Information Dispersal Algorithm

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Information Dispersal Algorithm

• 11/17/2006
• Sangwon Hyun

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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.

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• Basic Idea of IDA

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Dispersal(F, m, n)

• Let F be a data of size N in byte (FN).
• 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).
• FiF/m
• Thus, the size of F, N, should be a multiple of m.

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Dispersal(F, m, n) Example 1

• F32 bytes, m4, n8

F

• Fi 32/4 8 bytes (1 i n)

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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))

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Recovery(Fij (1 j m), (1 ij n), m, n)
Example 2

• F32 bytes, m4, n8, Fi8 bytes (1 i 8)
• Let us assume that the following 4(m) pieces are
  received.

F
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• Detailed Operations

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Dispersal(F, m, n)

• F b1,b2,,bN
• FN, and bi represents each byte in F (0 bi
  255(28-1)).
• 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),(bm1,,b2m),,(bN-m1,,bN)
• Si (b(i-1)m1,,bim) (1 i N/m)
• The matrix, M (m N/m), is constructed as
  follows
• M S1 S2 SN/m

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Dispersal(F, m, n)

• The matrix, A (nm), is constructed as follows
• ai (ai1, ,aim) (1 i n)
• chosen such that every subset of m different
  vectors are linearly independent.

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Dispersal(F, m, n)

• The following Vandermonde matrix satisfies the
  property required for A.

• m n, and all xis 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.

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Dispersal(F, m, n)

• n pieces, Fi (1 i n), are computed as
  follows

• aiSk (ai1b(k-1)m1 aimbkm)

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Dispersal(F, m, n) Example 3

• F32 bytes, m4, n8
• F b1,b2,,b32
• F (b1,,b4),(b5,,b8),,(b29,,b32)
• M (48)

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Dispersal(F, m, n) Example 3

• A (84)

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Dispersal(F, m, n) Example 3

• Fi (1 i 8) are computed as follows

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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.

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Recovery(Fij (1 j m), (1 ij n), m, n)
Example 4

• F32 bytes, m4, n8
• In example 3, Fi (1 i 8) pieces of 8 bytes
  are resulted.
• Assume that F1,F3,F4,F7 are received among
  them.

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Recovery(Fij (1 j m), (1 ij n), m, n)
Example 4

• The original data M can be recovered by the
  following computation

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Reference

• Michael O. Rabin. Efficient Dispersal of
  Information for Security, Load Balancing, and
  Fault Tolerance. Journal of ACM, 1989.
• Jung Min Park, Edwin K. P. Chong, and Howard Jay
  Siegel. Efficient Multicast Packet Authentication
  Using Signature Amortization. IEEE Security and
  Privacy, 2002.