1. Array BP-XOR Codes for Reliable Cloud Storage Systems Yongge Wang UNC Charlotte, USA IEEE ISIT(International Symposium on Information Theory) 2013 1

2. Outlines Introduction Edge-colored graphs Array BP-XOR codes Flat non-MDS BP-XOR codes Conclusion 2

3. EaaS and Remote Computation on Data IaaS, PaaS, SaaS, NaaS, EaaS, etc. data (services) are stored at remote client we may need the remote cloud server to process some query (processing) on these data instead of downloading the data to local computer and process the data 3

4. Where is the privacy? Data is stored on the remote server in clear? we do not trust the remote server what is the solution? encrypt the data and store the cipher text? how can do “computation on the data remotely”? 4

5. Examples many choices for personal cloud data storage Dropbox, SkyDrive, Google Drive, Amazon Cloud drive, Apple iCloud, Ubuntu One, etc. do you trust any one of these server and put your data (your memory) there? reliability? privacy? 5

6. Our Solution XOR-MDS codes are converted to XOR-based Secret Sharing Schemes 2-out-of-6 SSS(secret sharing scheme) Register accounts at: Dropbox, SkyDrive, Google Drive, Amazon Cloud drive, Apple iCloud, Ubuntu One, etc. data from any two servers are sufficient, but each single server learns zero information about data 6

7. Array Codes Mainly used for data storage system example array codes – Blaum, et al: EVENODD (2 disk faults) – Blaum, et al: extended EVENODD (3 disk faults). – [2k, k, d] chain code – Simple Product Code (SPC) – Row-Diagonal Parity (RDP) – Blaum RDP(p, i) for i ≦ 8 7 [1] M. Blaum, J. Brady, J. Bruck, and J. Menon. EVENODD: An efficient scheme for tolerating double disk failures in raid architectures. IEEE Trans. Computers, 44(2):192–202, 1995.

8. Sample EVENODD code 8

9. Array Code Definition Message set: M = {0, 1} and fixed n, k, t, and b Information variables: let v 1,…, v bk A t-erasure tolerating [n, k] array code is a b x n matrix C = [α i,j ] 1≤i≤b,1≤j≤n Each α i,j {0, 1} is XOR of information symbols v 1,…, v bk recovered from any n - t columns of the matrix For, call v ij a neighbor of α i,j and σ the degree of α i,j. A t-erasure tolerating [n,k] b x n array code C is said to be Maximum Distance Separable (MDS) if k = n - t 9

10. Array BP-XOR Code [Definition] A t-erasure tolerating [n, k] array code C = [α i,j ] 1≤i≤b,1≤j≤n is called an [n, k] array BP-XOR code if all information symbols v 1,…, v bk can be recovered from any n − t columns of encoding symbols using the BP-decoding process. 10

11. Degree 2 Array BP-XOR Code [Theorem] If each encoding symbol in C = [α i,j ] 1≤i≤b,1≤j≤n has degree at most 2, then the restricted array BP-XOR codes are equivalent to edge-colored graphs introduced by Wang and Desmedt for tolerating network homogeneous faults. 11

12. Edge-colored Graph Definition Definition : (Wang and Desmedt [15]) An edge-colored graph is a tuple G = (V, E,C, f), with V the node set, E the edge set, C the color set, and f a map from E onto C. The structure Z C,t = {Z : Z ⊆ E and |f(Z)| ≤ t}. is called a t-color adversary structure. Let A,B ∈ V be distinct nodes of G. A and B are called (t+1)-color connected for t ≥ 1 if for any color set C t ⊆ C of size t, there is a path p from A to B in G such that the edges on p do not contain any color in C t. An edge-colored graph G is (t+1)-color connected if and only if for any two nodes A and B in G, they are (t+1)-color connected. 12 [15] Yongge Wang and Yvo Desmedt. Edge-colored graphs with applications to homogeneous faults. Inf. Process. Lett., 111(13):634–641, 2011.

13. 13 3-color connected graph G 4,2 with 7 nodes, 12 edges, and 4 colors. Removal of any two colors in the graph will not disconnect the graph.

14. Definition Let K n = (V,E) be the complete graph with n nodes. For an even n, a one-factor of K n is a spanning 1- regular subgraph (or a perfect matching) of K n. A one-factorization of K n (n is even) is a set of one-factors that partition the set of edges E. A one-factorization is called perfect (or P1F) if the union of every two distinct one-factors is a Hamiltonian circuit. 14

15. P1F Perfect one-factorizations for K p+1, K 2p, and certain K 2n do exist, where p is a prime number. It is conjectured that P1F exist for all K 2n. 15

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17. P1F Example 17

18. P1F of K 8 18

19. Edge-colored graphs from P1F 19

20. Edge-colored graphs from P1F: Proof Proof. Let v 1,..., v n+1 be a list of nodes for K n+1 and V = {v 1, …, v n }. Let F’ i = F i \ { : j = 1, …, n}, E = F’ 1 ∪ ・ ・ ・ ∪ F’ t+2, and color all edges in F’ i with the color c i for i ≤ t + 2. Then it is straightforward to check that the edge-colored graph (V,E) is (t+1)-color connected, |V| = n, and |E| = (t + 2)(n − 1)/2. 20

21. 21 Choose a fixed node v 7 and remove all occurrences of v 7 to get the [4,2] 3 x 4 array BP-XOR code:

22. Edge-colored graphs from array BP- XOR code G = (V, E,C, f) be a (t + 1)-color connected edge-colored graph with V = {v 1, …, v bk, v bk+1 } and C = {c 1, c 2, …, c n } and b = max c ∈ C {|Z| : Z ⊆ E, f(Z) = c}. – 1) For 1 ≤ i ≤ n, let β i be defined as – 2) If | β i | is smaller than b, duplicate elements in β i to make it a b-element set. – 3) The array BP-XOR code is specified by the b × n matrix 22

23. Array BP-XOR codes from edge-colored graphs Theorem : Let C be an b × n array BP-XOR code with the following properties: 1) C is t-erasure tolerating, 2) C contains bk information symbols; and 3) C contains only degree one and two encoding symbols. Then there exists a (t+1)-color connected edge- colored graph G = (V, E,C, f) with|V|=bk+1, |E|=bn, and |C|=n. 23

24. MDS [n,2] array BP-XOR codes First find the smallest p (or 2p) such that n ≤ p (or n ≤ 2p-1), where p is an odd prime. Using P1F of K p+1 to construct the edge-colored graphs and then design the following array BP- XOR code 24

25. Efficient XOR-based secret sharing scheme As an example, design SSS based on the previous codes Let secret data file Now assume that the first bit of F is flipped. This is equivalent to flipping the first bit of v p-1. Thus the data owner only needs to inform each server to flip one bit at certain location without leaking any other information. Other remote computation is possible also (e.g., remote search or database query) 25

26. Flat BP-XOR codes A b x n array BP-XOR code is called a flat BP- XOR code if b = 1. Furthermore, a 1 x n BP-XOR code with k information symbols and distance d is called an [n, k, d] BP-XOR code. Fact : Let n ≥ k + 2, k ≥ 2, and d = n − k + 1. Then there is no flat [n, k, d] BP-XOR code. 26

27. Tolerating one erasure fault 27

28. Tolerating two erasure faults Fact 4.1 shows that two parity check symbols are not sufficient for tolerating two erasure faults for flat BP-XOR codes. In order to tolerate two erasure, we have to consider codes with n ≥ k + 3. 28 Theorem : For n ≥ k + 3 and k ≥ 3, there exists a flat [n, k, 3] BP-XOR code if and only if k ≤ 2 n−k −(n−k)−1. Proof. The truncated version (or non-truncated version if k = 2 n-k -(n-k)-1 ) of the Hamming code could be used to prove the theorem.

29. Tolerating two erasure faults Flat [5, 2, 3],[6, 3, 3], and [7, 4, 3] BP-XOR codes for tolerating two erasure faults. The above three codes are the only flat [k+3, k, 3] BP-XOR codes tolerating two erasure faults with three redundancy columns. 29

30. Tolerating two erasure faults 30

31. Tolerating three erasure faults 31

32. Tolerating three erasure faults 32 βi are distinct elements from X.

33. Tolerating three erasure faults Let n = 7, k = 3, d = 4, and β 1 = (1, 1, 1, 0), β 2 = (0, 1, 1, 1), and β 3 = (1, 0, 1, 1). Then the corresponding code has the following generator matrix: 33

34. Tolerating three erasure faults 34

35. Tolerating four or more erasure faults 35

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37. Conclusion We used edge-colored graphs to design degree one-and-two encoding symbol based array BP-XOR codes. Degree one and two encoding symbols could be used to design MDS array BP-XOR codes with t = 2 or k = 2. 37

38. References [1] M. Blaum, J. Brady, J. Bruck, and J. Menon. EVENODD: An efficient scheme for tolerating double disk failures in raid architectures. IEEE Trans. Computers, 44(2):192–202, 1995. [2] M. Blaum, J. Bruck, and E. Vardy. MDS array codes with independent parity symbols. IEEE Trans. on Information Theory, 42:529–542, 1996. [3] M. Blaum and R. M. Roth. On lowest-density MDS codes. IEEE Trans. on Information Theory, 45:46–59, 1999. [7] N. Cao, S. Yu, Z. Yang, W. Lou, and T. Hou. Lt codes-based secure and reliable cloud storage service. In Proceedings of INFOCOM, 2012. [8] C. Huang and L. Xu. STAR: an efficient coding scheme for correcting triple storage node failures. In FAST, pages 197–210, 2005. [13] M. Paterson, D. Stinson, and Yongge Wang. On encoding symbol degrees of array bp-xor codes. Submitted for publication, 2013. [15] Yongge Wang and Yvo Desmedt. Edge-colored graphs with applications to homogeneous faults. Inf. Process. Lett., 111(13):634–641, 2011. 38

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