Fast interpolation algorithms are presented for sparse sums of characters on the product monoid U(n) with values in a field K in two special cases. In the first case, U is a finite cyclic group of order e, and K is a ...
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Fast interpolation algorithms are presented for sparse sums of characters on the product monoid U(n) with values in a field K in two special cases. In the first case, U is a finite cyclic group of order e, and K is a field that contains a root of unity of order e. Here linear codes over Z/eZ can be used to construct sets of evaluation points that allow efficient interpolation by decoding Reed-Solomon codes. In the second case, K is the binary field GF(2), and U is the multiplicative monoid of GF(2). Here sparse sums of characters coincide with sparse Boolean polynomials and can be interpolated using the smallest set of evaluation points by decoding Reed-Muller codes.
We have considered a class of square Fibonacci matrix of order (p + 1) whose elements are based on the Fibonacci p numbers with determinant equal to +1 or -1. There is a relation between Fibonacci numbers with initial...
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We have considered a class of square Fibonacci matrix of order (p + 1) whose elements are based on the Fibonacci p numbers with determinant equal to +1 or -1. There is a relation between Fibonacci numbers with initial terms which is known as cassini formula. Fibonacci series and the golden mean plays a very important role in the construction of a relatively new space-time theory, which is known as E-infinity theory. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. There already exists a relation between the code matrix elements for the case p = 1 [Stakhov AP. Fibonacci matrices, a generalization of the cassini formula and a new coding theory. Chaos, Solitons and Fractals 2006;30:56-66.]. In this paper, we have established generalized relations among the code matrix elements for all values of p. For p = 2, the correct ability of the method is 99.80%. In general, correct ability of the method increases as p increases. (C) 2008 Elsevier Ltd. All rights reserved.
This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If RG is a group ring, where R is commutative and S is a set of generators of G then necessary an...
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This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If RG is a group ring, where R is commutative and S is a set of generators of G then necessary and sufficient conditions on a map from S to RG are established, such that the map can be extended to an R-derivation of RG. Derivations are shown to be trivial for semisimple group algebras of abelian groups. The derivations of finite group algebras are constructed and listed in the commutative case and in the case of dihedral groups. In the dihedral case, the inner derivations are also classified. Lastly, these results are applied to construct well known binary codes as images of derivations of group algebras. (C) 2018 The Authors. Published by Elsevier Inc.
A powerful strategy for the classification of multiple classes is to create a classifier ensemble that decomposes the polychotomy into several dichotomies. The central issue when designing a multiclass-to-binary decom...
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A powerful strategy for the classification of multiple classes is to create a classifier ensemble that decomposes the polychotomy into several dichotomies. The central issue when designing a multiclass-to-binary decomposition scheme is the definition of both the coding matrix and the decoding algorithm. In this study, we propose a new classification system based on low-density parity-check codes, which is a very effective class of binary block codes. The main idea is to exploit the algebraic properties of the codes to generate the codewords for the coding matrix and to define two decoding approaches, which allow us to detect and recover possible errors or rejects produced by the dichotomizers. Experiments based on benchmark datasets demonstrated that the proposed approach provides a statistically significant improvement in terms of the classification performance compared with state-of-the-art decomposition strategies. (C) 2016 Elsevier Inc. All rights reserved.
Masking is one of the most popular countermeasures to protect cryptographic implementations against side-channel analysis since it is provably secure and can be deployed at the algorithm level. To strengthen the origi...
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Masking is one of the most popular countermeasures to protect cryptographic implementations against side-channel analysis since it is provably secure and can be deployed at the algorithm level. To strengthen the original Boolean masking scheme, several works have suggested using schemes with high algebraic complexity. The Inner Product Masking (IPM) is one of those. In this paper, we propose a unified framework to quantitatively assess the side-channel security of the IPM in a coding-theoretic approach. Specifically, starting from the expression of IPM in a coded form, we use two defining parameters of the code to characterize its side-channel resistance. In order to validate the framework, we then connect it to two leakage metrics (namely signal-to-noise ratio and mutual information, from an information-theoretic aspect) and one typical attack metric (success rate, from a practical aspect) to build a firm foundation for our framework. As an application, our results provide ultimate explanations on the observations made by Balasch et al. at EUROCRYPT'15 and at ASIACRYPT'17, Wang et al. at CARDIS'16 and Poussier et al. at CARDIS'17 regarding the parameter effects in IPM, like higher security order in bounded moment model. Furthermore, we show how to systematically choose optimal codes (in the sense of a concrete security level) to optimize IPM by using this framework. Eventually, we present a simple but effective algorithm for choosing optimal codes for IPM, which is of special interest for designers when selecting optimal parameters for IPM.
Galois geometries and coding theory are two research areas which have been interacting with each other for many decades. From the early examples linking linear MDS codes with arcs in finite projective spaces, linear c...
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Galois geometries and coding theory are two research areas which have been interacting with each other for many decades. From the early examples linking linear MDS codes with arcs in finite projective spaces, linear codes meeting the Griesmer bound with minihypers, covering radius with saturating sets, links have evolved to functional codes, generalized projective Reed-Muller codes, and even further to LDPC codes, random network codes, and distributed storage. This article reviews briefly the known links, and then focuses on new links and new directions. We present new results and open problems to stimulate the research on Galois geometries, coding theory, and on their continuously developing and increasing interactions.
An integer-valued function f (x) on the integers that is periodic of period p(e), p prime, can be matched, modulo p(m), by a polynomial function w(x);we show that w(x) may be taken to have degree at most (m (p - 1) + ...
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An integer-valued function f (x) on the integers that is periodic of period p(e), p prime, can be matched, modulo p(m), by a polynomial function w(x);we show that w(x) may be taken to have degree at most (m (p - 1) + 1) p(e-1) - 1. Applications include a short proof of the theorem of McEliece on the divisibility of weights of codewords in p-ary cyclic codes by powers of p, an elementary proof of the Ax-Katz theorem on solutions of congruences modulo p, and results on the numbers of codewords in p-ary linear codes with weights in a given congruence class modulo p(e). (c) 2006 Elsevier B.V. All rights reserved.
This paper focuses on rough approximation operators in group mapping. The relationships between rough set theory and group theory are considered from a novel perspective. The necessary and sufficient conditions for th...
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This paper focuses on rough approximation operators in group mapping. The relationships between rough set theory and group theory are considered from a novel perspective. The necessary and sufficient conditions for the upper approximation and lower approximation of a group to be groups are analyzed. In addition, the homomorphism and isomorphism between two groups which have related upper or lower approximations are investigated. Finally, the applications of rough approximation operators in group mapping to coding theory are developed.
We consider the compressive sensing of a sparse or compressible x is an element of R-M signal. We explicitly construct a class of measurement matrices inspired by coding theory, referred to as low density frames, and ...
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We consider the compressive sensing of a sparse or compressible x is an element of R-M signal. We explicitly construct a class of measurement matrices inspired by coding theory, referred to as low density frames, and develop decoding algorithms that produce an accurate estimate (x) over cap even in the presence of additive noise. Low density frames are sparse matrices and have small storage requirements. Our decoding algorithms can be implemented in O(Md-v(2)) complexity, where d(v) is the left degree of the underlying bipartite graph. Simulation results are provided, demonstrating that our approach outperforms state-of-the-art recovery algorithms for numerous cases of interest. In particular, for Gaussian sparse signals and Gaussian noise, we are within 2-dB range of the theoretical lower bound in most cases.
We consider a new class of square Fibonacci (p + 1) x (p + 1)-matrices, which are based on the Fibonacci p-numbers (p = 0, 1, 2, 3....), with a determinant equal to + 1 or - 1. This unique property leads to a generali...
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We consider a new class of square Fibonacci (p + 1) x (p + 1)-matrices, which are based on the Fibonacci p-numbers (p = 0, 1, 2, 3....), with a determinant equal to + 1 or - 1. This unique property leads to a generalization of the "Cassini formula" for Fibonacci numbers. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. In contrast to classical redundant codes a basic peculiarity of the method is that it allows to correct matrix elements that can be theoretically unlimited integers. For the simplest case the correct ability of the method is equal 93.33 % which exceeds essentially all well-known correcting codes. (c) 2006 Elsevier Ltd. All rights reserved.
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