Exact-regenerating Codes Research Articles

Codes Between MBR and MSR Points With Exact Repair Property

In this paper distributed storage systems with exact repair are studied. A construction for regenerating codes between the minimum storage regenerating (MSR) and the minimum bandwidth regenerating (MBR) points is given. To the best of author's knowledge, no previous construction of exact-regenerating codes between MBR and MSR points is done except in the work by Tian et al. On contrast to their work, the methods used here are elementary. In this paper it is shown that in the case that the parameters $n$, $k$, and $d$ are close to each other, the given construction is close to optimal when comparing to the known functional repair capacity. This is done by showing that when the distances of the parameters $n$, $k$, and $d$ are fixed but the actual values approach to infinity, the fraction of the performance of constructed codes with exact repair and the known capacity of codes with functional repair, approaches to one. Also a simple variation of the constructed codes with almost the same performance is given.

Efficient Exact Regenerating Codes for Byzantine Fault Tolerance in Distributed Networked Storage

Today's large-scale distributed storage systems are commonly built using commodity software and hardware. As a result, crash-stop and Byzantine failures in such systems become more and more prevalent. In the literature, regenerating codes have been shown to be a more efficient way to disperse information across multiple storage nodes and recover from crash-stop failures. In this paper, we propose a novel decoding design of product-matrix constructed regenerating codes in conjunction with integrity check that allows exact regeneration of failed nodes and data reconstruction in the presence of Byzantine failures. A progressive decoding mechanism is incorporated in both procedures to leverage computation performed thus far. Unlike previous works, our new regenerating code decoding has the advantage that its building blocks, such as Reed-Solomon codes and standard cryptographic hash functions, are relatively well-understood because of their widespread applications. The fault tolerance and security properties of the proposed schemes are also analyzed. In addition, the performance of the proposed schemes, in terms of the average number of access nodes and the reconstruction failure probability versus the node failure probability, are also evaluated by Monte Carlo simulations.

Secure Regenerating Codes Based on Rashmi-Shah-Kumar MBR Codes

SUMMARY In this paper, we present a construction of (n,k,d,m )s ecure regenerating codes for distributed storage systems against eavesdroppers that can observe either data stored in at most m storage nodes or downloaded data for repairing at most m failed nodes in a network where m < k ≤ d ≤ n − 1. The (n,k,d,m) secure regenerating code is based on an (n,k,d) minimum bandwidth regenerating (MBR) code, which was proposed by Rashmi, Shah and Kumar as optimal exact-regenerating codes, for all values of the parameters (n,k,d). The (n,k,d,m) secure regenerating codes have the security as a secret sharing scheme such that even if an eavesdropper knows either data stored in at most m storage nodes or downloaded data for repairing at most m failed nodes, no information about data leaks to the eavesdropper.

Optimal Exact-Regenerating Codes for Distributed Storage at the MSR and MBR Points via a Product-Matrix Construction

Regenerating codes are a class of distributed storage codes that allow for efficient repair of failed nodes, as compared to traditional erasure codes. An <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$[n, k, d]$</tex></formula> regenerating code permits the data to be recovered by connecting to any <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$k$</tex></formula> of the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> nodes in the network, while requiring that a failed node be repaired by connecting to any <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$d$</tex></formula> nodes. The amount of data downloaded for repair is typically much smaller than the size of the source data. Previous constructions of exact-regenerating codes have been confined to the case <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n=d+1$</tex> </formula> . In this paper, we present optimal, explicit constructions of (a) Minimum Bandwidth Regenerating (MBR) codes for all values of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$[n, k, d]$</tex></formula> and (b) Minimum Storage Regenerating (MSR) codes for all <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$[n, k, d\geq 2k-2]$</tex></formula> , using a new product-matrix framework. The product-matrix framework is also shown to significantly simplify system operation. To the best of our knowledge, these are the first constructions of exact-regenerating codes that allow the number <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> of nodes in the network, to be chosen independent of the other parameters. The paper also contains a simpler description, in the product-matrix framework, of a previously constructed MSR code with <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$[n=d+1, k, d\geq 2k-1]$</tex></formula> .