Improving the Coding Speed of Erasure Codes with Polynomial Ring Transforms

Erasure codes are widely used in today’s storage systems to cope with failures. Most of them use the finite field arithmetic. In this paper, we propose an implementation and a coding speed evaluation of an original method called PYRIT (PolYnomial RIng Transform) to perform operations between elements of a finite field into a bigger ring by using fast transforms between these two structures. Working in such a ring is much easier than working in a finite field. Firstly, it reduces the coding complexity by design. Secondly, it allows simple but efficient xor-based implementations by unrolling the operations thanks to the properties of the ring structure. We evaluate this proposition for Maximum Distance Separable erasure codes and we show that our method has better performances than common codes. Compared to the best known implementations, the coding speeds are increased by a factor varying from 1.5 to 2

Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes

In this paper, we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes, algebraic geometry codes, and affine variety codes. Our lemma corresponds to the non-systematic encoding of affine variety codes, and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a Grobner basis and a generalized inverse discrete Fourier transform. We clarify that our lemma yields the error-value estimation in the fast erasure-and-error decoding of a class of dual affine variety codes. Moreover, we show that systematic encoding corresponds to a special case of erasure-only decoding. The lemma enables us to reduce the computational complexity of error-evaluation from O(n^3) using Gaussian elimination to O(qn^2) with some mild conditions on n and q, where n is the code length and q is the finite-field size.Comment: 37 pages, 1 column, 10 figures, 2 tables, resubmitted to IEEE Transactions on Information Theory on Jan. 8, 201

Concatenation of convolutional and block codes Final report

Comparison of concatenated and sequential decoding systems and convolutional code structural propertie

Erasure Techniques in MRD codes

This book is organized into six chapters. The first chapter introduces the basic algebraic structures essential to make this book a self contained one. Algebraic linear codes and their basic properties are discussed in chapter two. In chapter three the authors study the basic properties of erasure decoding in maximum rank distance codes. Some decoding techniques about MRD codes are described and discussed in chapter four of this book. Rank distance codes with complementary duals and MRD codes with complementary duals are introduced and their applications are discussed. Chapter five introduces the notion of integer rank distance codes. The final chapter introduces some concatenation techniques.Comment: 162 pages; Published by Zip publishing in 201

Coding for Privacy in Distributed Computing

I et distribuert datanettverk samarbeider flere enheter for å løse et problem. Slik kan vi oppnå mer enn summen av delene: samarbeid gjør at problemet kan løses mer effektivt, og samtidig blir det mulig å løse problemer som hver enkelt enhet ikke kan løse på egen hånd. På den annen side kan enheter som bruker veldig lang tid på å fullføre sin oppgave øke den totale beregningstiden betydelig. Denne såkalte straggler-effekten kan oppstå som følge av tilfeldige hendelser som minnetilgang og oppgaver som kjører i bakgrunnen på de ulike enhetene. Straggler-problemet blokkerer vanligvis hele beregningen siden alle enhetene må vente på at de treigeste enhetene blir ferdige. Videre kan deling av data og delberegninger mellom de ulike enhetene belaste kommunikasjonsnettverket betydelig. Spesielt i et trådløst nettverk hvor enhetene må dele en enkelt kommunikasjonskanal, for eksempel ved beregninger langs kanten av et nettverk (såkalte kantberegninger) og ved føderert læring, blir kommunikasjonen ofte flaskehalsen. Sist men ikke minst gir deling av data med upålitelige enheter økt bekymring for personvernet. En som ønsker å bruke et distribuert datanettverk kan være skeptisk til å dele personlige data med andre enheter uten å beskytte sensitiv informasjon tilstrekkelig. Denne avhandlingen studerer hvordan ideer fra kodeteori kan dempe straggler-problemet, øke effektiviteten til kommunikasjonen og garantere datavern i distribuert databehandling. Spesielt gir del A en innføring i kantberegning og føderert læring, to populære instanser av distribuert databehandling, lineær regresjon, et vanlig problem som kan løses ved distribuert databehandling, og relevante ideer fra kodeteori. Del B består av forskningsartikler skrevet innenfor rammen av denne avhandlingen. Artiklene presenterer metoder som utnytter ideer fra kodeteori for å redusere beregningstiden samtidig som datavernet ivaretas ved kantberegninger og ved føderert læring. De foreslåtte metodene gir betydelige forbedringer sammenlignet med tidligere metoder i litteraturen. For eksempel oppnår en metode fra artikkel I en 8%-hastighetsforbedring for kantberegninger sammenlignet med en nylig foreslått metode. Samtidig ivaretar vår metode datavernet, mens den metoden som vi sammenligner med ikke gjør det. Artikkel II presenterer en metode som for noen brukstilfeller er opp til 18 ganger raskere for føderert læring sammenlignet med tidligere metoder i litteraturen.In a distributed computing network, multiple devices combine their resources to solve a problem. Thereby the network can achieve more than the sum of its parts: cooperation of the devices can enable the devices to compute more efficiently than each device on its own could and even enable the devices to solve a problem neither of them could solve on its own. However, devices taking exceptionally long to finish their tasks can exacerbate the overall latency of the computation. This so-called straggler effect can arise from random effects such as memory access and tasks running in the background of the devices. The effect typically stalls the whole network because most devices must wait for the stragglers to finish. Furthermore, sharing data and results among devices can severely strain the communication network. Especially in a wireless network where devices have to share a common channel, e.g., in edge computing and federated learning, the communication links often become the bottleneck. Last but not least, offloading data to untrusted devices raises privacy concerns. A participant in the distributed computing network might be weary of sharing personal data with other devices without adequately protecting sensitive information. This thesis analyses how ideas from coding theory can mitigate the straggler effect, reduce the communication load, and guarantee data privacy in distributed computing. In particular, Part A gives background on edge computing and federated learning, two popular instances of distributed computing, linear regression, a common problem to be solved by distributed computing, and the specific ideas from coding theory that are proposed to tackle the problems arising in distributed computing. Part B contains papers on the research performed in the framework of this thesis. The papers propose schemes that combine the introduced coding theory ideas to minimize the overall latency while preserving data privacy in edge computing and federated learning. The proposed schemes significantly outperform state-of-the-art schemes. For example, a scheme from Paper I achieves an 8% speed-up for edge computing compared to a recently proposed non-private scheme while guaranteeing data privacy, whereas the schemes from Paper II achieve a speed-up factor of up to 18 for federated learning compared to current schemes in the literature for considered scenarios.Doktorgradsavhandlin