On AMD Codes for Storage Systems. ITW 2015 ... This presentation is on privacy and security aspects of distributed ... Security and Integrity in Storage Networks.
Introduction
AMD codes
Our contributions
Discussion
On Algebraic Manipulation Detection (AMD) Codes from Linear Codes and their Application to Storage Systems J. Harshan and Fr´ed´erique Oggier Division of Mathematical Sciences, Nanyang Technological University, Singapore
Supported by the MoE Tier-2 grant “eCode: erasure codes for data center environments”
Oct. 2015 1/13 On AMD Codes for Storage Systems
ITW 2015
Introduction
AMD codes
Our contributions
Discussion
So far...
This presentation is on privacy and security aspects of distributed storage 2/13 On AMD Codes for Storage Systems
ITW 2015
Introduction
AMD codes
Our contributions
Discussion
Security and Integrity in Storage Networks
The model is such that message stored is private and cannot be seen by adversary, but adversary can manipulate the stored messages using a priori information. On AMD Codes for Storage Systems
3/13 ITW 2015
Introduction
AMD codes
Our contributions
Discussion
Security and Integrity in Storage Networks
Error control codes (K.V Rashmi et al. 2012), Hash functions (F. Oggier and A. Datta 2011, Y.S. Han et al. 2011), Message authentication codes (Chen and Lee, 2013)
4/13 On AMD Codes for Storage Systems
ITW 2015
Introduction
AMD codes
Our contributions
Discussion
Highlights of our work ◮
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We apply AMD codes to distributed storage ◮
Information theoretic security approach
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Low storage overhead
We have developed on the work of ◮
E. Jongsma, Algebraic Manipulation Detection Codes, 2008
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R. Cramer et al. Detection of Algebraic Manipulation with Applications to Robust Secret Sharing and Fuzzy Extractors, 2008
We provide new insights on construction of AMD codes 5/13
On AMD Codes for Storage Systems
ITW 2015
Introduction
AMD codes
Our contributions
Discussion
AMD Codes
6/13 On AMD Codes for Storage Systems
ITW 2015
Introduction
AMD codes
Our contributions
Discussion
AMD codes from Linear Codes (E. Jongsma, 2008) A (q m , N q m+1 )-AMD code from a linear (N, m) code over Fq E : Fm → Fm q q × ZN × Fq , v 7→ (v, i, [vG]i ) where G is the generator matrix of the linear code. Manipulation E(v) + δ = (v, i, f (i, v)) + (∆(v), g1 , g2 ) = (v + ∆(v), i + g1 , f (i, v) + g2 ). Deception Event D(E(v) + δ) = D(E(v′ )) = v′ for v′ 6= v On AMD Codes for Storage Systems
7/13 ITW 2015
Introduction
AMD codes
Our contributions
Discussion
A Relevant Metric (E. Jongsma, 2008) µ(S) = max max |{i, (c − c′ )i = a}| ′ c6=c ∈S a∈Fq
8/13 On AMD Codes for Storage Systems
ITW 2015
Introduction
AMD codes
Our contributions
Discussion
Performance of AMD codes
Theorem (E. Jongsma, 2008) A (q m , N q m+1 )-AMD code E is ǫ-secure for ǫ=
µ(cl(C)) where cl(C) := ∪t∈ZN rott (C) N
if for all t ∈ ZN , t 6= 0, rott (C) ∩ C = 0, and ǫ = 1 otherwise.
9/13 On AMD Codes for Storage Systems
ITW 2015
Introduction
AMD codes
Our contributions
Discussion
Preliminary Results Rate of the linear code
Lemma 1 m ≤ . N 2
Lower bound on µ(cl(C))
Lemma µ(cl(C)) ≥ 2m. 10/13 On AMD Codes for Storage Systems
ITW 2015
Introduction
AMD codes
Our contributions
Discussion
Example code for m = 2 Parameters: N = 16, q = 17 E : F217 7→ F217 × Z16 × F17 , v 7→ (v, i, [vG]i ).
G=
16 8 2 15 4 1 4 11 4 10 12 6 11 15 13 1 10 4 5 1 8 15 2 8 7 1 6 1 6 4 15 6
Properties ◮
The parameter µ(cl(C)) = 7 ≥ 4, 11/13
On AMD Codes for Storage Systems
ITW 2015
Introduction
AMD codes
Our contributions
Discussion
New Results Theorem A (q m , N q m+1 )-AMD code is ǫ-secure for ǫ=
µ(C) 1 + m N q
Theorem The code C generated by G has µ(C) = m if and only if any ˜ = G , is full rank. (m + 1) × (m + 1) submatrix of G 1 12/13 On AMD Codes for Storage Systems
ITW 2015
Introduction
AMD codes
Our contributions
Discussion
Future Work
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Improved bounds on the probability of deception
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Connection to erasure codes from Vandermonde matrices
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Implementation aspects
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Relevant to other applications such as robust secret sharing schemes and fuzzy extractors.
13/13 On AMD Codes for Storage Systems
ITW 2015
