Reed-Solomon Research Articles

NEW THRESHOLD PRIVATE SET INTERSECTION PROTOCOLS

With the rising amount of digital technologies that we use on a daily basis, it is more important than ever to handle and process private data securely. Research and academic communities are becoming increasingly interested in multi-party computation, with a focus on the field of Private Set Intersection (PSI). In this regard, this work introduces a novel technique that successfully converts the Cid-Davidson Private Set Intersection protocol into a Threshold Private Set Intersection. It achieves this conversion by introducing two new protocols, TPSI-1 and TPSI-2, and utilizing two previously developed methodologies while the Reed-Solomon codes and the Shamir-secret sharing scheme are the foundations of TPSI-1, whereas Secure Comparison Protocols serve as the foundation for TPSI-2. Specifically, our suggested protocols perform better asymptotically than previous threshold PSI protocols because they have a fixed number of rounds and linear communication and computation complexity that increase with data set size. This study adds to the continuous effort to strengthen the security and effectiveness of private data calculations, highlighting how safe data processing is changing in an era where digital technologies are ingrained in every aspect of our lives.

Efficient DNA-based data storage using shortmer combinatorial encoding

Data storage in DNA has recently emerged as a promising archival solution, offering space-efficient and long-lasting digital storage solutions. Recent studies suggest leveraging the inherent redundancy of synthesis and sequencing technologies by using composite DNA alphabets. A major challenge of this approach involves the noisy inference process, obstructing large composite alphabets. This paper introduces a novel approach for DNA-based data storage, offering, in some implementations, a 6.5-fold increase in logical density over standard DNA-based storage systems, with near-zero reconstruction error. Combinatorial DNA encoding uses a set of clearly distinguishable DNA shortmers to construct large combinatorial alphabets, where each letter consists of a subset of shortmers. We formally define various combinatorial encoding schemes and investigate their theoretical properties. These include information density and reconstruction probabilities, as well as required synthesis and sequencing multiplicities. We then propose an end-to-end design for a combinatorial DNA-based data storage system, including encoding schemes, two-dimensional (2D) error correction codes, and reconstruction algorithms, under different error regimes. We performed simulations and show, for example, that the use of 2D Reed-Solomon error correction has significantly improved reconstruction rates. We validated our approach by constructing two combinatorial sequences using Gibson assembly, imitating a 4-cycle combinatorial synthesis process. We confirmed the successful reconstruction, and established the robustness of our approach for different error types. Subsampling experiments supported the important role of sampling rate and its effect on the overall performance. Our work demonstrates the potential of combinatorial shortmer encoding for DNA-based data storage and describes some theoretical research questions and technical challenges. Combining combinatorial principles with error-correcting strategies, and investing in the development of DNA synthesis technologies that efficiently support combinatorial synthesis, can pave the way to efficient, error-resilient DNA-based storage solutions.

Private information retrieval from locally repairable databases with colluding servers

We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups g, locality r, local distance δ and dimension k. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed–Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only r nodes (out of r+δ−1) are downloaded per server. The PIR scheme achieves the (download) rate R=(N−k−rt+1)/N, where N=gr is the length of the MDS code obtained after removing the local parities, and for any t colluding servers such that k+rt≤N. For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed–Solomon codes, but Reed–Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications.

Storage and Data Processing Technology in the Era of Information Explosion: from Traditional to Edge Computing

Data-intensive IoT gradually occupying the market, and efficient and reliable storage algorithms of signals and information are becoming increasingly important. The old-era storage algorithms no longer meet the needs of digital and intelligent cloud storage due to large space usage and low operating efficiency. One of the important topics in IoT cloud computing is to find a turning point from traditional storage methods to develop data-intensive and efficient storage methods. This research analyzed the principles, features, advantages, and disadvantages of many classic storage algorithms, including RS code, EVENODD code, Pyramid code, etc. It is to find storage algorithms suitable with higher storage efficiency in the IoT. Finally, Pyramid code is leading in data recovery efficiency and scalability and is suitable for higher and faster storage needs of the IoT. However, the shortcoming of Pyramid code also has challenges in the large number of failures that occur, due to its slower recovery speed.

Improved List-Decodability and List-Recoverability of Reed–Solomon Codes via Tree Packings

Improved List-Decodability and List-Recoverability of Reed–Solomon Codes via Tree Packings

Secure transmission and storage of Internet of Things data

The Internet of Everything is the ultimate level form of the Internet of Things. The result of digitizing the objects in life is a huge amount of data that needs to be stored and transmitted. This paper will briefly introduce the IoT and the current status of data processing. Different coding methodologies, Reed Solomon (RS) codes, Low-Density Parity-Check (LDPC) codes, and Low Complexity Parity Check (LCPC) codes will also be explained in detail from their encoding and decoding aspect. Then different codes are specifically analyzed for their usage scenarios in IoT data transmission and storage based on their particular characteristics. In the experimental section, different codes are compared from different perspectives. The main comparisons are made in terms of memory usage, operation time, energy consumption per bit of encoding, and error correction performance. The results of the analysis are RS code is simple but lacks error correction, the LDPC code has good error correction but is more complex, LCPC code strikes a balance between power efficiency and performance.

Several Efficient Alternative Solutions for RS Codes in RAID 6

The RAID-6 specification requires a storage system with multiple storage devices to be able to endure the failure of any two parities. RAID-6 can be implemented using a variety of erasure coding methods, but each has drawbacks, like low fault tolerance, bad writing performance low decoding speed, and so on. In this paper, we research a variety of materials, explore RAID 6 disks to comprehend their purpose and prerequisites, and understand how to decode RS codes. In order to help people find a decoding method that is more appropriate to their own systems and actual needs, we research and compare numerous high-quality alternating decoding techniques, and the RAID 6 Liberation codes and Row-Diagonal Parity emerge as the winners. We introduce their decoding methods briefly and list their advantages and disadvantages in this paper for people to request based on their needs. Finally, we compare different decoding methods, propose practical application scenarios and future development directions for these more efficient decoding methods, and also point out their existing shortcomings.

Data Storage and Reliability Analysis on the Internet of Things Based on Pyramid Code

As the IoT (Internet of Things) develops, computers and devices generate a lot of data that needs to be processed and stored. Pyramid code, as an update of the Reed-Solomon code, is suitable for correcting data failure on various sides as an erasure code. Not only does Pyramid code reduce the cost of computation, but it can also meet the reliability and security requirements of IoT. The convenience of the Pyramid code is evaluated by comparing the (15, 12) RS code to the (16, 12) Pyramid code in this paper. Then, section 4 investigates how Pyramid code can match the needs of reliability, security, cost-effectiveness, etc. After that, the following section aims to give the applications of Pyramid code in IoT. There are many applications and this section concludes the several outstanding scenarios, such as healthcare, smart city, etc.

Single trace HQC shared key recovery with SASCA

This paper presents practicable single trace attacks against the Hamming Quasi-Cyclic (HQC) Key Encapsulation Mechanism. These attacks are the first Soft Analytical Side-Channel Attacks (SASCA) against code-based cryptography. We mount SASCA based on Belief Propagation (BP) on several steps of HQC’s decapsulation process. Firstly, we target the Reed-Solomon (RS) decoder involved in the HQC publicly known code. We perform simulated attacks under Hamming weight leakage model, and reach excellent accuracies (superior to 0.9) up to a high noise level (σ = 3), thanks to a re-decoding strategy. In a real case attack scenario, on a STM32F407, this attack leads to a perfect success rate. Secondly, we conduct an analogous attack against the RS encoder used during the re-encryption step required by the Fujisaki-Okamoto-like transform. Both in simulation and practical instances, results are satisfactory and this attack represents a threat to the security of HQC. Finally, we analyze the strength of countermeasures based on masking and shuffling strategies. In line with previous SASCA literature targeting Kyber, we show that masking HQC is a limited countermeasure against BP attacks, as well as shuffling countermeasures adapted from Kyber. We evaluate the “full shuffling” strategy which thwarts our attack by introducing sufficient combinatorial complexity. Eventually, we highlight the difficulty of protecting the current RS encoder with a shuffling strategy. A possible countermeasure would be to consider another encoding algorithm for the scheme to support a full shuffling. Since the encoding subroutine is only a small part of the implementation, it would come at a small cost.

Solving polynomial systems over non-fields and applications to modular polynomial factoring

We study the problem of solving a system of m polynomials in n variables over the ring of integers modulo a prime-power pk. The problem over finite fields is well studied in varied parameter settings. For small characteristic p=2, Lokshtanov et al. (SODA'17) initiated the study, for degree d=2 systems, to improve the exhaustive search complexity of O(2n)⋅poly(m,n) to O(20.8765n)⋅poly(m,n); which currently is improved to O(20.6943n)⋅poly(m,n) in Dinur (SODA'21). For large p but constant n, Huang and Wong (FOCS'96) gave a randomized poly(d,m,log⁡p) time algorithm. Note that for growing n, system-solving is known to be intractable even with p=2 and degree d=2.We devise a randomized poly(d,m,log⁡p)-time algorithm to find a root of a given system of m integral polynomials of degrees bounded by d, in n variables, modulo a prime power pk; when n+k is constant. In a way, we extend the efficient algorithm of Huang and Wong (FOCS'96) for system-solving over Galois fields (i.e., characteristic p) to system-solving over Galois rings (i.e., characteristic pk); when k>1 is constant. The challenge here is to find a lift of singularFp-roots (exponentially many); as there is no efficient general way known in algebraic-geometry for resolving singularities.Our algorithm has applications to factoring univariate polynomials over Galois rings. Given f∈Z[x] and a prime-power pk (k≥2), finding factors of fmodpk has a curious state-of-the-art. It is solved for large k by p-adic factoring algorithms (von zur Gathen, Hartlieb, ISSAC'96); but unsolved for small k. In particular, no nontrivial factoring method is known for k≥5 (Dwivedi, Mittal, Saxena, ISSAC'19). One issue is that degree-δ factors of f(x)modpk could be exponentially many, as soon as k≥2. We give the first randomized poly(deg⁡(f),log⁡p)-time algorithm to find a degree-δ factor of f(x)modpk, when k+δ is constant. Our method has potential application in algebraic coding theory. In particular, extending algebraic geometric and Reed-Solomon codes to Galois rings could enable new and improved bounds on their underlying efficiency parameters.

A dual-rule encoding DNA storage system using chaotic mapping to control GC content.

DNA as a novel storage medium is considered an effective solution to the world's growing demand for information due to its high density and long-lasting reliability. However, early coding schemes ignored the biologically constrained nature of DNA sequences in pursuit of high density, leading to DNA synthesis and sequencing difficulties. This article proposes a novel DNA storage coding scheme. The system encodes half of the binary data using each of the two GC-content complementary encoding rules to obtain a DNA sequence. After simulating the encoding of representative document and image file formats, a DNA sequence strictly conforming to biological constraints was obtained, reaching a coding potential of 1.66 bit/nt. In the decoding process, a mechanism to prevent error propagation was introduced. The simulation results demonstrate that by adding Reed-Solomon code, 90% of the data can still be recovered after introducing a 2% error, proving that the proposed DNA storage scheme has high robustness and reliability. Availability and implementation: The source code for the codec scheme of this paper is available at https://github.com/Mooreniah/DNA-dual-rule-rotary-encoding-storage-system-DRRC.

A Comparative Analysis of Energy Consumption in Various Wireless Sensor Network Techniques

The objective of this study is to analyze the energy consumption associated with modern methodologies utilized in wireless sensor networks and to conduct a comparative assessment with Reed Solomon (RS) codes. This paper presents three discrete techniques for wireless sensor networks. The strategies mentioned include the Self-Evolving Sensor System (SESS), the Secure and Adaptive Key Management utilizing Multipath Routing Protocol (SAKM-MRP), and the National Instruments Secure Reference-based Data Aggregation (NI-SRDA). A distinct algorithm was developed for each method to examine the energy use. Based on the experimental results, it has been shown that the RS-codes approach consumes a considerably greater quantity of energy compared to the SESS methods, which, in contrast, exhibit a significantly lower energy consumption. When comparing the efficiency of RS-codes and SESS methods, it is observed that the SAKN-MRP technique exhibits a more significant decrease in energy consumption. Compared to the RS-Codes system, the SESS scheme stands out with a significant 45.5% reduction in energy usage at the maximum delivery node. Similarly, the SAKM-MRP scheme showcases an average decrease of 35.7% in energy consumption. Notably, the NI-SRDA scheme achieves an impressive 60% reduction in energy consumption, underscoring its remarkable impact on energy efficiency. In a broader sense, it can be inferred that the NI-SDRA technique holds promise as an energy-efficient solution for wireless sensor networks in comparison to alternative strategies suggested in the current study.

Three New Classes of Subsystem Codes

In this paper, we construct three classes of Clifford subsystem maximum distance separable (MDS) codes based on Reed-Solomon codes and extended generalized Reed-Solomon codes over finite fields [see formula in PDF] for specific code lengths. Moreover, our Clifford subsystem MDS codes are new because their parameters differ from the previously known ones.

ACRS-Raft: A Raft Consensus Protocol for Adaptive Data Maintenance in the Metaverse Based on Cauchy Reed–Solomon Codes

ACRS-Raft: A Raft Consensus Protocol for Adaptive Data Maintenance in the Metaverse Based on Cauchy Reed–Solomon Codes

Advanced Elastic Reed–Solomon Codes for Erasure-Coded Key–Value Stores

Advanced Elastic Reed–Solomon Codes for Erasure-Coded Key–Value Stores

Optimal Asymmetric Quantum Codes from the Euclidean Sums of Linear Codes

In this paper, we first give the definition of the Euclidean sums of linear codes, and prove that the Euclidean sums of linear codes are Euclidean dual-containing. Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes, and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields. Moreover, these optimal asymmetric quantum error-correcting codes constructed in this paper are different from the ones in the literature.

Optimizing RAID 6 performance and reliability using Reed-Solomon codes: Implementation, analysis, and exploration of alternative methods

This manuscript offers an exhaustive exploration of Reed-Solomon codes and their role in Redundant Array of Independent Disks level 6 (RAID 6) systems. The use of these codes is instrumental in enhancing data reliability and bolstering system fault tolerance. A detailed examination and analysis of Reed-Solomon codes deliver pivotal insights into the performance optimization process of RAID 6 systems. Despite the widespread usage of Reed-Solomon codes, these codes come with their own set of challenges, such as increased computational complexity, which could potentially hamper system performance. Considering this, this paper explores potential alternatives to Reed-Solomon codes, specifically - Row-Diagonal Parity (RDP), Liberation Codes for RAID 6, and Johnston's Code (J-Code). Each of these alternatives carries its unique set of benefits and limitations, which calls for a careful selection process based on specific application scenarios. The manuscript also probes into how these alternatives can be effectively related to RAID 6 systems. With the rapid expansion of data centers and cloud computing, the demand for data reliability and fault tolerance continues to surge. Future research is anticipated to revolve around the optimization of RAID 6 systems, particularly focusing on the selection and implementation of the most suitable coding scheme. Furthermore, studying and applying a wider range of alternatives to meet diverse requirements will also be a significant area of focus.

An Investigation on Reed-Solomon Codes as Erasure Coding Technique on Its Properties and Utilizations

Erasure coding is an essential part of cloud computing, which is an important technology for effective data storage for recovering data that may be lost due to various reasons, there are various erasure coding techniques in the market. In this paper, linear MDS codes, which is a branch of erasure coding, will be investigated on their performance and usage. This paper will focus on the Reed-Solomon Code, which is the most implemented form of linear MDS codes, on three different aspects: 1) the methodologies of the encoding and decoding operations; 2) the pros and cons of different forms of Reed-Solomon Codes; 3) the different ways that different Reed-Solomon Codes are being employed. Moreover, the paper includes the definition of general Cloud Computing for the audience to understand its importance, how the erasure coding acts like a fault tolerance system of Cloud Computing, and how different kinds of Reed-Solomon code perform on tolerating erasures in cloud storage failures.

Comparison Of Two Encoding Methods in RAID6 Storage Architecture and Analysis of Improvement of EVENODD Encodings

This paper provides a comparative analysis of two widely used error correction codes: RS (Reed-Solomon) and EVENODD. The analysis covers principles, algebraic forms, computational complexities, minimum Hamming distances, decoding processes, and specific application scenarios. RS encoding is known for its versatility, allowing for the flexible addition of parity symbols. However, it comes with a high computational cost due to polynomial evaluations. On the other hand, EVENODD encoding offers simplicity and computational efficiency through mere XOR operations but has limitations on the number of parity symbols it can generate. The paper explores the algebraic representations of both codes, discusses their minimum Hamming distances, and evaluates their decoding processes. Additionally, it analyzes the computational requirements of EVENODD under small write operations and provides a comprehensive comparison of the computational complexities of RS and EVENODD, aiding practitioners in choosing the most suitable error correction scheme for specific applications. A seeming limitation in our building process is because of the number of information disks must fall into a prime number. Nevertheless, if the preferred quantity of data sectors isn't a prime number, we can easily posit the existence of additional disks filled entirely with zeros, and this won't impact the encoding and decoding procedures.

The Comparison of RS Codes, LDPC Codes, And Turbo Codes

Error correction codes have been widely used in various fields in recent years, including the communication field, storage field, digital media field, computer network field, and so on. In this paper, the development history of error correction codes is briefly summarized; and the principles and performance of three kinds of error correction control codes are systematically summarized starting from RS codes, LDPC codes, and Turbo codes. Three types of error correction codes are compared and analyzed in the field of wireless optical communication, and the error correction characteristics of different codes in three different weather conditions are compared and summarized for the applicable environments. The error correction codes are also compared in the field of aviation flight telemetry, and which codes are suitable for uplink and downlink telemetry in flight telemetry are summarized. At the same time, the paper provides an outlook on the future development trend of error correction codes.