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On the key equation over a commutative ring

DESIGNS CODES AND CRYPTOGRAPHY 2000年第2期20卷 125-141页

作者： Norton, GH Salagean, A Univ Bristol Ctr Commun Res Algebra Coding Res Grp Bristol BS8 1TH Avon England Nottingham Trent Univ Dept Math Nottingham England

We define alternant codes over a commutative ring R and a corresponding key equation. We show that when the ring is a domain, e.g. the p-adic integers, the error-locator polynomial is the unique monic minimal polynomial (equivalently, the unique shortest linear recurrence) of the finite sequence of syndromes and that it can be obtained by algorithm MR of Norton. When R is a local ring, we show that the syndrome sequence may have more than one (monic) minimal polynomial, but that all the minimal polynomials coincide modulo the maximal ideal of R. We characterise the set of minimal polynomials when R is a Hensel ring. We also apply these results to decoding alternant codes over a local ring R: it is enough to find any monic minimal polynomial over R and to find its roots in the residue field. This gives a decoding algorithm for alternant codes over a finite chain ring, which generalizes and improves a method of Interlando et. al. for BCH and Reed-Solomon codes over a Galois ring.

关键词： domain local ring alternant code polynomial decoding algorithm error-locator

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Skew differential Goppa codes and their application to Mceliece cryptosystem

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DESIGNS CODES AND CRYPTOGRAPHY 2023年第12期91卷 3995-4017页

作者： Gomez-Torrecillas, Jose Lobillo, F. J. Navarro, Gabriel Univ Granada Dept Algebra Ave Fuentenueva S-N Granada 18071 Spain Univ Granada Dept Comp Sci & Artificial Intelligence Periodista Daniel Saucedo Aranda S-N Granada 18071 Spain Univ Granada IMAG Ventanilla 11 Granada 18001 Spain Univ Granada CITIC Periodista Rafael Gomez Montero 2 Granada 18071 Spain

A class of linear codes that extends classical Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem.

关键词： Skew-differential Goppa code decoding algorithm McEliece-type cryptosystem

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Modified BP decoding algorithms Combined with GA for Low-Density Parity Check Codes

Modified BP Decoding Algorithms Combined with GA for Low-Den...

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International Conference on Computational Intelligence and Security

作者： Zerong Deng Xingcheng Liu Man Teng Department of Electronics and Communication Engineering Sun Yat-Sen University

ISBN: (纸本)9781424441969

In this paper we propose algorithms which combine Belief-propagation (BP) decoding algorithm with Genetic algorithm (GA). The main idea is to improve the performance of traditional BP decoding algorithm by efficiently using the belief of variable nodes passing to check nodes. By considering the belief as genes, the proposed algorithms mainly employ the mutation operation in GA, and change the genes' amplitudes (either increase or decrease) when updating the check nodes' information. First, we explore that the marriage of the GA and BP decoding algorithm is reasonable. Three algorithms are proposed accordingly. Proposed algorithm I mutates chosen genes by amplitude enhancement, while proposed algorithm II by amplitude reduction. Finally, proposed algorithm III is a mixture of the former two. From computer simulation results, we show that for short and middle length LDPC codes, our proposed algorithms can improve both BER (Bit Error Rate) and FER (Frame Error Rate) performance over traditional BP decoding algorithm, with slight modification of traditional BP decoding algorithm and little increase of computation complexity.

关键词： low density parity check codes Gallium decoding algorithm algorithms Bit error rate Ganglia Beliefs Marriage

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A new, fast, and efficient image codec based on set partitioning in hierarchical trees

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY 1996年第3期6卷 243-250页

作者： Said, A Pearlman, WA RENSSELAER POLYTECH INST DEPT ELECT COMP & SYST ENGNTROYNY 12180

Embedded zerotree wavelet (EZW) coding, introduced by J. M. Shapiro, is a very effective and computationally simple technique for image compression, Here we offer an alternative explanation of the principles of its operation, so that the reasons for its excellent performance can be better understood, These principles are partial ordering by magnitude with a set partitioning sorting algorithm, ordered bit plane transmission, and exploitation of self-similarity across different scales of an image wavelet transform, Moreover, we present a new and different implementation based on set partitioning in hierarchical trees (SPIHT), which provides even better performance than our previously reported extension of EZW that surpassed the performance of the original EZW, The image coding results, calculated from actual file sizes and images reconstructed by the decoding algorithm, are either comparable to or surpass previous results obtained through much more sophisticated and computationally complex methods, In addition, the new coding and decoding procedures are extremely fast, and they can be made even faster, with only small loss in performance, by omitting entropy coding of the bit stream by arithmetic code.

关键词： set partitioning Partial ordering Image compression decoding algorithm Data compression Entropy coding Hierarchical arithmetic coding codecs bit stream

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A Dynamic Hybrid Decoder Apprroach Using EG-LDPC Codes for Signal Processing Applications

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WIRELESS PERSONAL COMMUNICATIONS 2022年第2期122卷 1435-1454页

作者： Babu, J. Chinna Rao, Nuka Mallikharjuna Ramana, Kadiyala Bhaskar, Vidhyacharan Annamacharya Inst Technol & Sci Dept Elect & Commun Engn Rajampet Andhra Pradesh India Annamacharya Inst Technol & Sci Dept Comp Sci & Engn Rajampet Andhra Pradesh India San Francisco State Univ Dept Elect & Comp Engn San Francisco CA 94132 USA

The Low-Density Parity Check (LDPC) codes of Euclidean Geometry (EG) are encrypted and decrypted in numerous ways, namely Soft Bit Flipping (SBF), Sequential Peeling Decoder (SPD), Belief Propagation Decoder (BPD), Majority Logic Decoder/Detector (MLDD), and Parallel Peeling Decoder (PPD) decoding algorithms. These algorithms provide aextensive range of trade-offs between latency decoding, power consumption, hardware complexity-required resources, and error rate performance. Therefore, the problem is to communicate a sophisticated technique specifying the both soft and burst errors for effective information transmission. In this research, projected a technique named as Hybrid SBF (HSBF) decoder for EG-LDPC codes, which reduces the decoding complexity and maximizes the signal transmission and reception. In this paper, HSBF is also known as Self Reliability based Weighted Soft Bit Flipping (SRWSBF) Decoder. It is obvious from the outcomes that the proposed technique is better than the decoding algorithms SBF, MLDD, BPD, SPD and PPD. Using Xilinx synthesis and SPARTAN 3e, a simulation model is designed to investigate latency, hardware utilization and power consumption. Average latency of 16.65 percent is found to be reduced. It is observed that in considered synthesis parameters such as number of 4-input LUTs, number of slices, and number of bonded IOBs, excluding number of slice Flip-Flops, hardware utilization is minimized to an average of 4.25 percent. The number of slices Flip-Flops resource use in the proposed HSBF decoding algorithm is slightly higher than other decoding algorithms, i.e. 1.85%. It is noted that, over the decoding algorithms considered in this study, the proposed research study minimizes power consumption by an average of 41.68%. These algorithms are used in multimedia applications, processing systems for security and information.

关键词： Low Density Parity Check (LDPC) EG-LDPC Soft errors Burst errors decoding algorithm HSBF (SRWSBF) Multimedia applications Security

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A QC-LDPC Code Based Digital Signature algorithm

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JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 2019年第5期35卷 1061-1074页

作者： Ren, Fang Yang, Xue-Fei Zheng, Dong Xian Univ Posts & Telecommun Sch Telecommun & Informat Engn Xian 710121 Shaanxi Peoples R China Xian Univ Posts & Telecommun Natl Engn Lab Wireless Secur Xian 710121 Shaanxi Peoples R China

Courtois-Finiasz-Sendrier (CFS) digital signature algorithm, which proposed in 2001, is the most important code based digital signature algorithm and can resist the known attack of quantum algorithms such as Shor algorithm and Grover algorithm. But the efficiency of CFS is very low because of the extremely low signing speed and the large public key size. In this paper, a variation of CFS algorithm is presented. Instead of the Goppa code and the Patterson decoding algorithm, the new algorithm selects the Quasi-Cyclic Low Density Parity Check (QC-LDPC) code and the Belief Propagation (BP) decoding algorithm in the signing process. Compared with CFS algorithm, the new algorithm greatly reduces the storage space of public key and improves the efficiency of signature without compromising the security.

关键词： code based cryptography digital signature QC-LDPC code decoding algorithm CFS algorithm

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Two-Dimensional Array Codes Correcting Rectangular Burst Errors

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PROBLEMS OF INFORMATION TRANSMISSION 2006年第2期42卷 90-105页

作者： Boyarinov, I. RAS Inst Syst Anal Moscow Russia

Two-dimensional array codes correcting rectangular burst errors are considered. We give a construction and examples of linear two-dimensional array codes correcting rectangular burst errors of size b(1) x b(2) with minimum redundancy r = 2b(1) b(2). We present constructions of cyclic two-dimensional array codes correcting phased and arbitrary rectangular burst errors;their encoding and decoding algorithms are also given. A class of cyclic two-dimensional array codes correcting rectangular burst errors with asymptotically minimal redundancy is described. We construct a class of linear two-dimensional array codes correcting cyclic rectangular b(1) x b(2) burst errors with asymptotic excess redundancy (r) over tilde (C)(b(1), b(2)) = 2b(1) b(2) - 3.

关键词： Rectangular coding Codes two-dimensional array BURSTS decoding algorithm Phase

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Optimisation of convergence-aware coded PDMA for 5G wireless systems

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IET COMMUNICATIONS 2019年第2期13卷 243-250页

作者： Xu, Jin Han, Xueqin Ding, Hanqing Li, Zhigang Yu, Zeqi Dai, Xiaoming Zhengzhou Univ Light Ind Sch Comp & Commun Engn 136 Kexue Rd Zhengzhou Henan Peoples R China Univ Sci & Technol Beijing Sch Comp & Commun Engn 30 Xueyuan Rd Beijing Peoples R China

Orthogonal frequency division multiplexing-based pattern division multiple access (PDMA) is considered as a promising multiple access scheme for future wireless communication systems due to its high spectral efficiency and powerful ability to support massive users. In this study, it is shown that two pattern matrices with unequal diversity gain in the PDMA system have better convergence performance compared with the signature matrices in a traditional low-density signature multiple access system by simulations and extrinsic information transfer (EXIT) chart analysis. For a low-density parity check (LDPC)-coded PDMA system, the EXIT characteristics between the front-end multiple-user detector (MUD) and the LDPC decoder may be mismatch, which usually leads to degraded performance when an iterative detection and decoding algorithm is used. To overcome this problem, a two-stage iterative optimisation algorithm, which is easy to implement is proposed to find an optimal (near optimal) degree distribution pair for the LDPC code. Both EXIT chart analysis and simulation results demonstrate that the proposed algorithm exhibits further performance gain compared with regular (3,6) LDPC-coded systems under both an additive white Gaussian noise channel and a Rayleigh fading channel.

关键词： Rayleigh channels multi-access systems decoding OFDM modulation AWGN channels parity check codes 5G mobile communication optimisation matrix algebra iterative methods pattern matrices unequal diversity gain signature matrices low-density signature multiple access system extrinsic information transfer chart analysis low-density parity check-coded PDMA system EXIT characteristics multiple-user detector LDPC decoder iterative detection decoding algorithm two-stage iterative optimisation algorithm optimal degree distribution pair 5G wireless systems orthogonal frequency division multiplexing Rayleigh fading channel additive white Gaussian noise channel front-end multiple-user detector LDPC-coded PDMA system EXIT chart analysis convergence-aware coded PDMA pattern division multiple access spectral efficiency wireless communication systems multiple access scheme

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Turbo equalization

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IEEE SIGNAL PROCESSING MAGAZINE 2004年第1期21卷 67-80页

作者： Koetter, R Singer, AC Tüchler, M Univ Illinois Coordinated Sci Lab Urbana IL 61801 USA Univ Illinois Dept Elect & Comp Engn Urbana IL 61801 USA Tech Univ Munich Inst Commun Engn D-8000 Munich Germany

Turbo equalization is an iterative equalization and decoding technique that can achieve equally impressive performance gains for communication systems that send digital data over channels that require equalization, i.e., those that suffer from intersymbol interference (ISI). In this article, we discuss the turbo equalization approach to coded data transmission over ISI channels, with emphasis on the basic ideas and some of the practical details. The original system introduced by Douillard et al. can be viewed as an extension of the turbo decoding algorithm by considering the effect of the ISI channel as another form of error protection, i.e., as a rate-1 convolutional code.

关键词： turbo codes equalisers telecommunication channels iterative decoding data communication intersymbol interference convolutional codes channel coding turbo equalization iterative equalization iterative decoding technique digital data coded data transmission ISI channels turbo decoding algorithm convolutional code digital data data communication decoding algorithm data transmission Communication channels convolutional coding intersymbolic interference Equalizers Iterative decoding turbo codes Channel coding Equalization

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Structure, Property, and Design of Nonbinary Regular Cycle Codes

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IEEE TRANSACTIONS ON COMMUNICATIONS 2010年第4期58卷 1060-1071页

作者： Huang, Jie Zhou, Shengli Willett, Peter Univ Connecticut Dept Elect & Comp Engn Storrs CT 06269 USA

In this paper, we study nonbinary regular LDPC cycle codes whose parity check matrix H has fixed column weight j = 2 and fixed row weight d. Through graph analysis, we show that the parity check matrix H of a regular cycle code can be put into an equivalent structure in the form of concatenation of row-permuted block-diagonal matrices if.. is even, or, if.. is odd and the code's associated graph contains at least one spanning subgraph that consists of disjoint edges. This equivalent structure of H enables: i) parallel processing in linear-time encoding;ii) considerable resource reduction on the code storage for encoding and decoding;and iii) parallel processing in sequential belief-propagation decoding, which increases the throughput without compromising performance or complexity. On the code's structure design, we propose a novel design methodology based on the equivalent structure of H. Finally, we present various numerical results on the code performance and the decoding complexity.

关键词： LDPC regular cycle code Galois field graph theory decoding algorithm code design

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