Correcting adversarial errors with generalized regenerating codes

Abstract

<p style='text-indent:20px;'>Traditional regenerating codes are efficient tools to optimize both storage and repair bandwidth in storing data across a distributed storage system, particularly in comparison to erasure codes and data replication. In traditional regenerating codes, the collection of any <inline-formula><tex-math id="M1">\begin{document}$ k $\end{document}</tex-math></inline-formula> nodes can reconstruct all stored information and is called the reconstruction set, <inline-formula><tex-math id="M2">\begin{document}$ \aleph _R $\end{document}</tex-math></inline-formula>. A failed node can be regenerated from any <inline-formula><tex-math id="M3">\begin{document}$ d $\end{document}</tex-math></inline-formula> surviving nodes. These collections of <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula> nodes are called the regeneration sets, <inline-formula><tex-math id="M5">\begin{document}$ \aleph _H $\end{document}</tex-math></inline-formula>. The number of reconstruction sets and the number of regeneration sets satisfy <inline-formula><tex-math id="M6">\begin{document}$ |\aleph _R| = C_n^k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ |\aleph _H| = C_{n-1}^d $\end{document}</tex-math></inline-formula>. In generalized regenerating codes, we will have, <inline-formula><tex-math id="M8">\begin{document}$ 1\le|\aleph_R|\le C^k_n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ 1\le|\aleph_H|\le C_{n-1}^d $\end{document}</tex-math></inline-formula>. In this paper, we address the problem of secure generalized regenerating codes and present a coding scheme by focusing on the features of the generalized regenerating codes that protects data in the distributed storage system in presence of an active omniscient adversary. This adversary can maliciously alter the data stored on the nodes under its control and send erroneous outgoing message when contacted for the repair of failed nodes. In our scheme notwithstanding the presence of an adversary in distributed storage system, a data collector can still obtain the original file using a classical minimum distance decoder.</p>