Polynomial Ring Transforms for Efficient XOR-based Erasure Coding

The complexity of software implementations of MDS erasure codes mainly depends on the efficiency of the finite field operations implementation. In this paper, we propose a method to reduce the complexity of the finite field multiplication by using simple transforms between a field and a ring to perform the multiplication in a ring. We show that moving to a ring reduces the complexity of the operations. Then, we show that this construction allows the use of simple scheduling to reduce the number of operation

PYRIT: Polynomial Ring Transforms for Fast Erasure Coding

Pyrit make erasure coding operations as fast as possible by performing finite field multiplications in a ring

RS + LDPC-Staircase Codes for the Erasure Channel: Standards, Usage and Performance

Application-Level Forward Erasure Correction (AL-FEC) codes are a key element of telecommunication systems. They are used to recover from packet losses when retransmission are not feasible and to optimize the large scale distribution of contents. In this paper we introduce Reed-Solomon/LDPCStaircase codes, two complementary AL-FEC codes that have recently been recognized as superior to Raptor codes in the context of the 3GPP-eMBMS call for technology [1]. After a brief introduction to the codes, we explain how to design high performance codecs which is a key aspect when targeting embedded systems with limited CPU/battery capacity. Finally we present the performances of these codes in terms of erasure correction capabilities and encoding/decoding speed, taking advantage of the 3GPP-eMBMS results where they have been ranked first

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

Codage et décodage correcteur d'erreurs par matrice génératrice avec multiplications simplifiées dans coprs de galois

Procédé d'encodage d'un vecteur de donnée en un vecteur encodé transformé, selon un code correcteur d'erreur linéaire défini par une matrice génératrice (G), dans lequel : - le vecteur de donnée et la matrice génératrice (G) sont dans le corps fini de polynômes défini par : Bi=GF2(x)/(Pi(x)), p,(x) étant un polynôme irréductible qui divise le polynôme xn+1, i et n étant des entiers, - Le vecteur encodé transformé appartenant à un anneau et image par un isomorphisme du produit de la matrice génératrice (G) par le vecteur de donnée, l'isomorphisme, du corps fini B, dans un ensemble Ai, étant défini par : Fi, (ρ(χ))=ρ(χ)Qi(χ); où Ai, est l'idéal principal de l'anneau engendré par le polynôme χη+1/ρ,(χ), Q,(x) est l'unique idempotent dudit idéal principal, et p(x) un polynôme appartenant au corps B,, l'anneau étant défini par : R2,n=GF2(x)/(xn+1) ledit procédé comprenant : - une étape de calcul de l'image par l'isomorphisme F, du vecteur de donnée pour obtenir un vecteur de donnée transformé, - une étape de détermination d'une matrice transformée, dans laquelle chaque élément situé à une ligne et à une colonne est la somme de l'image de l'élément situé à ladite ligne et à ladite colonne d'une matrice brute (G) par l'isomorphisme F, et d'un élément de l'anneau qui n'a aucune composante dans l'idéal principal, la matrice brute (G) étant la matrice génératrice (G), - une étape de multiplication de la matrice transformée par le vecteur de donnée transformée pour obtenir ledit vecteur encodé transformé

Point-to-point transmetting method based on the use of an erasure coding scheme and TCP/IP protocol

The present invention concerns a reliable point-to-point transmitting method based on the joint use of an error recovery mechanism with the TCP protocol. The present invention also concerns a telecommunication system using any telecommunication network that implements the reliable point-to-point transmitting method according to the invention

Erasure code-based low storage blockchain node

The concept of a decentralized ledger usually implies that each node of a blockchain network stores the entire blockchain. However, in the case of popular blockchains, which each weigh several hundreds of GB, the large amount of data to be stored can incite new or low-capacity nodes to run lightweight clients. Such nodes do not participate to the global storage effort and can result in a centralization of the blockchain by very few nodes, which is contrary to the basic concepts of a blockchain. To avoid this problem, we propose new low storage nodes that store a reduced amount of data generated from the blockchain by using erasure codes. The properties of this technique ensure that any block of the chain can be easily rebuilt from a small number of such nodes. This system should encourage low storage nodes to contribute to the storage of the blockchain and to maintain decentralization despite of a globally increasing size of the blockchain. This system paves the way to new types of blockchains which would only bemanaged by low capacity nodes

Tetrys, an On-the-Fly Network Coding protocol

Working document of the NWCRG (Network Coding Research Group) group of IRTF (Internet Research Task Force)This document describes Tetrys, an On-The-Fly Network Coding (NC) protocol which can be used to transport delay and loss sensitive data over a network. This protocol can be used for unicast, multicast or anycast communications. It recovers lost packets by using added coded packets. Tetrys allows to recover all missing data within a time independent of the RTT and does not rely on a reliable feedback path to transmit data

Generic Application Programming Interface (API) for Sliding Window FEC Codes

Internet Research Task Force - Working document of the Network Coding Research Group (NWCRG), draft-roca-nwcrg-generic-fec-api-04 (work in progress), https://datatracker.ietf.org/doc/draft-roca-nwcrg-generic-fec-api/This document introduces a generic Application Programming Interface (API) for sliding window FEC codes. This API is meant to be compatible with any sliding window FEC code. It defines the core procedures and functions meant to control the codec (i.e., implementation of the FEC code). However, it leaves out all upper layer aspects that are the responsibility of the application or protocol making use of the codec. As a consequence, this is not an API for a FEC Scheme since certain mechanisms that must be defined by any FEC Scheme (e.g., signalling and FEC Payload IDs) are the responsibility of the caller instead of being addressed by the codec. A first goal of this document is to pave the way for a future open-source implementation of such codes, another goal is to simplify the development of content delivery protocols that rely on sliding window FEC codes for robust transmissions