Random Linear Coding Research Articles

Weights of Codewords in Random Binary Linear Codes and Their Correlation Properties

Weights of Codewords in Random Binary Linear Codes and Their Correlation Properties

Decoding error probability of random parity-check matrix ensemble over the erasure channel

AbstractIn this paper we carry out an in-depth study on the average decoding error probability of the random parity-check matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random parity-check matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random parity-check matrix ensemble and the error exponent of the variance, which implies a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random linear code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.

Linear Codes for Hyperdimensional Computing.

Hyperdimensional computing (HDC) is an emerging computational paradigm for representing compositional information as high-dimensional vectors and has a promising potential in applications ranging from machine learning to neuromorphic computing. One of the long-standing challenges in HDC is factoring a compositional representation to its constituent factors, also known as the recovery problem. In this article, we take a novel approach to solve the recovery problem and propose the use of random linear codes. These codes are subspaces over the Boolean field and are a well-studied topic in information theory with various applications in digital communication. We begin by showing that hyperdimensional encoding using random linear codes retains favorable properties of the prevalent (ordinary) random codes; hence, HD representations using the two methods have comparable information storage capabilities. We proceed to show that random linear codes offer a rich subcode structure that can be used to form key-value stores, which encapsulate the most used cases of HDC. Most important, we show that under the framework we develop, random linear codes admit simple recovery algorithms to factor (either bundled or bound) compositional representations. The former relies on constructing certain linear equationsystems over the Boolean field, the solution to which reduces the search space dramatically and strictly outperforms exhaustive search in many cases. The latter employs the subspace structure of these codes to achieve provably correct factorization. Both methods are strictly faster than the state-of-the-art resonator networks, often by an order of magnitude. We implemented our techniques in Python using a benchmark software library and demonstrated promising experimental results.

Weight distribution of random linear codes and Krawtchouk polynomials

AbstractFor and pick uniformly at random vectors in and let be the orthogonal complement of their span. Given with , let be the random variable that counts the number of words in of Hamming weight (where is assumed to be an even integer). Linial and Mosheiff [Random Struct. Algorithms. 62 (2023), 68Ű‐130] determined the asymptotics of the moments of of all orders . In this paper we extend their estimates up to moments of linear order. Our key observation is that the behavior of the suitably normalized moment of is essentially determined by the norm of the Krawtchouk polynomial .

A method for recovering linear block codes over an arbitrary finite field from sets of distorted code words

The article is devoted to one of the practically important problems of information security and cryptanalysis, which consists in recovering an unknown linear block code over an arbitrary field from a set of distorted code words. This is a hard computational problem, and the known problem-solving methods are proposed only for codes over the field of two elements and are based on the algorithms for searching words of small weight in (undistorted) linear block codes.
 The main result of the article is a method for solving the problem posed, which differs in essence from the known ones and consists in recovering the desired code by solving the LPN (Learning Parity with Noise) problem, namely, recovering the solutions of systems of linear equations with distorted right-hand sides and a random equally probable matrix of coefficients over specified field. The LPN problem is well known from the Theory of Computational Algorithms and Cryptanalysis. It is equivalent to the problem of random linear block code decoding, and the security of many modern post-quantum cryptosystems are based on its hardness.
 The proposed method provides an opportunity to apply a wider class of algorithms for recovering linear block codes in comparison with the previously known methods, in particular, algorithms like BKW and also the low weight words search algorithms in co-sets of linear block codes. Moreover, in contrast to previously known ones, the complexity of the proposed method depends linearly on the length of the required code (and increases with increasing of its dimension according to which algorithm for the LPN problem-solving is applied). Thus, the basic parameter determined the complexity of recovering a linear block code is its dimension (not its length), which, in principle, makes it possible to speed up known algorithms for recovering linear block codes from a set of corrupted code words.

AES as Error Correction: Cryptosystems for Reliable Communication

In this paper, we show that the Advanced Encryption Standard (AES) cryptosystem can be utilized as an error-correcting code to obtain reliability over noisy communication and data systems. Specifically, we show that by composing a simple padding followed by encrypting with AES, we can achieve error-correcting performance similar to random codes. We conduct an empirical comparison between our AES-based error correction approach, Random Linear Codes, and CA-Polar codes to assess their performance in practical scenarios. Our results show that the AES-based approach achieves nearly identical performance to the other well-established error-correcting codes.

Expanded low-rank parity-check codes and their application to cryptography

In this paper, we define expanded LRPC codes from LRPC codes with a decoding algorithm using the one of the underlying LRPC codes. Next, we propose to use these codes for cryptography by deriving two cryptosystems in a McEliece setting. in order to reduce the key sizes, we use a generator matrix in systematic form for the first scheme, and an m − order quasi-cyclic LRPC code that we define for the second scheme. The obtained code has a very poor structure and is more likely to be a random linear code. Next, we give some security parameters and compare the key sizes of our public key with the key sizes of some cryptosystems.

Algorithm 1033: Parallel Implementations for Computing the Minimum Distance of a Random Linear Code on Distributed-memory Architectures

The minimum distance of a linear code is a key concept in information theory. Therefore, the time required by its computation is very important to many problems in this area. In this article, we introduce a family of implementations of the Brouwer–Zimmermann algorithm for distributed-memory architectures for computing the minimum distance of a random linear code over 𝔽 2 . Both current commercial and public-domain software only work on either unicore architectures or shared-memory architectures, which are limited in the number of cores/processors employed in the computation. Our implementations focus on distributed-memory architectures, thus being able to employ hundreds or even thousands of cores in the computation of the minimum distance. Our experimental results show that our implementations are much faster, even up to several orders of magnitude, than current implementations widely used nowadays.

Statistical zero-knowledge and analysis of rank-metric zero-knowledge proofs of knowledge

A series of Stern-like Code-Based Zero-Knowledge Proofs of Knowledge (CBZKPoKs) in the rank setting have been proposed since 2011. These CBZKPoKs (RStern, RJKPT, RVDC, RankId, RCVE, RVéronID) are rank metric adaptations of Stern, JKPT, AGS, CVE, and Véron protocols in the Hamming setting. RVéronID has been analyzed and the witness can be recovered because improper permutation leaks the information of witness. However, there are several open problems in the rest of rank metric CBZKPoKs: (1) statistical zero-knowledge property; (2) security and completeness. In this paper, we deeply analyze rank metric permutation and its cryptographic properties, and rigorously show that RStern and RJKPT can achieve statistical zero-knowledge property. We then analyze RVDC, RankId, and RCVE and show that they do not satisfy completeness and RVDC can be broken by the rank support learning attack. To repair and strengthen security, we reconstruct the rank metric protocols (RAGS and RVéron), in which two protocols work on random linear codes without the limitation of cyclic structure. The performance analysis shows that: their communication costs (20 KB and 26 KB) are the lowest among existing Stern-like CBZKPoKs for 128-bit security. When compared with lattice-based zero-knowledge proofs of knowledge with a non-negligible soundness error such as improved Stern (PKC 2013), Bootle et al. (CRYPTO 2019), and Beullens (EUROCRYPT 2020), our protocols perform a significant advantage in communication costs.

Fourth moments of the code-weights

In this paper we give the fourth joint moments of the weights of random linear codes. We calculate the fourth moment using mutual position of 4 lines. The solution of the 4-subspace problem is known and we obtain the information on line configurations from the set of indecomposables of the 4-subspace problem.

Random Linear Streaming Codes in the Finite Memory Length and Decoding Deadline Regime—Part I: Exact Analysis

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Streaming codes</i> take a string of source symbols as input and output a string of coded symbols in real time, which eliminate the queueing delay of traditional <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">block codes</i> and are thus especially appealing for delay sensitive applications. Existing works on streaming code performance either focused on the asymptotic error-exponent analyses, or on the optimal code construction under <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deterministic adversarial channel models</i> . In contrast, this work analyzes the exact error probability of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">random linear streaming codes</i> (RLSCs) in the large field size regime over the stochastic i.i.d. symbol erasure channel model. A closed-form expression of the error probability of large-field-size RLSCs is derived under, simultaneously, the finite memory length and decoding deadline constraints. The result is then used to examine the intricate tradeoff between memory length (complexity), decoding deadline (delay), code rate (throughput), and error probability (reliability). Numerical evaluation shows that under the same code rate and error probability requirements, the end-to-end delay of RLSCs is 40–48% of that of the optimal block codes (i.e., MDS codes). This implies that switching from block codes to streaming codes not only eliminates the queueing delay completely (which accounts for the initial 50% of the delay reduction) but also improves the reliability (which accounts for the additional 2–10% delay reduction).

Distribution of the Minimum Distance of Random Linear Codes

Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q\geq 2$ </tex-math></inline-formula> be a prime power. In this paper, we study the distribution of the minimum distance (in the Hamming metric) of a random linear code of dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}^{n}$ </tex-math></inline-formula> . We provide quantitative estimates showing that the distribution function of the minimum distance is close ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">superpolynomially</i> in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> ) to the cumulative distribution function of the minimum of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(q^{k}-1)/(q-1)$ </tex-math></inline-formula> independent binomial random variables with parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {1}{q}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> . The latter, in turn, converges to a Gumbel distribution at integer points when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {k}{n}$ </tex-math></inline-formula> converges to a fixed number in (0, 1). Our result confirms in a strong sense that apart from identification of the weights of proportional codewords, the probabilistic dependencies introduced by the linear structure of the random code, produce a negligible effect on the minimum code weight. As a corollary of the main result, we obtain an improvement of the Gilbert–Varshamov bound for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2&lt; q&lt; 49$ </tex-math></inline-formula> .

Adaptive Packet Coding for Reliable Underwater Acoustic Communications

This work investigates adaptive random linear packet coding (RLPC) for reliable underwater acoustic (UWA) communications. Our goal is to minimize the total transmission time of data blocks by adjusting the packet coding rate. We first consider the application of RLPC with the conventional automatic repeat request (ARQ) scheme. We dynamically adjust the coding rate to fit the time variations of UWA channels by choosing the optimal number of packets in each transmission. The optimal number of packets in each transmission is obtained based on a dynamic programming (DP) algorithm according to the feedback messages, which contain the number of successfully transmitted packets in the last transmission and the channel state information. Furthermore, considering the long propagation delay of UWA communications, we propose a modified juggling-like ARQ (J-ARQ) for the RLPC scheme, for which the duration of each transmission can be adjusted based on the characteristics of RLPC. A two-step DP algorithm is proposed to find out the optimal solutions for this case. Simulation results show that the proposed schemes can improve the throughput efficiency and reduce the outage probability.

Keep the Bursts and Ditch the Interleavers

While many communications media, such as wireless and certain classes of wireline channels, typically lead to bursty errors, most decoders are designed assuming memoryless channels. Consequently, communication systems generally rely on interleaving over tens of thousands of bits to match decoder assumptions. Even for short high rate codes, awaiting sufficient data in interleaving and de-interleaving is a significant source of unwanted latency. We construct an extension to the recently proposed Guessing Random Additive Noise Decoding (GRAND) algorithm, which we call GRAND-MO for GRAND Markov Order. By foregoing interleaving and instead making use of the bursty nature of noise, low-latency communication is possible with block error rates outperforming their interleaved counterparts by a substantial margin. We establish that certain well-known binary codes with structured code-word patterns are ill-suited for use in bursty channels, but Random Linear Codes (RLCs) prove robust to correlated noise. We further demonstrate that by operating directly on modulated symbols rather than de-mapped bits, GRAND-MO achieves further performance and complexity gains by exploiting information that is lost in demodulation. As a result, GRAND-MO provides one potential solution for applications that require ultra-reliable low latency communication.

VQProtect: Lightweight Visual Quality Protection for Error-Prone Selectively Encrypted Video Streaming.

Mobile multimedia communication requires considerable resources such as bandwidth and efficiency to support Quality-of-Service (QoS) and user Quality-of-Experience (QoE). To increase the available bandwidth, 5G network designers have incorporated Cognitive Radio (CR), which can adjust communication parameters according to the needs of an application. The transmission errors occur in wireless networks, which, without remedial action, will result in degraded video quality. Secure transmission is also a challenge for such channels. Therefore, this paper’s innovative scheme “VQProtect” focuses on the visual quality protection of compressed videos by detecting and correcting channel errors while at the same time maintaining video end-to-end confidentiality so that the content remains unwatchable. For the purpose, a two-round secure process is implemented on selected syntax elements of the compressed H.264/AVC bitstreams. To uphold the visual quality of data affected by channel errors, a computationally efficient Forward Error Correction (FEC) method using Random Linear Block coding (with complexity of ) is implemented to correct the erroneous data bits, effectively eliminating the need for retransmission. Errors affecting an average of 7–10% of the video data bits were simulated with the Gilbert–Elliot model when experimental results demonstrated that 90% of the resulting channel errors were observed to be recoverable by correctly inferring the values of erroneous bits. The proposed solution’s effectiveness over selectively encrypted and error-prone video has been validated through a range of Video Quality Assessment (VQA) metrics.

Bounds for List-Decoding and List-Recovery of Random Linear Codes

A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input $\ell$ if for every sufficiently large subset of codewords (of $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the list size in either case. The capacity, i.e., the largest possible rate for these notions as the $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet of the code family. In this work, we study the of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $\epsilon > 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log_q(\ell) - \epsilon$ requires $L \ge \ell^{\Omega(1/\epsilon)}$ for list-recovery from input $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/\epsilon)$ suffices w.h.p. (2) A random linear code of rate $1 - h_q(p) - \epsilon$ requires $L \ge \lfloor h_q(p)/\epsilon+0.99 \rfloor$ for list-decoding from error fraction $p$, when $\epsilon$ is sufficiently small. (3) A random binary linear code of rate $1 - h_2(p) - \epsilon$ is list-decodable from average error fraction $p$ with with $L \leq \lfloor h_2(p)/\epsilon \rfloor + 2$. The second and third results together precisely pin down the sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values.

Guessing Random Additive Noise Decoding With Symbol Reliability Information (SRGRAND)

The design and implementation of error correcting codes has long been informed by two fundamental results: Shannon&#x2019;s 1948 capacity theorem, which established that long codes use noisy channels most efficiently; and Berlekamp, McEliece, and Van Tilborg&#x2019;s 1978 theorem on the NP-completeness of decoding linear codes. These results shifted focus away from creating code-independent decoders, but recent low-latency communication applications necessitate relatively short codes, providing motivation to reconsider the development of universal decoders. We introduce a scheme for employing binarized symbol soft information within Guessing Random Additive Noise Decoding, a universal hard detection decoder. We incorporate codebook-independent quantization of soft information to indicate demodulated symbols to be reliable or unreliable. We introduce two decoding algorithms: one identifies a conditional Maximum Likelihood (ML) decoding; the other either reports a conditional ML decoding or an error. For random codebooks, we present error exponents and asymptotic complexity, and show benefits over hard detection. As empirical illustrations, we compare performance with majority logic decoding of Reed-Muller codes, with Berlekamp-Massey decoding of Bose-Chaudhuri-Hocquenghem codes, with CA-SCL decoding of CA-Polar codes, and establish the performance of Random Linear Codes, which require a universal decoder and offer a broader palette of code sizes and rates than traditional codes.

Federated Learning Over Wireless Device-to-Device Networks: Algorithms and Convergence Analysis

The proliferation of Internet-of-Things (IoT) devices and cloud-computing applications over siloed data centers is motivating renewed interest in the collaborative training of a shared model by multiple individual clients via federated learning (FL). To improve the communication efficiency of FL implementations in wireless systems, recent works have proposed compression and dimension reduction mechanisms, along with digital and analog transmission schemes that account for channel noise, fading, and interference. The prior art has mainly focused on star topologies consisting of distributed clients and a central server. In contrast, this paper studies FL over wireless device-to-device (D2D) networks by providing theoretical insights into the performance of digital and analog implementations of decentralized stochastic gradient descent (DSGD). First, we introduce generic digital and analog wireless implementations of communication-efficient DSGD algorithms, leveraging random linear coding (RLC) for compression and over-the-air computation (AirComp) for simultaneous analog transmissions. Next, under the assumptions of convexity and connectivity, we provide convergence bounds for both implementations. The results demonstrate the dependence of the optimality gap on the connectivity and on the signal-to-noise ratio (SNR) levels in the network. The analysis is corroborated by experiments on an image-classification task.

Coded Computing via Binary Linear Codes: Designs and Performance Limits

We consider the problem of coded distributed computing where a large linear computational job, such as a matrix multiplication, is divided into <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> smaller tasks, encoded using an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,k)$ </tex-math></inline-formula> linear code, and performed over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> distributed nodes. The goal is to reduce the average execution time of the computational job. We provide a connection between the problem of characterizing the average execution time of a coded distributed computing system and the problem of analyzing the error probability of codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> used over erasure channels. Accordingly, we present closed-form expressions for the execution time using binary random linear codes and the best execution time any linear-coded distributed computing system can achieve. It is also shown that there exist <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">good</i> binary linear codes that not only attain (asymptotically) the best performance that any linear code (not necessarily binary) can achieve but also are numerically stable against the inevitable rounding errors in practice. We then develop a low-complexity algorithm for decoding Reed-Muller (RM) codes over erasure channels. Our decoder only involves additions, subtractions, and inversion of relatively small matrices of dimensions at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\log n+1$ </tex-math></inline-formula> , and enables coded computation over real-valued data. Extensive numerical analysis of the fundamental results as well as RM- and polar-coded computing schemes demonstrate the excellence of the RM-coded computation in achieving close-to-optimal performance while having a low-complexity decoding and explicit construction. The proposed framework in this paper enables efficient designs of distributed computing systems given the rich literature in the channel coding theory.

Network Communication Data Encryption Method Based on Wireless Channel Characteristics

In order to improve the secure transmission capability of wireless transmission network communication data, a network communication data encryption design is required. An encryption and secure transmission method of wireless transmission network communication data based on wireless channel feature detection is proposed. The ECC (Elliptic Curves Cryptography) algorithm analyzes by monitoring single-channel and multi-channel samples separately. The ciphertext protocol for data secure transmission is constructed, the Hash dynamic transmission protocol is used for data access control, the data dynamic symmetric key, the key construction and arithmetic coding design are constructed. Combined with the ellipse linear mapping method, the data is encrypted safely. According to the intensity of the wireless channel feature distribution, the scrambling degree is rearranged for the random scrambling and encryption of the data, and the random linear coding method is used to realize the random scrambling and encryption of the data, so as to realize the secure transmission of network communication data and the secure storage of information. The simulation test results show that using this method to encrypt and transmit wireless transmission network communication data has better security and stronger anti-attack ability, thereby improving the secure transmission performance of network communication data.