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Secure Storage using Maximally Recoverable Locally Repairable Codes

Tim Janz, Hedongliang Liu, Rawad Bitar, Frank R. Kschischang

TL;DR

This paper addresses data secrecy in distributed storage systems encoded with Maximally Recoverable Locally Repairable Codes (MR-LRCs) under an $(l_1,l_2)$-eavesdropper. It proposes a per-group central processing unit (CPU) framework that uses local polynomials to enable secure global repairs and introduces two schemes, direct and forwarded, to control information leakage during repair. The authors derive explicit secrecy-dimension expressions for both repair schemes and show that positive secrecy dimension is achievable in several parameter regimes, improving security for MR-LRC-based DSSs. The work provides a practical repair framework with formal secrecy guarantees and points to future explorations of more general repair topologies and the full secrecy capacity of MR-LRC DSSs.

Abstract

This paper considers data secrecy in distributed storage systems (DSSs) using maximally recoverable locally repairable codes (MR-LRCs). Conventional MR-LRCs are in general not secure against eavesdroppers who can observe the transmitted data during a global repair operation. This work enables nonzero secrecy dimension of DSSs encoded by MR-LRCs through a new repair framework. The key idea is to associate each local group with a central processing unit (CPU), which aggregates and transmits the contribution from the intact nodes of their group to the CPU of a group needing a global repair. The aggregation is enabled by so-called local polynomials that can be generated independently in each group. Two different schemes -- direct repair and forwarded repair -- are considered, and their secrecy dimension using MR-LRCs is derived. Positive secrecy dimension is enabled for several parameter regimes.

Secure Storage using Maximally Recoverable Locally Repairable Codes

TL;DR

This paper addresses data secrecy in distributed storage systems encoded with Maximally Recoverable Locally Repairable Codes (MR-LRCs) under an $(l_1,l_2)$-eavesdropper. It proposes a per-group central processing unit (CPU) framework that uses local polynomials to enable secure global repairs and introduces two schemes, direct and forwarded, to control information leakage during repair. The authors derive explicit secrecy-dimension expressions for both repair schemes and show that positive secrecy dimension is achievable in several parameter regimes, improving security for MR-LRC-based DSSs. The work provides a practical repair framework with formal secrecy guarantees and points to future explorations of more general repair topologies and the full secrecy capacity of MR-LRC DSSs.

Abstract

This paper considers data secrecy in distributed storage systems (DSSs) using maximally recoverable locally repairable codes (MR-LRCs). Conventional MR-LRCs are in general not secure against eavesdroppers who can observe the transmitted data during a global repair operation. This work enables nonzero secrecy dimension of DSSs encoded by MR-LRCs through a new repair framework. The key idea is to associate each local group with a central processing unit (CPU), which aggregates and transmits the contribution from the intact nodes of their group to the CPU of a group needing a global repair. The aggregation is enabled by so-called local polynomials that can be generated independently in each group. Two different schemes -- direct repair and forwarded repair -- are considered, and their secrecy dimension using MR-LRCs is derived. Positive secrecy dimension is enabled for several parameter regimes.
Paper Structure (18 sections, 8 theorems, 30 equations, 5 figures)

This paper contains 18 sections, 8 theorems, 30 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linearized Reed--Solomon Codes
  4. Skew Lagrange Polynomials and Lagrange Basis
  5. Maximally Recoverable Locally Repairable Codes
  6. Novel Global Repair of MR-LRCs
  7. The Naive Global Repair of MR-LRC
  8. Direct and Forwarded Global Repair
  9. Local Polynomials
  10. Secrecy Dimension of MR-LRCs
  11. Secrecy Dimension for Direct and Forwarded Global Repair
  12. Direct Global Repair
  13. Forwarded Global Repair
  14. Comparison of Direct and Forwarded Global Repair
  15. Conclusion and Outlook
  16. ...and 3 more sections

Key Result

Theorem 1

We follow the notations in def:local_polynomial. Let $f$ be an encoding polynomial of the outer code $\mathcal{C}_{\mathrm{out}}=\mathcal{C}_{\mathrm{LRS}}^{\sigma, k}(\mathbf{b},\boldsymbol{\beta})$. Let $\Delta_{\mathrm{gl},1}:=\{i\mid (i,j)\in \Delta_{\mathrm{gl}}\}$. It holds that

Figures (5)

  • Figure 1: Illustration of a DSS with $N = 15$ nodes and storing $k=7$ independent symbols. The DSS is encoded by an MR-LRC with $g=3$ groups, locality $r=3$, local distance $\delta=3$ (parities in light gray) and $h=2$ global parities (in dark gray). The failed nodes marked with diamonds can be repaired locally while the failed nodes marked with stars need data from other groups to be repaired. The DSS is in presence of a $(1,1)$-eavesdropper who can read the data stored on one node (marked by a blue circle) and the downloaded and stored data of any node in the top group (marked by a red triangle).
  • Figure 2: Illustration of two different global repair schemes where an erasure (star) in the first group is repaired with global repair. Each circle depicts a CPU of a group that coordinates a repair. The nodes in the groups are depicted as the little squares. The forwarding list for (b) is $\mathcal{F}=\{2,3,4,5,1\}$
  • Figure 3: Illustration of the global repair schemes for the DSS from \ref{['fig:MR-LRC_example']}. The failed nodes marked with stars need to be repaired globally. In (a) they are repaired by direct global repair and in (b) forwarded global repair is used with the forwarding list $\mathcal{F}=\{2,3,1\}$. Both repairs are coordinated by the CPUs of the groups, depicted by the circles on the left. The secrecy rates of the DSS with respective repair schemes are given in \ref{['ex:1']}.
  • Figure 4: Plot of the secrecy dimension of a DSS that uses an MR-LRC with forwarded global repair (blue) and direct global repair (red) for fixed parameters $l_2=1$, $l_1=0$$r=7$, $h=3$. The secrecy dimensions are the same for $g\leq 3$. For $g>3$ forwarded global repair has a higher secrecy dimension. In addition, the secrecy dimension of an LRC-coded DSS without global repair, i.e., $h=0$ is plotted.
  • Figure 5: Illustration of a DSS with $N = 9$ nodes and storing $k=5$ independent symbols. The DSS is encoded by an MR-LRC with $g=3$ groups, locality $r=2$, local distance $\delta=2$ (parities in light gray) and $h=1$ global parities (in dark gray). The DSS is observed by a $(1,1)$-eavesdropper who can read the downloaded and stored data of any node in the top group (marked by a red triangle) and the data stored on one node (marked by a blue circle).

Theorems & Definitions (15)

  • Definition 1: Linearized Reed--Solomon (LRS) Codes
  • Definition 2: Skew Lagrange Polynomials
  • Definition 3: MR-LRC blaum2013partialgopalan2014explicit
  • Definition 4: Local Polynomial
  • Theorem 1
  • Lemma 1: Secrecy Lemmashah2011information
  • Lemma 2
  • Theorem 2
  • Theorem 3: Secrecy Dimension with Direct Global Repair
  • Theorem 4: Secrecy Dimension with Forwarded Global Repair
  • ...and 5 more