing today’s technological setting is disclosed. The paper demonstrates that the purpose of the codes is to enhance data integrity and security and underlines that this is why they should be useful for modern communication systems and much more. [ABSTRACT FROM AUTHOR]

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This paper proposes reliability-based decoding for complex low-density lattice codes (CLDLC) which can be applied to both Gaussian and Eisenstein integers. Two major contributions are: first, a decoding algorithm for CLDLC using a likelihood-based reliability function is used to determine the number of complex Gaussian functions at the variable node. This allows each message to be approximated by a variable number of Gaussian functions depending upon its reliability. An upper bound on the Kullback-Leibler (KL) divergence of the approximation is formed to find selection thresholds via linear regression. Second, a construction of CLDLC using Eisenstein integers is given. Compared to Gaussian integers, this reduces the complexity of CLDLC decoding by exploiting the structure of the Eisenstein integers. The proposed CLDLC decoding algorithm has higher performance and lower complexity compared to existing algorithms. When the reliability-based algorithm is applied to Eisenstein integer CLDLC decoding, the complexity is reduced to $\mathcal {O}(n\cdot t \cdot 1.35^{d-1})$ at the volume-to-noise ratio of 6 dB, for lattice dimension n, with degree d inverse generator matrix and t decoding iterations. Decoding CLDLC using Eisenstein integers has lower complexity than CLDLC using Gaussian integers when $n \geq 49$ .

First, we will go through the theory behind the Eisenstein field (EF) and its extension field. In contrast, we provide a detailed framework for building BCH codes over the EF in the second stage. BCH codes over the EF are decoded using the Berlekamp-Massey algorithm (BMA) in this article. We investigate the error-correcting capabilities of these codes and provide expressions for minimal distance. We provide researchers and engineers creating and implementing robust error-correcting codes for digital communication systems with detailed information on building, decoding and performance assessment.

In block ciphers, the nonlinear components, also known as substitution boxes (S-boxes), are used with the purpose to induce confusion in cryptosystems. For the last decade,most of the work on designing S-boxes over the points of elliptic curves, chaotic maps, and Gaussian integers has been published. The main purpose of these studies is to hide data and improve the security levels of crypto algorithms. In this work, we design pair of nonlinear components of a block cipher over the residue class of Eisenstein integers (EI). The fascinating features of this structure provide S-boxes pair at a time by fixing three parameters. However, in the same way, by taking three fixed parameters only one S-box is obtained through a prime field-dependent Elliptic curve (EC), chaotic maps, and Gaussian integers. The newly designed pair of S-boxes are assessed by various tests like nonlinearity, bit independence criterion, strict avalanche criterion, linear approximation probability, and differential approximation probability. [ABSTRACT FROM AUTHOR]

First, we will go through the theory behind the Eisenstein field (EF) and its extension field. In contrast, we provide a detailed framework for building BCH codes over the EF in the second stage. BCH codes over the EF are decoded using the Berlekamp-Massey algorithm (BMA) in this article. We investigate the error-correcting capabilities of these codes and provide expressions for minimal distance. We provide researchers and engineers creating and implementing robust error-correcting codes for digital communication systems with detailed information on building, decoding and performance assessment. [ABSTRACT FROM AUTHOR]

Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are considered. However, in communications, complex-valued processing is usually of interest. Besides, by the use of dual-polarized transmission as well as by the combination of two time slots or frequencies, four-dimensional (quaternion-valued) approaches become more and more important. Hence, to account for this fact, well-known lattice algorithms and related concepts are generalized in this work. To this end, a brief review of complex arithmetic, including the sets of Gaussian and Eisenstein integers, and an introduction to quaternion-valued numbers, including the sets of Lipschitz and Hurwitz integers, are given. On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice by performing a special variant of the Euclidean algorithm defined for matrices, and second, of an algorithm to calculate the successive minima—the norms of the shortest independent vectors of a lattice—and its related lattice points. Generalized bounds for the quality of the particular results are established and the asymptotic complexities of the algorithms are assessed. These findings are extensively compared to conventional real-valued processing. It is shown that the generalized approaches outperform their real-valued counterparts in complexity and/or quality aspects. Moreover, the application of the generalized algorithms to MIMO communications is studied, particularly in the field of lattice-reduction-aided and integer-forcing equalization. [ABSTRACT FROM AUTHOR]

In this paper, a new MIMO scheme termed as precoded Spatially Modulated Non-orthogonal Space Time Block Code (precoded SM-NSTBC) is proposed. The primary concept of precoded SM-NSTBC is to activate a subset of receive antennas in a pre-defined manner and choose specific activated patterns to represent information symbols. We have synthesized schemes derived from full rank Cyclic codes defined over Galois field Rank preserving transformations are used to map the full rank codewords over a finite field to full rank Space Time Block Codes. Due to the characteristics of full rank Cyclic codes employed, a performance improvement of approximately 2 dB to 7 dB is observed. This advantage is realized when the performance of these schemes is compared with precoded SM-OSTBC and precoded STBC-SM. The improvement due to the coding gain is observed in both uncorrelated as well as correlated Rayleigh fading environments. An upper bound on the average bit error rate (ABER) is derived. Close correspondence between Monte-Carlo simulations and analytic values are observed. [ABSTRACT FROM AUTHOR]

We define a Mendelsohn triple system (MTS) of order coprime with 3, and having multiplication affine over an abelian group, to be affine, nonramified. By exhibiting a one‐to‐one correspondence between isomorphism classes of affine MTS and those of modules over the Eisenstein integers, we solve the isomorphism problem for affine, nonramified MTS and enumerate these isomorphism classes (extending the work of Donovan, Griggs, McCourt, Opršal, and Stanovský). As a consequence, all entropic MTSs of order coprime with 3 and distributive MTS of order coprime with 3 are classified. Partial results on the isomorphism problem for affine MTS with order divisible by 3 are given, and a complete classification is conjectured. We also prove that for any affine MTS, the qualities of being nonramified, pure, and self‐orthogonal are equivalent. [ABSTRACT FROM AUTHOR]

NTRU is a lattice-based cryptosystem built on a convolutional ring of polynomials. There are many generalizations of NTRU in the literature; however, group ring NTRU, or GR-NTRU, is the most reasonable description of NTRU as a general framework to design its variants. Most versions are commutative and are obtained by changing the ring of coefficients while keeping the cyclic structure intact. In this work, we analyze the noncommutative version of GR-NTRU designed with the group ring of dihedral group over the ring of Eisenstein integers. We experimentally test the size of the keyspace of this new variant and find that it is comparable and even larger for certain parameters compared to the existing commutative counterpart. We observe that although it is slightly slow in terms of speed of encryption and decryption, it has higher lattice security. [ABSTRACT FROM AUTHOR]

In this paper, we propose two new signal constellation designs employing Gaussian and Eisenstein Integers for Enhanced Spatial Modulation (ESM). ESM is a novel technique which was propounded by Cheng et al. The advantage of ESM over other Spatial Modulation (SM) schemes lies in its ability to enhance spectral efficiency while keeping the energy efficiency intact. This is done by activating either one or two antennas judiciously depending upon the required trade-off. In ESM, information radiated from the antennas depends upon index of the active transmit antenna combination(s) and also on the set of constellation points chosen, which may include points from multiple constellations. In this paper, we propose signal constellations based on multiplicative groups of Gaussian and Eisenstein integers. The set comprising of Gaussian and Eisenstein integers serves as primary and secondary constellation points for Gaussian Enhanced Spatial Modulation (GESM) scheme. The secondary constellation points are deduced from a single geometric interpolation from the primary constellation points. The Monte Carlo simulation results indicate that the proposed nonuniform constellations achieve impressive SNR gains compared to conventional constellation points used in the design of ESM. This new design has been described for MIMO employing 4 × 4 and 8 × 8 antenna configurations with only two active antennas. Furthermore, a closed form expression for the pairwise error probability (PEP) for the GESM scheme has been deduced. The PEP is utilized to determine the upper bound on the average bit error probability (ABEP). Our simulations indicate that the proposed GESM from Gaussian and Eisenstein integers scheme outperforms all the other variants of SM including conventional ESM by at least 2.5 dB at an average bit error ratio (ABER) of 10 − 5 . Close correspondence between the theoretical analysis and the Monte Carlo simulation results are observed. [ABSTRACT FROM AUTHOR]

This letter introduces signal constellations based on multiplicative groups of Eisenstein integers, i.e., hexagonal lattices. These sets of Eisenstein integers are proposed as signal constellations for generalized spatial modulation. The algebraic properties of the new constellations are investigated and a set partitioning technique is developed. This technique can be used to design coded modulation schemes over hexagonal lattices. [ABSTRACT FROM PUBLISHER]

Integer coefficient selection is an important decoding step in the implementation of compute-and-forward (C-F) relaying scheme. Choosing the optimal integer coefficients in C-F has been shown to be a shortest vector problem, which is known to be NP-hard in its general form. Exhaustive search of the integer coefficients is only feasible in complexity for small number of users while approximation algorithms, such as Lenstra–Lenstra–Lovász lattice reduction algorithm, only find a vector within an exponential factor of the shortest vector. An optimal deterministic algorithm was proposed for C-F by Sahraei and Gastpar specifically for the real valued channel case. In this paper, we adapt their idea to the complex valued channel and propose an efficient search algorithm to find the optimal integer coefficient vectors over the ring of Gaussian integers and the ring of Eisenstein integers. A second algorithm is then proposed that generalizes our search algorithm to the integer-forcing multiple-input multiple-output (MIMO) C-F receiver. Performance and efficiency of the proposed algorithms are evaluated through simulations and theoretical analysis. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the use of lattice codes over Eisenstein integers for implementing a compute-and-forward protocol in wireless networks when channel state information is not available at the transmitter. We extend the compute-and-forward paradigm of Nazer and Gastpar to decoding Eisenstein integer combinations of transmitted messages at relays by proving the existence of a sequence of pairs of nested lattices over Eisenstein integers in which the coarse lattice is good for covering and the fine lattice can achieve the Poltyrev limit. Using this result, we show that both the outage performance and error-correcting performance of the nested lattice codebooks over Eisenstein integers surpass those of lattice codebooks over integers considered by Nazer and Gastpar with no additional computational complexity. [ABSTRACT FROM PUBLISHER]

NTRU is a public-key cryptosystem based on polynomial rings over $$\mathbb Z .$$ Replacing $$\mathbb Z $$ with the ring of Eisenstein integers yields ETRU. We prove through both theory and implementation that ETRU is faster and has smaller keys for the same or better level of security than does NTRU. [ABSTRACT FROM AUTHOR]

We describe a method to perform scalar multiplication on two classes of ordinary elliptic curves, namely E:y² = x³ + Ax in prime characteristic p ≡ 1 mod 4, and E:y² = x³ + B in prime characteristic p ≡ 1 mod 3. On these curves, the 4-th and 6-th roots of unity act as (computationally efficient) endomorphisms. In order to optimise the scalar multiplication, we consider a width-w-NAF (Non-Adjacent Form) digit expansion of positive integers to the complex base of τ, where τ is a zero of the characteristic polynomial x²-tx+p of the Frobenius endomorphism associated to the curve. We provide a precomputationless algorithm by means of a convenient factorisation of the unit group of residue classes modulo τ in the endomorphism ring, whereby we construct a digit set consisting of powers of subgroup generators, which are chosen as efficient endomorphisms of the curve. [ABSTRACT FROM AUTHOR]

Abstract: We classify all cyclotomic matrices over the Eisenstein and Gaussian integers, that is, all Hermitian matrices over the Eisenstein and Gaussian integers that have all their eigenvalues in the interval . [Copyright &y& Elsevier]

It is well known that a function can be decomposed uniquely into the sum of an odd and an even function. This notion can be extended to the unique decomposition into the sum of four functions - two of which are even and two odd. These four functions are eigenvectors of the Fourier Transform with four different eigenvalues. That is, the Fourier transformof each of the four components is simply that component multiplied by the corresponding eigenvalue. Some eigenvectors of the discrete Fourier transform of particular interest find application in coding, communication and imaging. Some of the underlying mathematics goes back to the times of Carl Friedrich Gauss. [ABSTRACT FROM AUTHOR]

Abstract: We give a computer-free proof of a theorem of Basak, describing the group generated by 16 complex reflections of order 3, satisfying the braid and commutation relations of the diagram. The group is the full isometry group of a certain lattice of signature over the Eisenstein integers . Along the way we enumerate the cusps of this lattice and classify the root and Niemeier lattices over . [Copyright &y& Elsevier]

Abstract: We prove lower bounds for the complexity of deciding several relations in imaginary, norm-Euclidean quadratic integer rings, where computations are assumed to be relative to a basis of piecewise-linear operations. In particular, we establish lower bounds for deciding coprimality in these rings, which yield lower bounds for gcd computations. In each imaginary, norm-Euclidean quadratic integer ring, a known binary-like gcd algorithm has complexity that is quadratic in our lower bound. [Copyright &y& Elsevier]

We consider three-variable analogues of the theta series of Borwein and Borwein. We prove various identities involving these theta series including a generalization of the cubic identity of Borwein and Borwein. [ABSTRACT FROM AUTHOR]

The RSA public-key encryption system of Rivest, Shamir, and Adelman can be broken if the modulus, R say, can be factorized. However, it is still not known if this system can be broken without factorizing R. A version of the RSA scheme is presented with encryption exponent e ≡ 3 (mod 6). For this modified version, the equivalence of decryption and factorization of R can be demonstrated. [ABSTRACT FROM AUTHOR]

A new class of non-binary Spatially Modulated Non-orthogonal Space Time Block Code designs (SM-NSTBC) has been proposed. These designs employ full rank, length n , (n | q m − 1 , m ≤ n) cyclic codes defined over G F (q m). The underlying cyclic code constructions have the property that the codewords when viewed as m × n matrices over G F (q) have rank equal to m (Full rank). These codes are punctured to yield m × m full rank matrices over G F (q). Rank preserving transformations are used to map the codewords of full rank codes over a finite field to full rank Space Time Block Codes. The proposed scheme can be generalized to handle any number of transmit antenna greater than two. Due to the characteristics of Full rank cyclic codes employed, a coding gain of approximately 1.5 dB to 5 dB is obtained over conventional STBC-SM and SM-OSTBC schemes. This is demonstrated for spectral efficiencies of 4, 5, 7 and 8 bpcu. Analytical as well as Monte-Carlo simulations show that proposed SM-NSTBC outperforms STBC-SM and its variants. The upper bound on average bit error rate has been derived and the computation complexity for ML detection has been estimated. [ABSTRACT FROM AUTHOR]