CSC 774 Advanced Network Security

CSC 774 Dr. Peng Ning 1

Computer Science

CSC 774 Advanced Network Security

Topic 2.4 Rabin’s Information Dispersal Algorithm

Slides by Sangwon Hyun

CSC 774 Dr. Peng Ning 2Computer Science

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.


Basic Idea of IDA


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.


Dispersal(F, m, n) – Example 1

• |F|=32 bytes, m=4, n=8

F

Dispersal(F, 4, 8)

F1 F2 F3 F4 F5 F6 F7 F8

– |Fi| = 32/4 = 8 bytes (1 ≤ i ≤ n)


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))


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.

Recovery({F1, F3, F4, F7}, 4, 8)

F

F1 F3 F4 F7


Detailed Operations


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 ]


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.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

n

2

1

a

...

a

a

A


Dispersal(F, m, n)

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

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−−−−

−

−

−

12

11

211

13

233

12

222

11

211

...1

...1

...............

...1

...1

...1

mnnn

mnnn

m

m

m

xxx

xxx

xxx

xxx

xxx

– 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.


Dispersal(F, m, n)

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

[ ]

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅

⋅⋅⋅⋅⋅⋅

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=⋅

nF

F

F

...

Sa...SaSa

............

Sa...SaSa

Sa...SaSa

S...SS

a

...

a

a

M A

2

1

N/mn2n1n

N/m22212

N/m12111

N/m21

n

2

1

– ai ・ Sk = (ai1b(k−1)m+1 + … + aimbkm)



• |F|=32 bytes, m=4, n=8– F = b1,b2,…,b32

– F = (b1,…,b4),(b5,…,b8),…,(b29,…,b32)

– M (4×8)

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

==

3284

3173

3062

2951

821

b...bb

b...bb

b...bb

b...bb

S...SS M



– A (8×4)

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

38

288

32

222

31

211

8

2

1

1

............

1

1

a

...

a

a

A

xxx

xxx

xxx



• Fi (1 ≤ i ≤ 8) are computed as follows:

[ ]

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅

⋅⋅⋅⋅⋅⋅

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=⋅

8

2

1

882818

822212

812111

821

8

2

1

...

Sa...SaSa

............

Sa...SaSa

Sa...SaSa

S...SS

a

...

a

a

M A

F

F

F


Recovery({Fij |(1≤ j ≤m), (1≤ ij ≤n)}, m, n)

• Given m pieces Fij ( (1≤ j ≤m), (1≤ ij ≤n) ),

M A' M

a

...

a

a

F

...

F

F

m

2

1

m

2

1

i

i

i

i

i

i

⋅=⋅

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

• M can be recovered from the given m pieces Fij ( (1≤ j ≤m), (1≤ ij ≤n) ) because A’ is invertible.

M

F

...

F

F

a

...

a

a

m

2

1

m

2

1

i

i

i

1

i

i

i

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡−


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.

M

a

a

a

a

Sa...SaSa

Sa...SaSa

Sa...SaSa

Sa...SaSa

7

4

3

1

872717

842414

832313

812111

7

4

3

1

⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

FFFF


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

computation:

M

a

a

a

a

7

4

3

1

1

7

4

3

1

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

FFFF

CSC 774 Advanced Network Security

Documents

Transcript of CSC 774 Advanced Network Security