Linear Error-correcting Codes Research Articles

Designing an ultra-efficient Hamming code generator circuit for a secure nano-telecommunication network

Communication links forming secure telecommunications networks rely on various technologies such as message switching, circuit switching, or packet switching to transmit messages and data. Hamming codes, a family of linear error-correcting codes, are commonly used in communication networks to detect and correct one-bit and two-bit errors. However, reducing power consumption, occupied area, and latency in secure telecommunication networks remains a challenge for future information and communication technology. To address these challenges, emerging technologies like quantum dots offer potential solutions. Quantum-dot cellular automata (QCA) stands as a promising frontier in nanotechnology for enhancing secure telecommunications networks. It opens up the possibility of crafting high-performance, energy-efficient digital circuits. This research harnesses the potential of QCA and introduces groundbreaking innovations: a 3-8 decoder employing a single-layer layout and a 3-input XOR gate with a multi-layer configuration. These components are utilized in the design of an electronic circuit for Hamming codes, incorporating the QCA-based approach. It is important to note that practical implementation in real-world scenarios presents challenges due to the nature of QCA technology. As a result, the evaluation and validation of the proposed designs heavily rely on simulations using QCADesigner. While experimental validation in real-world scenarios is limited, the simulations provide insights into the functionality and feasibility of the suggested designs. By leveraging QCA, the proposed Hamming code circuit significantly enhances cell count, occupied area, and clock latency. The suggested design can be adapted to fit different generating matrices in Hamming codes without requiring drastic modifications to the underlying architecture.

Twisted Goppa Codes With an Efficient Decoding Algorithm and Quasi-Cyclic Properties

In this paper, we introduce twisted Goppa codes, which generalize classical Goppa codes by adding a twisted term. Then we provide an efficient decoding algorithm for twisted Goppa codes. The Niederreiter cryptosystem is bassed on linear error-correcting codes in which the public key is a parity check matrix. When twisted Goppa codes are applied to the Niederreiter cryptosystem, the public key size is overlarge. To reduce the public key size, we construct quasi-cyclic twisted Goppa codes via a non-trivial automorphism group carefully selecting the defining set and the matched polynomial. Moreover, we obtain a family of cyclic twisted Goppa codes.

Some hypersurfaces over finite fields, minimal codes and secret sharing schemes

Linear error-correcting codes can be used for constructing secret sharing schemes; however, finding in general the access structures of these secret sharing schemes and, in particular, determining efficient access structures is difficult. Here we investigate the properties of certain algebraic hypersurfaces over finite fields, whose intersection numbers with any hyperplane only takes a few values; these varieties give rise to q-divisible linear codes with at most 5 weights. Furthermore, for q odd, these codes turn out to be minimal and we characterize the access structures of the secret sharing schemes based on their dual codes. Indeed, the secret sharing schemes thus obtained are democratic, that is each participant belongs to the same number of minimal access sets and can easily be described.

Efficient Construction of Successive Cancellation Decoding of Polar Codes using Logistic regression Algorithm

In recent days, polar codes have attracted the attention of Industry and Academia. It is one of a linear block error correction codes. Polar codes are one of the capacity achieving algorithms over a wide range of channels, making them very attractive from a theoretical perspective. Polar codes are shown to be instances of concatenated codes. It has been shown that the effect of a polar code can be enhanced by showing the multistage decoding algorithm with log-likelihood based Successive List decoding (SCL). However, SCL decoders are no longer optimal in terms of frame error rate. It is believed that the machine learning based algorithm can improve the FER for the decoded bit. Thus, the logistic regression algorithm is proposed in this paper. The proposed method offers an optimized solution to the decoded bits. The simulation result shows that with sufficient training, this method can provide a lower frame error rate for different code lengths.

The simplification of computationals in error correction coding

The object of research is the processes of error correction transformation of information in automated systems. The research is aimed at reducing the complexity of decoding cyclic codes by combining modern mathematical models and practical tools. The main prerequisite for the complication of computations in deterministic linear error-correcting codes is the use of the algebraic representation as the main mathematical apparatus for these types of codes. Despite the universalism of the algebraic approach, its main drawback is the impossibility of taking into account the characteristic features of all subclasses of linear codes. In particular, the cyclic property is not taken into account at all for cyclic codes. Taking this property into account, one can go to a fundamentally different mathematical representation of cyclic codes – the theory of linear automata in Galois fields (linear finite-state machine). For the automaton representation of cyclic codes, it is proved that the problem of syndromic decoding of these codes in the general case is an NP-complete problem. However, if to use the proposed hierarchical approach to problems of complexity, then on its basis it is possible to carry out a more accurate analysis of the growth of computational complexity. Correction of single errors during one time interval (one iteration) of decoding has a linear decoding complexity on the length of the codeword, and error correction during m iterations of permutations of codeword bits has a polynomial complexity. According to three subclasses of cyclic codes, depending on the complexity of their decoding: easy decoding (linear complexity), iteratively decoded (polynomial complexity), complicate decoding (exponential complexity). Practical ways to reduce the complexity of computations are considered: alternate use of probabilistic and deterministic linear codes, simplification of software and hardware implementation by increasing the decoding time, use of interleaving. A method of interleaving is proposed, which makes it possible to simultaneously generate the burst errors and replace them with single errors. The mathematical apparatus of linear automata allows solving together the indicated problems of error correction coding.

Multiple Observations for Secret-Key Binding with SRAM PUFs.

We present a new Multiple-Observations (MO) helper data scheme for secret-key binding to an SRAM-PUF. This MO scheme binds a single key to multiple enrollment observations of the SRAM-PUF. Performance is improved in comparison to classic schemes which generate helper data based on a single enrollment observation. The performance increase can be explained by the fact that the reliabilities of the different SRAM cells are modeled (implicitly) in the helper data. We prove that the scheme achieves secret-key capacity for any number of enrollment observations, and, therefore, it is optimal. We evaluate performance of the scheme using Monte Carlo simulations, where an off-the-shelf LDPC code is used to implement the linear error-correcting code. Another scheme that models the reliabilities of the SRAM cells is the so-called Soft-Decision (SD) helper data scheme. The SD scheme considers the one-probabilities of the SRAM cells as an input, which in practice are not observable. We present a new strategy for the SD scheme that considers the binary SRAM-PUF observations as an input instead and show that the new strategy is optimal and achieves the same reconstruction performance as the MO scheme. Finally, we present a variation on the MO helper data scheme that updates the helper data sequentially after each successful reconstruction of the key. As a result, the error-correcting performance of the scheme is improved over time.

A survey on single server private information retrieval in a coding theory perspective

In this paper, we present a new perspective of single server private information retrieval (PIR) schemes by using the notion of linear error-correcting codes. Many of the known single server schemes are based on taking linear combinations between database elements and the query elements. Using the theory of linear codes, we develop a generic framework that formalizes all such PIR schemes. This generic framework provides an appropriate setup to analyze the security of such PIR schemes. In fact, we describe some known PIR schemes with respect to this code-based framework, and present the weaknesses of the broken PIR schemes in a unified point of view.

Niederreiter cryptosystems using quasi-cyclic codes that resist quantum Fourier sampling

<p style='text-indent:20px;'>McEliece and Niederreiter cryptosystems are robust and versatile cryptosystems. These cryptosystems work with many linear error-correcting codes. They are popular these days because they can be quantum-secure. In this paper, we study the Niederreiter cryptosystem using non-binary quasi-cyclic codes. We prove, if these quasi-cyclic codes satisfy certain conditions, the corresponding Niederreiter cryptosystem is resistant to the hidden subgroup problem using weak quantum Fourier sampling. Though our work uses the weak Fourier sampling, we argue that its conclusions should remain valid for the strong Fourier sampling as well.</p>

Multilayer Codes for Synchronization From Deletions and Insertions

Consider two remote nodes (encoder and decoder), each with a binary sequence. The encoder's sequence X differs from the decoder's sequence Y by a small number of edits (deletions and insertions). The goal is to construct a message M, to be sent via a one-way error free link, such that the decoder can reconstruct X using M and Y. In this paper, we devise a coding scheme for this one-way synchronization model. The scheme is based on multiple layers of Varshamov-Tenengolts (VT) codes combined with off-the-shelf linear error-correcting codes, and uses a list decoder. We bound the expected list size of the decoder under certain assumptions, and validate its performance via numerical simulations. We also consider an alternative decoder that uses only the constraints from the VT codes (i.e., does not require a linear code), and has a smaller redundancy at the expense of a slightly larger average list size.

A Library for Formalization of Linear Error-Correcting Codes

Error-correcting codes add redundancy to transmitted data to ensure reliable communication over noisy channels. Since they form the foundations of digital communication, their correctness is a matter of concern. To enable trustful verification of linear error-correcting codes, we have been carrying out a systematic formalization in the Coq proof-assistant. This formalization includes the material that one can expect of a university class on the topic: the formalization of well-known codes (Hamming, Reed–Solomon, Bose–Chaudhuri–Hocquenghem) and also a glimpse at modern coding theory. We demonstrate the usefulness of our formalization by extracting a verified decoder for low-density parity-check codes based on the sum-product algorithm. To achieve this formalization, we needed to develop a number of libraries on top of Coq’s Mathematical Components. Special care was taken to make them as reusable as possible so as to help implementers and researchers dealing with error-correcting codes in the future.

Squares of matrix-product codes

The component-wise or Schur product C⁎C′ of two linear error-correcting codes C and C′ over certain finite field is the linear code spanned by all component-wise products of a codeword in C with a codeword in C′. When C=C′, we call the product the square of C and denote it C⁎2. Motivated by several applications of squares of linear codes in the area of cryptography, in this paper we study squares of so-called matrix-product codes, a general construction that allows to obtain new longer codes from several “constituent” codes. We show that in many cases we can relate the square of a matrix-product code to the squares and products of their constituent codes, which allow us to give bounds or even determine its minimum distance. We consider the well-known (u,u+v)-construction, or Plotkin sum (which is a special case of a matrix-product code) and determine which parameters we can obtain when the constituent codes are certain cyclic codes. In addition, we use the same techniques to study the squares of other matrix-product codes, for example when the defining matrix is Vandermonde (where the minimum distance is in a certain sense maximal with respect to matrix-product codes).

Tensor Product DFT Codes vs Standard DFT Codes

We present a new class of linear error-correcting codes taking numerical data into codewords with numerical symbols. These codes can correct large random numerical errors added to codewords. The goal is for numerical data to be protected directly as numbers whether in storage or transmission. The coding structures are based on discrete Fourier transform (DFT) codes, defined by parity-check matrices where rows are consecutively indexed DFT vectors. The new tensor codes use the Kronecker product of two parity-check DFT matrices of shorter length codes. We construct all necessary processing matrices. Error correction methods involve two levels of syndromes and use several stochastic Berlekamp-Massey Algorithms (sBMA) to find big syndromes, locate and evaluate large errors. Decoding is done in two stages with stage one producing corrected syndromes. Stage two determines errors within a well-defined segment for the codewords. Tensor codes have simpler and more efficient decoding operations. The characteristics of the tensor codes are compared to standard DFT codes of equal length. The processing operations are significantly less for the tensor codes but the standard DFT codes sometimes have better correcting performance. Nevertheless, tensor product codes can have acceptable levels of correction, more efficiently.

A Cryptographic System Based on a New Class of Binary Error-Correcting Codes

Abstract In this paper we introduce a new cryptographic system which is based on the idea of encryption due to [McEliece, R. J. A public-key cryptosystem based on algebraic coding theory, DSN Progress Report. 44, 1978, 114–116]. We use the McEliece encryption system with a new linear error-correcting code, which was constructed in [Hannusch, C.—Lakatos, P.: Construction of self-dual binary 22 k, 22 k−1, 2 k-codes, Algebra and Discrete Math. 21 (2016), no. 1, 59–68]. We show how encryption and decryption work within this cryptosystem and we give the parameters for key generation. Further, we explain why this cryptosystem is a promising post-quantum candidate.

A Discrete Polymatroidal Framework for Differential Error-Correcting Index Codes

In the conventional index coding problem, the messages transmitted by the source to satisfy the demands of the receivers are error-free. However, in practice, the messages may be error prone which led to the introduction of error-correcting index codes. An error-correcting index code is an encoding scheme which enables every receiver to decode its demanded message even in the presence of certain number of transmission errors. Formally, an index code capable of correcting at most $\delta $ errors at all its receivers is defined as a $\delta $ -error-correcting index code. In this paper, we explore the connections between vector linear error-correcting index codes and discrete polymatroids. This is motivated from the connections between linear network error-correcting codes and matroids. It is shown that a vector linear error-correcting index code exists if and only if there exists a representable discrete polymatroid satisfying certain conditions. Differential error-correcting index codes which allow different receivers to have different error-correcting capabilities are also studied. Note that the $\delta $ -error-correcting code is a special case of a differential error-correcting index code. Error correction at a subset of receivers is another special case which is discussed.

A Memristor-Based In-Memory Computing Network for Hamming Code Error Correction

Hamming code is a linear error-correcting code widely used in memory and telecommunication. For the first time, here we put forth an approach to analyzing Hamming codes in hardware using networks consisting of memristors. Such networks are capable of detecting and correcting potential bit errors in Hamming codes in a high-speed and energy-efficient way, where the parity-check matrix is stored in the memristor crossbar array and the syndrome vector is represented by the states of the output unipolar memristors whose resistance flipping naturally implements vector-matrix multiplication with modulo 2. Our experimental results demonstrate that the output syndrome vector is able to correctly represent the position of the bit error whenever it exists, and subsequent correction can thus be applied by flipping the bit identified. The energy consumption using the present network has decreased by >100 times compared with GPU while >1000 times compared with CPU.

Problems and solutions from the fourth International Students’ Olympiad in Cryptography (NSUCRYPTO)

Mathematical problems and their solutions from the fourth International Students’ Olympiad in cryptography (NSUCRYPTO-2017) are presented. We consider problems related to attacks on ciphers and hash functions, cryptographic Boolean functions, linear branch numbers, addition chains, and error correction codes, among others. We discuss several open problems involving the algebraic structures of cryptographic functions, useful proof-of-work algorithms, the Boolean hidden shift problem, and quantum computing.

On Privacy Amplification, Lossy Compression, and Their Duality to Channel Coding

We examine the task of privacy amplification from information-theoretic and coding-theoretic points of view. In the former, we give a one-shot characterization of the optimal rate of privacy amplification against classical adversaries in terms of the optimal type-II error in asymmetric hypothesis testing. This formulation can be easily computed to give finite-blocklength bounds and turns out to be equivalent to smooth min-entropy bounds by Renner and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a bound in terms of the $E_\gamma$ divergence by Yang, Schaefer, and Poor [arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy amplification based on linear codes can be easily repurposed for channel simulation. Combined with known relations between channel simulation and lossy source coding, this implies that privacy amplification can be understood as a basic primitive for both channel simulation and lossy compression. Applied to symmetric channels or lossy compression settings, our construction leads to proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing to the notion of channel duality recently detailed by us in [IEEE Trans. Info. Theory 64, 577 (2018)], we show that linear error-correcting codes for symmetric channels with quantum output can be transformed into linear lossy source coding schemes for classical variables arising from the dual channel. This explains a "curious duality" in these problems for the (self-dual) erasure channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and partly anticipates recent results on optimal lossy compression by polar and low-density generator matrix codes.

Securing Smart Electricity Meters Against Customer Attacks

In this paper, we propose and discuss the challenges of a novel method for securing the readings of smart electricity meters against tampering or malfunctioning. We apply the idea of linear error-correcting block codes, which has originally been used in detecting and correcting errors in data transmission, to detect and correct customer attacks aimed at changing the readings of smart electricity meters. We propose to use these codes, with some modifications, in order to detect and correct deceptive readings of a group of electricity meters measuring the energy delivered by cables\feeders fed from the same node\bus. We start by considering a single meter attack, and we show that the detection and correction of a single attack in the readings of ${ 2}^{ M}{ -}{ M}{ -}{ 1}$ meters, connected to cables fed from the same node, requires only using ${M}$ current transformers in addition to knowing the voltage at the common node. We demonstrate the theoretical implementation of this coding scheme, and then we discuss some practical challenges and their proposed solutions. Furthermore, extension to multi-error correcting codes, which are capable of detecting and correcting several simultaneous deceptive readings of a group of meters, is also discussed.

On the Mollard code as a partially robust code

In this paper a generalization of classic Mollard construction for any code length is given. It is shown that such generalized codes have property of partial robustness. This construction has less undetectable and miscorrected errors than the traditional linear error-correcting codes, just as than the generalized Vasil'ev code for some code parameters, thereby providing better protection against multiple bit errors.

Ring‐based linear network coding on erroneous cyclic networks

In this study, the authors study ring-based linear network coding over erroneous cyclic networks over commutative rings such as a principal ideal domain or discrete valuation ring. In the first part, they study coherent field-based linear error-correcting network codes (LENCs) over cyclic networks. By changing alphabet symbols from fields to commutative rings, they extend Zhang's formulation for LENCs restricted on acyclic networks to cyclic networks, and generalise fundamental results and concepts such as the minimum rank distance and the refined Singleton bound, and show that this bound is tight. In the second part, they generalise some main results such as the free distance and the generalised Singleton bound from convolutional codes to ring-based LENCs over cyclic networks. In the third part, they propose an algebraic method for calculating sink bit error probability of ring-based linear network codes over all erroneous networks by using the authors’ formulations.