CSC 774 Advanced Network Security

1. CSC 774 Advanced Network Security Topic 2.4 Rabin’s Information Dispersal Algorithm Slides by Sangwon Hyun Dr. Peng Ning

2. Motivation • IDA was developed to provide safe and reliable transmission of information in distributed systems. • Inefficiency of retransmission of lost packets • In multicast transmission, different receivers lose different sets of packets. • Re-request and retransmission increases delays. • Forward error correction technique might be desirable in distributed systems. Dr. Peng Ning

3. Basic Idea of IDA Dr. Peng Ning

4. Dispersal(F, m, n) • Let F be a data of size N in byte (|F|=N). • m should be less than or equal to n (m ≤ n). • Dispersal(F, m, n): • splitting the data F with some amount of redundancy resulting in n pieces Fi(1 ≤ i ≤ n). • |Fi|=|F|/m • Thus, the size of F, N, should be a multiple of m. Dr. Peng Ning

5. Dispersal(F, 4, 8) F1 F2 F3 F4 F5 F6 F7 F8 Dispersal(F, m, n) – Example 1 • |F|=32 bytes, m=4, n=8 F • |Fi| = 32/4 = 8 bytes (1 ≤ i ≤ n) Dr. Peng Ning

6. Recovery({Fij |(1≤ j ≤m), (1≤ ij ≤n)}, m, n) • Recovery({Fij |(1≤ j ≤m), (1≤ ij ≤n)}, m, n): • reconstructing the original data F from any m pieces among n pieces (Fi(1 ≤ i ≤ n)) Dr. Peng Ning

7. F1 F3 F4 F7 Recovery({F1, F3, F4, F7}, 4, 8) Recovery({Fij |(1≤ j ≤m), (1≤ ij ≤n)}, m, n) – Example 2 • |F|=32 bytes, m=4, n=8, |Fi|=8 bytes(1 ≤ i ≤ 8) • Let us assume that the following 4(=m) pieces are received. F Dr. Peng Ning

8. Detailed Operations Dr. Peng Ning

9. Dispersal(F, m, n) • F = b1,b2,…,bN • |F|=N, and bi represents each byte in F (0 ≤ bi ≤ 255). • All computations should be done in GF(28). • GF(28) is closed under addition and multiplication. • Every nonzero element in GF(28) has a multiplicative inverse. • F = (b1,…,bm),(bm+1,…,b2m),…,(bN-m+1,…,bN) • Si = (b(i-1)m+1,…,bim) T(1 ≤ i ≤ N/m) • The matrix, M (m× N/m), is constructed as follows: • M = [ S1 S2 … SN/m] Dr. Peng Ning

10. Dispersal(F, m, n) • The matrix, A (n×m), is constructed as follows: • ai = (ai1, …,aim) (1 ≤ i ≤ n) • chosen such that every subset of m different vectors are linearly independent. Dr. Peng Ning

11. Dispersal(F, m, n) • The following Vandermonde matrix satisfies the property required for A. • m ≤ n, and all xi’s are nonzero elements in GF(28) and pairwise different. • Any m different rows are linearly independent, so any matrix composed of a set of any m different rows is invertible. Dr. Peng Ning

12. Dispersal(F, m, n) • n pieces, Fi (1 ≤ i ≤ n), are computed as follows: • ai・Sk = (ai1b(k−1)m+1 + …+ aimbkm) Dr. Peng Ning

13. Dispersal(F, m, n) – Example 3 • |F|=32 bytes, m=4, n=8 • F = b1,b2,…,b32 • F = (b1,…,b4),(b5,…,b8),…,(b29,…,b32) • M (4×8) Dr. Peng Ning

14. Dispersal(F, m, n) – Example 3 • A (8×4) Dr. Peng Ning

15. Dispersal(F, m, n) – Example 3 • Fi (1 ≤ i ≤ 8) are computed as follows: Dr. Peng Ning

16. Recovery({Fij |(1≤ j ≤m), (1≤ ij ≤n)}, m, n) • Given m pieces Fij ( (1≤ j ≤m), (1≤ ij ≤n) ), • M can be recovered from the given m pieces Fij ( (1≤ j ≤m), (1≤ ij ≤n) ) because A’ is invertible. Dr. Peng Ning

17. Recovery({Fij |(1≤ j ≤m), (1≤ ij ≤n)}, m, n) – Example 4 • |F|=32 bytes, m=4, n=8 • In example 3, Fi (1 ≤ i ≤ 8) pieces of 8 bytes are resulted. • Assume that {F1,F3,F4,F7} are received among them. Dr. Peng Ning

18. Recovery({Fij |(1≤ j ≤m), (1≤ ij ≤n)}, m, n) – Example 4 • The original data M can be recovered by the following computation: Dr. Peng Ning