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PKC-PC: A variant of the McEliece public-key cryptosystem based on polar codes

IET COMMUNICATIONS 2020年第12期14卷 1883-1893页

作者： Hooshmand, Reza Koochak Shooshtari, Masoumeh Reza Aref, Mohammad Shahid Sattari Aeronaut Univ Sci & Technol Dept Elect Engn Tehran Iran Sharif Univ Technol Dept Elect Engn Tehran Iran

Polar codes are novel and efficient error-correcting codes with low encoding and decoding complexities. These codes have a channel-dependent generator matrix, which is determined by the code dimension, code length and transmission channel parameters. A variant of the McEliece public-key cryptosystem based on polar codes, called the PKC-PC, is studied. Since the structure of the polar codes' generator matrix depends on the parameters of the channel, the authors have used an efficient approach to conceal their generator matrix. The proposed approach is based on a random selection of rows of the matrix by which a random generator matrix is constructed. Using the characteristics of polar codes and introducing an efficient approach, they could reduce the public and secret key sizes, and computational complexity compared to the McEliece cryptosystem. Moreover, they show that PKC-PC yields an increased security level against conventional attacks as well as possible vulnerabilities to the code-based public-key cryptosystems. Furthermore, they prove the security of the authors' cryptosystem and show that its security is reduced to solve NP-complete problems, called polar parameterised syndrome decoding and polar parameterised codeword existence.

关键词： public key cryptography decoding error correction codes computational complexity polar codes channel coding matrix algebra PKC-PC McEliece public-key cryptosystem polar codes error-correcting codes channel-dependent generator matrix code dimension code length transmission channel parameters random generator matrix public key sizes secret key sizes code-based public-key cryptosystems polar parameterised syndrome decoding decoding complexity encoding complexity computational complexity polar parameterised codeword existence

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decoding of expander codes at rates close to capacity

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IEEE TRANSACTIONS ON INFORMATION THEORY 2006年第12期52卷 5475-5485页

作者： Ashikhmin, Alexei Skachek, Vitaly Lucent Technol Bell Labs Math Commun Dept Murray Hill NJ 07974 USA Technion Israel Inst Technol Dept Comp Sci IL-32000 Haifa Israel

The decoding error probability of codes is studied as a function of their block length. It is shown that the existence of codes with a polynomially small decoding error probability implies the existence of codes with an exponentially small decoding error probability. Specifically, it is assumed that there exists a family of codes of length N and rate R = (1 - epsilon)C (C is a capacity of a binary-symmetric channel), whose decoding probability decreases inverse polynomially in N. It is shown that if the decoding probability decreases sufficiently fast, but still only inverse polynomially fast in N, then there exists another such family of codes whose decoding error probability decreases exponentially fast in N. Moreover, if the decoding time complexity of the assumed family of codes is polynomial in N and 1/epsilon, then the decoding time complexity of the presented family is linear in N and polynomial in 1/epsilon. These codes are compared to the recently presented codes of Barg and Zemor, "Error Exponents of Expander Codes," IEEE Transactions on Information Theory, 2002, and "Concatenated Codes: Serial and Parallel," 11 IEEE Transactions on Information Theory, 2005. It is shown that the latter families cannot be tuned to have exponentially decaying (in N) error probability, and at the same time to have decoding time complexity linear in N and polynomial in 1/epsilon.

关键词： concatenated codes decoding complexity decoding error probability error exponent expander codes irregular repeat accumulative (IRA) codes iterative decoding linear-time decoding low-density paritycheck (LDPC) codes

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Low complexity trellis representations of convolutional codes via sectionalization of the minimal trellis

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TELECOMMUNICATION SYSTEMS 2015年第4期59卷 491-500页

作者： Benchimol, Isaac Pimentel, Cecilio Souza, Richard Demo CMDI IFAM Manaus AM Brazil CODEC DES UFPE Recife PE Brazil CPGEI UTPPR Curitiba PE Brazil

It has been shown by McEliece and Lin that convolutional codes can be represented by a minimal trellis structure in order to reduce the decoding complexity of the Viterbi algorithm. This trellis module has an irregular structure presenting sections with different number of states. In this paper we present the sectionalization of the minimal trellis module which yields a more compact and regular trellis representation (in terms of maximum number of states and total number of sections) with the same decoding complexity and distance spectrum of the minimal trellis module. We investigate the effects of the trellis sectionalization over the trellis and merge complexity measures. A set of rules are constructed to govern these effects. A list of the most compact sectionalized trellis modules with the same complexities than the minimal module is shown for codes of several rates. We show that various trellis topologies proposed in the literature are specific cases of the sectionalized minimal trellis.

关键词： Convolutional codes Minimal trellis Sectionalization decoding complexity

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Fast decoding of Expander Codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2018年第2期64卷 972-978页

作者： Dowling, Michael Gao, Shuhong Clemson Univ Dept Elect & Comp Engn Clemson SC 29634 USA Clemson Univ Dept Math Sci Clemson SC 29634 USA

Expander codes are Tanner codes defined on sparse graphs that have good expansion properties. Sipser and Spielman (1996) showed that there is a linear-time decoding algorithm for expander codes when the vertex expansion is at least 3/4 and the number of errors corrected is a constant fraction of the code length. Later, Feldman et al. (2007) gave a decoding algorithm that allows the expansion to be 2/3 + 1/(3c), where c is the left degree of the underlying bipartite graph, at the expense of polynomial-time decoding complexity. Recently, Viderman (2013) further improved the expansion parameter to 2/3 - 1/(6c), and the decoding algorithm runs in linear time. These results are for expander codes whose inner codes are parity-check codes. By using stronger inner codes, Chilappagari et al. (2010) showed that there is a linear-time decoding algorithm for every vertex expansion greater than 1/2. In this paper, it is shown that for every vertex expansion, there is a linear-time decoding algorithm for expander codes (using inner codes with minimum distance depending on the vertex expansion), and that the number of errors corrected is a constant fraction of the code length.

关键词： Expander codes expander graphs asymptotically good codes decoding complexity linear-time decoding low-density parity check (LDPC) codes

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Full-Diversity Dispersion Matrices From Algebraic Field Extensions for Differential Spatial Modulation

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2017年第1期66卷 385-394页

作者： Rajashekar, Rakshith Ishikawa, Naoki Sugiura, Shinya Hari, K. V. S. Hanzo, Lajos Univ Southampton Sch Elect & Comp Sci Southampton SO17 1BJ Hants England Tokyo Univ Agr & Technol Dept Comp & Informat Sci Koganei Tokyo 1848588 Japan Indian Inst Sci Dept Elect Commun Engn Bangalore 560012 Karnataka India

We consider differential spatial modulation (DSM) operating in a block fading environment and propose sparse unitary dispersion matrices (DMs) using algebraic field extensions. The proposed DM sets are capable of exploiting full transmit diversity and, in contrast to the existing schemes, can be constructed for systems having an arbitrary number of transmit antennas. More specifically, two schemes are proposed: 1) field-extension- based DSM (FE-DSM), where only a single conventional symbol is transmitted per space-time block;and 2) FE-DSM striking a diversity-rate tradeoff (FE-DSM-DR), where multiple symbols are transmitted in each space-time block at the cost of a reduced transmit diversity gain. Furthermore, the FE-DSM scheme is analytically shown to achieve full transmit diversity, and both proposed schemes are shown to impose decoding complexity, which is independent of the size of the signal set. It is observed from our simulation results that the proposed FE-DSM scheme suffers no performance loss compared with the existing DM-based DSM (DM-DSM) scheme, whereas FE-DSM-DR is observed to give a better bit-error-ratio performance at higher data rates than its DM-DSM counterpart. Specifically, at data rates of 2.25 and 2.75 bits per channel use, FE-DSM-DR is observed to achieve about 1-and 2-dB signal-to-noise ratio (SNR) gain with respect to its DM-DSMcounterpart.

关键词： decoding complexity differential spatial modulation (DSM) dispersion matrices (DMs) diversity field extension

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Delay-complexity Trade-Off of Random Linear Network Coding in Wireless Broadcast

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IEEE TRANSACTIONS ON COMMUNICATIONS 2020年第9期68卷 5606-5618页

作者： Su, Rina Sun, Qifu Tyler Zhang, Zhongshan Univ Sci & Technol Beijing Sch Comp & Commun Engn Beijing 10083 Peoples R China Beijing Inst Technol Sch Informat & Elect Beijing 100081 Peoples R China

In wireless broadcast, random linear network coding (RLNC) over GF(2(L)) is known to asymptotically achieve the optimal completion delay with increasing L. However, the high decoding complexity hinders the potential applicability of RLNC schemes over large GF(2(L)). In this paper, a comprehensive analysis of completion delay and decoding complexity is conducted for field-based systematic RLNC schemes in wireless broadcast. In particular, we prove that the RLNC scheme over GF(2) can also asymptotically approach the optimal completion delay per packet when the packet number goes to infinity. Moreover, we introduce a new method, based on circular-shift operations, to design RLNC schemes which avoid multiplications over large GF(2(L)). Based on both theoretical and numerical analyses, the new RLNC schemes turn out to have a much better trade-off between completion delay and decoding complexity. In particular, numerical results demonstrate that the proposed schemes can attain average completion delay just within 5% higher than the optimal one, while the decoding complexity is only about 3 times the one of the RLNC scheme over GF(2).

关键词： Wireless broadcast random linear network coding circular-shift completion delay decoding complexity

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On the BCJR trellis for linear block codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 1996年第4期42卷 1072-1092页

作者： McEliece, RJ Department of Electrical Engineering California Institute of Technology Pasadena CA USA

In this semi-tutorial paper, we will investigate the computational complexity of an abstract version of the Viterbi algorithm on a trellis, and show that if the trellis has e edges, the complexity of the Viterbi algortithm is Theta(e), This result suggests that the ''best'' trellis representation for a given linear block code is the one with the fewest edges, We will then show that, among all trellises that represent a given code, the original trellis introduced by BahI, Cocke, Jelinek, and Raviv in 1974, and later rediscovered by Wolf, Massey, and Forney, uniquely minimizes the edge count, as well as several other figures of merit. Following Forney and Kschischang and Sorokine, we will also discuss ''trellis-oriented'' or ''minimal-span'' generator matrices, which facilitate the calculation of the size of the BCJR trellis, as well as the actual construction of it.

关键词： block code trellis Viterbi algorithm decoding complexity

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On Lattice Sequential decoding for the Unconstrained AWGN Channel

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IEEE TRANSACTIONS ON COMMUNICATIONS 2013年第6期61卷 2446-2456页

作者： Abediseid, Walid Alouini, Mohamed-Slim King Abdullah Univ Sci & Technol Comp Elect & Math Sci & Engn CEMSE Div Elect Engn Program Thuwal Makkah Province Saudi Arabia

In this paper, the performance limits and the computational complexity of the lattice sequential decoder are analyzed for the unconstrained additive white Gaussian noise channel. The performance analysis available in the literature for such a channel has been studied only under the use of the minimum Euclidean distance decoder that is commonly referred to as the lattice decoder. Lattice decoders based on solutions to the NP-hard closest vector problem are very complex to implement, and the search for low complexity receivers for the detection of lattice codes is considered a challenging problem. However, the low computational complexity advantage that sequential decoding promises, makes it an alternative solution to the lattice decoder. In this work, we characterize the performance and complexity tradeoff via the error exponent and the decoding complexity, respectively, of such a decoder as a function of the decoding parameter - the bias term. For the above channel, we derive the cut-off volume-to-noise ratio that is required to achieve a good error performance with low decoding complexity.

关键词： Lattice coding lattice decoding sequential decoding error exponent decoding complexity

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Multigroup Decodable STBCs From Clifford Algebras

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IEEE TRANSACTIONS ON INFORMATION THEORY 2009年第1期55卷 223-231页

作者： Karmakar, Sanjay Rajan, B. Sundar Univ Colorado Dept Comp Sci & Elect Engn Boulder CO 80309 USA Indian Inst Sci Dept Elect Commun Engn Bangalore 560012 Karnataka India

A space-time block code (STBC) in K symbols (Variables) is called a g-group decodable STBC if its maximum-likelihood (ML) decoding metric can be written as a sum of g terms, for some positive integer g greater than one, such that each term is a function of a subset of the K variables and each variable appears in only one term. In this paper, we provide a general structure of the weight matrices of multigroup decodable codes using Clifford algebras. Without assuming that the number of variables in each group is the same, a method of explicitly constructing the weight matrices of full-diversity, delay-optimal multigroup decodable codes is presented for arbitrary number of antennas. For the special case of 2(a) number of transmit antennas, we construct two subclass of codes: 1) a class of 2a-group decodable codes with rate a/2((a-1)), which is, equivalently, a class of single-symbol decodable codes, and 2) a class of (2a - 2)-group decodable codes with rate (a-1)/2((a-2)), i.e., a class of double-symbol decodable codes.

关键词： Clifford algebra decoding complexity diversity space-time block codes (STBCs)

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A new M-PSK code construction with good minimum Euclidean distance for AWGN channels

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2001年第6期E84A卷 1564-1571页

作者： Ibn, A Tanabe, H Umeda, H Fukui Univ Dept Elect & Elect Engn Informat & Commun Syst Lab Fukui 9109507 Japan

An important goal in communication theory is to construct good minimum squared Euclidean distance (MSED) codes for transmission over additive white Gaussian noise (AWGN) channels. In this paper, a new construction method for the M-ary phase-shift-keyed (M-PSK) codes over the ring structure Z(M) the ring of integers module M, with a good minimum Euclidean distance, is proposed. The proposed codes are linear when multiple coset leaders are considered. The characteristics and performance levels of the newly constructed codes are analyzed for code length up to n less than or equal to 8. It is found that the proposed codes compare favorably with Piret's codes and Graeffe's method codes on Gaussian channels in terms of decoding complexity, coding gain, and error performance.

关键词： coset codes ideal idempotents decoding complexity ring structure

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