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On Secret Sharing Schemes based on Regenerating Codes
Masazumi Kurihara
(University of Electro-Communications)
2012 IEICE General Conference, Okayama, Japan, 20-23, March, 2012.
(Proc. of 2012 IEICE General Conference, AS-2-1, pp.S17-S18)
1 Masazumi Kurihara(UEC, Tokyo, Japan)
Distributed Storage System (Secret Sharing Scheme ?)
Failed node (Repair the failed node) (Regeneration)
Data Collector (Reconstruction)
Data file
Helper node Helper node
2
encoding
Network
Masazumi Kurihara(UEC, Tokyo, Japan)
Helper node
Regenerating Codes
• A.G.Dimakis, P.B.Godfrey, Y.Wu, M.J.Wainwright, K.Ramchandran,
“Network Coding for Distributed Storage Systems,”
IEEE IT Trans. Vol.56, no.9, 2010. [Dimakis, et al.,2010]
• Repair Problem ( --- symmetry --- )
-- Repairing a failed node
-- Regenerating data that was stored in the failed node
3 Masazumi Kurihara(UEC, Tokyo, Japan)
Contents
1. Concept of regenerating codes
2. Relation between network-coding and distributed-storage (The results of Dimakis, et al.)
3. Several current regenerating codes
4. Secret sharing scheme based on regenerating codes
(Secure regenerating codes)
Masazumi Kurihara(UEC, Tokyo, Japan) 4
Keywords and their relation
• Codes for Distributed Storage – Reconstruction(Recovery) Erasure codes (MDS codes, RS codes, etc.)
– Reconstruction and Regeneration Regenerating codes
• Network Coding (for multicast, non-multicast) – Liner network Coding
– Random linear network Coding
• Secret Sharing – MDS-code-based Secret Sharing
– Regenerating-code-based Secret Sharing
(Secure Regenerating Codes)
5 Masazumi Kurihara(UEC, Tokyo, Japan)
Distributed Storage using ( n , k ) MDS codes
6
Message (one symbol)
( n , 1 ) MDS code ( n , k ) MDS code Share
Node 1
Node 2
Node n
Message
Share Node 1
Node 2
Node n
( k symbols)
Node k
at most n-k failed nodes
at most n-1 failed nodes
DC DC
Tradeoff between Redundancy and Reliability
k 1
Masazumi Kurihara(UEC, Tokyo, Japan)
Performance metric
Regenerating Codes
7
Erasure codes
Regenerating codes
Masazumi Kurihara(UEC, Tokyo, Japan)
• Collection of six parameters ( n, k, d, α, β, B )
• Reconstruction property – A data collector can reconstruct the original data by using the data
downloaded from any k active nodes.
• Regeneration property – A failed node can regenerate the data, which was stored in itself, by
using the data downloaded from any d active nodes.
Message, Encoding , and Shares
8
,,,,,,
65
43
21
uuuuuu
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc
Node 1
Node 2
Node 3
Node 4
Node 5
Share
message
B= 6 (Message size)
α=2 (Share size = storage )
encoding
n=5 (Total number of nodes in a network)
( n, k, d, α, β, B )
Masazumi Kurihara(UEC, Tokyo, Japan)
Reconstruction Property( k )
9
1
2
3
4
5
Share
message
B= 6 (message size)
α=2 (Share size = storage )
DC
2,11,1 ,cc
2,21,2 ,cc
2,41,4 ,ccReconstruction
k=3
encoding
n=5
,,,,,,
65
43
21
uuuuuu
( n, k, d, α, β, B )
Original message
Masazumi Kurihara(UEC, Tokyo, Japan)
k shares provide is the minimum data for reconstructing the original message.
,,,,,,
65
43
21
uuuuuu
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc
Maintaining the reliable distributed storage system (Repair a failed node, i.e., regenerate the share of it)
10
1
2
3
4
5
Share
message
B= 6
α=2
cut
Masazumi Kurihara(UEC, Tokyo, Japan)
Failed node ,,,,,,
65
43
21
uuuuuu
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc
Regeneration by reconstruction
11
1
2
3
4
5
Share
message
B= 6
α=2
2,11,1 ,cc
2,21,2 ,cc
2,41,4 ,cc
Reconstruction
encoding
k=3
encoding
3
cut Repair-Bandwidth kα=3x2=6(=B)
The original message
Share
Masazumi Kurihara(UEC, Tokyo, Japan)
,,,,,,
65
43
21
uuuuuu
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc ,,,,,,
65
43
21
uuuuuu
2,31,3 ,cc
Regeneration Property ( d and β )
12
1
2
3
4
5
Share
message
B= 6 1,3d
d=4
encoding
Repair the failed node 3
Regeneration
Piece β=1 (piece size )
2,3dPiece
4,3dPiece
5,3dPiece
),,,( 5,34,32,31,33 ddddd =Piece-vector
2,31,3 ,ccNode 3
( Exact-regeneration )
dβ=4 (piece-vector size )
Repair-Bandwidth dβ=4x1=4 (<6=B)
cut
Point ( d ≧ k )
α=2 (Share size = storage )
( n, k, d, α, β, B )
Masazumi Kurihara(UEC, Tokyo, Japan)
,,,,,,
65
43
21
uuuuuu
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc
Why d ≧ k ? If d<k, then …
13
1
2
3
4
5
Share
message
B= 6 1,3d
d=2
encoding
Repair the failed nodes 3,4,5
Piece β=1 (piece size )
2,3dPiece
2,31,3 ,ccNode 3
Repair-Bandwidth dβ x k =2 x 3
cut
α=2 (Share size = storage )
2,41,4 ,cc
2,51,5 ,ccNode 4
Node 5
( d=2<3=k )
k shares provide is the minimum data for reconstructing the message.
Masazumi Kurihara(UEC, Tokyo, Japan)
,,,,,,
65
43
21
uuuuuu
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc
skip
1. Reducing the repair-bandwidth 2. Tradeoff between Storage, α, and Repair-Bandwidth, dβ
14
1
2
3
4
5
Share
message
B 1,3d
d
encoding
Repair the failed node 3
Regeneration
Piece β (piece size )
2,3dPiece
4,3dPiece
5,3dPiece
),,,( 5,34,32,31,33 ddddd =Piece-vector
2,31,3 ,ccNode 3
dβ (piece-vector size )
Repair-Bandwidth dβ
cut
( Fixed k and d )
α (Share size = Storage )
Trade-off
Masazumi Kurihara(UEC, Tokyo, Japan)
,,,,,,
65
43
21
uuuuuu
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc
Tradeoff between Storage, α, and Repair-Bandwidth, dβ
15
( B=1, k=3, d=4 )
4/9
2/3
3/8 1/3 4/9
1/2
α
dβ
Storage per node α
Band
wid
th to
repa
ir o
ne n
ode
dβ
Minimum Storage Regenerating(MSR) point
Minimum Bandwidth Regenerating(MBR) point
(α,dβ) point
O
Masazumi Kurihara(UEC, Tokyo, Japan)
Contents
1. Concept of regenerating codes
2. Relation between network-coding and distributed-storage
3. Several current regenerating codes
4. Secret sharing scheme based on regenerating codes
(Secure regenerating codes)
Masazumi Kurihara(UEC, Tokyo, Japan) 16
(Linear) Network Coding for multicast
terminal Single source
terminal
terminal
s
1t
it
Capacity for network code for multicast
},...,2,1),,(maxflow{mim mits i =
mt
17 Masazumi Kurihara(UEC, Tokyo, Japan)
Max-flow min-cut theorem Min-cut
s
1t
Max-flow from source s to terminal t_{1}
2t
2),(maxflow 1 =ts
18
disjoint path
Masazumi Kurihara(UEC, Tokyo, Japan)
Information flow graph
Repairable distributed storage system ( Network coding for multicast )(1/3)
19
DC
DC
d
d
d
Message B
1
2
n
n-1
3
4
k
k
Repaired nodes
Keep n active nodes in a network
Masazumi Kurihara(UEC, Tokyo, Japan)
20
DC
DC
d
d
d
Message B
1
2
n
n-1
3
4
k
k
Masazumi Kurihara(UEC, Tokyo, Japan)
Repairable distributed storage system ( Network coding for multicast )(2/3)
21
DC
DC
d
d
d
Message B
1
2
n
n-1
3
4
k
k
Multicast
Masazumi Kurihara(UEC, Tokyo, Japan)
Repairable distributed storage system ( Network coding for multicast )(3/3)
Capacity for (n,k,d,α,β,B)Regenerating Codes(1/5)
22
DC
DC
d
d
d
Message B
1
2
n
n-1
3
4
k
k
disjoint path
Masazumi Kurihara(UEC, Tokyo, Japan)
Capacity for (n,k,d,α,β,B)Regenerating Codes(2/5)
23
DC
d
d
d
Message B
1
2
3
4
k
DC
d
d-1
k
d-2
α dβ≧α
in out
α in out
α in out
Zoom in
Masazumi Kurihara(UEC, Tokyo, Japan)
β
β β
β
β β β
β β
β
β
Capacity for (n,k,d,α,β,B)Regenerating Codes(3/5)
24
Message B
DC
d
d-1
k
d-2
α in out
α in out
α in out
β
β
β
∞
∞
∞
∞
∞
∞
∞
β
β
Flow from Source to DC
disjoint path
Masazumi Kurihara(UEC, Tokyo, Japan)
β
β
β
β
Capacity for (n,k,d,α,β,B)Regenerating Codes(4/5)
25
Message B
DC
d
d-1
k
d-2
α in out
α in out
α in out
β
β
β
∞
∞
∞
∞
∞
∞
∞
β β
Masazumi Kurihara(UEC, Tokyo, Japan)
disjoint path
Capacity for (n,k,d,α,β,B)Regenerating Codes(5/5)
26
Message B
DC
d
d-1
k
d-2
α in out
α in out
α in out
β
β
β
∞
∞
∞
∞
∞
∞
∞
β β min{ dβ , α }
min{ (d-1)β , α }
min{ (d-2)β , α }
Min-cut
∑=
≥+−=k
iBidC
1},)1min{( αβ
The Capacity for (n,k,d,α,β,B)Regenerating Codes C
Information flow graph
Masazumi Kurihara(UEC, Tokyo, Japan)
disjoint path
MSR and MBR points (α,dβ)
• Capacity for (n,k,d,α,β,B) regenerating codes
• Minimum Storage Regenerating(MSR) point
• Minimum Bandwidth Regenerating(MBR) point
27
∑=
+−=k
iidC
1},)1min{( αβ
+−
==)1(
,kd
ddkB αβα
+−
==)12(
2 ,kdk
Bddd ββα
Masazumi Kurihara(UEC, Tokyo, Japan)
Tradeoff between Storage, α, and Repair-Bandwidth, dβ
28
( B=1, k=3, d=4 )
4/9
2/3
3/8 1/3 4/9
1/2
α
dβ
Storage per node α
Band
wid
th to
repa
ir o
ne n
ode
dβ
Minimum Storage Regenerating(MSR) point
Minimum Bandwidth Regenerating(MBR) point
(α,dβ) point
O
Masazumi Kurihara(UEC, Tokyo, Japan)
Contents
1. Concept of regenerating codes
2. Relation between network-coding and distributed-storage
3. Several current regenerating codes
4. Secret sharing scheme based on regenerating codes
(Secure regenerating codes)
Masazumi Kurihara(UEC, Tokyo, Japan) 29
Exact vs. Functional Regeneration [Rashmi, et al.,2011]
• Functional-Regeneration(Functional-Repair) The resulting network of n nodes continues to possess reconstruction and
regeneration properties
• Exact-Regeneration(Exact-Repair) The repaired node stores exactly the same data as was stored in it prior to
failure.
30
Functional-Repair
Exact-Repair
Partial Exact-Repair
(Credit : Fig.3 in [Suh, et al., 2010]) Masazumi Kurihara(UEC, Tokyo, Japan)
Codes for regenerations • Functional-Regeneration
– Network Coding for multicast
(Multicasting Problem on information flow graph)
[Dimakis, et al.,2007],[Wu, et al.,2007] [Dimakis, et al.,2010] • Partial Exact-Regeneration
– Interference alignment (MSR codes, d=n-1)[Shah, et al.,2010]
( MSR codes with systematic-nodes and parity-nodes )
• Exact-Regeneration
– Interference alignment (MSR codes) [Suh, et al.,2011], [Shah, et al.,2012b]
– Complete graph and MDS code (MBR codes) [Shah, et al., 2012a]
– Product-Matrix framework [Rashmi, et al.,2011]
• (n,k,d) MBR codes for all values of (n,k,d)
• (n,k,d) MSR codes for all values of (n,k,d), d>=2k-2
(the first-, general-, explicit-, exact-constructions of MSR and MBR codes)
31 Masazumi Kurihara(UEC, Tokyo, Japan)
MSR codes by Interference alignment (Fig.4 in [Suh, et al.,2010])
32
2
1,aa
1
2
3
4
Share (size α=2)
Message B=4
( n=4, k=α=2, d=3, β=1, B=4 ) MSR code
21
21
,,,
bbaa
2
1,bb
22
11
2,ba
ba+
+
22
11 ,2baba
++
21 bb +
2121 2 bbaa +++
21212 bbaa +++
Regeneration d=3
Systematic node
Parity node
piece (size β=1)
piece (size β=1)
piece (size β=1)
2
1,aa
Node 1 Masazumi Kurihara(UEC, Tokyo, Japan)
example
MBR and MSR codes on Product-Matrix framework [Rashmi, et al.,2011]
• Using the property of symmetric matrices such that
if a matrix M is a symmetric matrix, then
where
since
33
matrix symmetric x : ddM
ttt MAMAAM ) () ( t ==
tMM =
tt BAABBAAB≠
≠
)(matrix x : ddA
tt MAAM =)(
Masazumi Kurihara(UEC, Tokyo, Japan)
example
( n, k )MBR codes by complete graph and ( , B )MDS code[Shah, et al.,2012](1/2)
(d=α=n-1, β=1, B= (n-1)k - )
34
321 ,, ccc
541 ,, ccc
642 ,, ccc
653 ,, ccc
1
2
3
4
Share
message 1c
3c
2c
,,,,,,
654
321
uuuuuu
6c
1 3
2
4
4c5c
Complete graph with n=4 nodes and edges
2n
2n
(n=4, k=d=α=3, β=1, B=6) MBR code
2k
Masazumi Kurihara(UEC, Tokyo, Japan)
example
MBR codes by Complete graph and MDS code (2/2)
35
321 ,, ccc
541 ,, ccc
642 ,, ccc
653 ,, ccc
1
2
3
4
Share
message
Regeneration
1cPiece
3cPiece
2cPiece
Node 1
,,,,,,
654
321
uuuuuu
321 ,, ccc
DC
Reconstruction
321 ,, ccc
54 ,cc
6c
1 3
2
4
1 3
2
4
(n=4, k=d=α=3, β=1, B=6) MBR code
Masazumi Kurihara(UEC, Tokyo, Japan)
example
Contents
1. Concept of regenerating codes
2. Relation between network-coding and distributed-storage
3. Several current regenerating codes
4. Secret sharing scheme based on regenerating codes
(Secure regenerating codes)
Masazumi Kurihara(UEC, Tokyo, Japan) 36
Information flow
37
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc
1
2
3
4
5
message ( size B )
1,3d
Regeneration
Piece
( size β )
2,3dPiece
4,3dPiece
5,3dPiece
),,,( 5,34,32,31,33 ddddd =Piece-vector
2,31,3 ,ccNode 3
( size α ) DC 2,11,1 ,cc
2,41,4 ,cc
Share
Share
Reconstruction
(size dβ )
Share
Masazumi Kurihara(UEC, Tokyo, Japan)
Share Piece
Piece-vector
,,,,,,
65
43
21
uuuuuu
Secure Regenerating Codes (Secret Sharing based on Regenerating Codes)
38
1
2
3
4
5
message ( size B )
1,3d
Regeneration
Piece
( size β )
2,3dPiece
4,3dPiece
5,3dPiece
),,,( 5,34,32,31,33 ddddd =Piece-vector
2,31,3 ,ccNode 3
( size α ) DC 2,11,1 ,cc
2,41,4 ,cc
Share
Share
Reconstruction
(size dβ )
Share
R
Ssecret
Random symbols
Masazumi Kurihara(UEC, Tokyo, Japan)
Share Piece
Piece-vector
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc
Regeneration by reconstruction
39
1
2
3
4
5
Share
message
B
α
2,11,1 ,cc
2,21,2 ,cc
2,41,4 ,cc
Reconstruction
encoding
k
encoding
2,31,3 ,cc3
cut
Entire message
Share
43
21
21
,,,,,
RRRRSS
Repair-Bandwidth kα=B
Masazumi Kurihara(UEC, Tokyo, Japan)
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc 43
21
21
,,,,,
RRRRSS
Regeneration by regenerating codes
40
1
2
3
4
5
Share
message
B 1,3d
d
encoding
Repair the failed node 3
Regeneration
Piece
2,3dPiece
4,3dPiece
5,3dPiece
),,,( 5,34,32,31,33 ddddd =Piece-vector
2,31,3 ,ccNode 3
( Exact-regeneration )
dβ (piece-vector size )
Repair-Bandwidth dβ<B)
cut
α
Masazumi Kurihara(UEC, Tokyo, Japan)
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc
43
21
21
,,,,,
RRRRSS
Secure Regenerating Codes (Secret sharing based on Regenerating codes)
• Exact-Regeneration(MSR and MBR codes)
• Informationally secure regenerating codes
– Secure MBR codes
• Complete graph and MDS code : [Pawar, et al., 2011]
• Product-Matrix framework :
[Kurihara and Kuwakado,2011d]( non-linear ramp scheme),
[Shah, et al., 2011]
– Secure MSR codes
• Product-Matrix framework : [Kurihara and Kuwakado,2011a,b], [Kuwakado and Kurihara,2011],
[Shah, et al., 2011]
• Computationally secure regenerating codes
[Kuwakado and Kurihara,2011]
41 Masazumi Kurihara(UEC, Tokyo, Japan)
42
Computationally-Secure Regenerating(CS-R) codes [Kuwakado and Kurihara,2011]
Regenerating code
secret
Informationally-Secure Regenerating(IS-R) code
Random symbols
share share
Credit: Kuwakado and Kurihara , “Computationally-Secure Regenerating code,” CSS2011 Masazumi Kurihara(UEC, Tokyo, Japan)
message
43
Computationally-Secure Regenerating(CS-R) codes [Kuwakado and Kurihara,2011]
ciphertext
Regenerating code Regenerating code
encripting
secret
Key
share
Improve Coding Rate
Credit: Kuwakado and Kurihara , “Computationally-Secure Regenerating code,” CSS2011
share share
Com
puta
tiona
l sec
urity
Info
rmat
iona
l se
curi
ty
(IS-R
cod
e)
Random symbols
Masazumi Kurihara(UEC, Tokyo, Japan)
message
Recall ( n, k )MBR codes by complete graph and ( , B )MDS code[Shah, et al.,2012a]
(d=α=n-1, β=1, B= (n-1)k - )
44
321 ,, ccc
541 ,, ccc
642 ,, ccc
653 ,, ccc
1
2
3
4
Share
message 1c
3c
2c
,,,,,,
654
321
uuuuuu
6c
1 3
2
4
4c5c
Complete graph with n=4 nodes and edges
2n
2n
(n=4, k=d=α=3, β=1, B=6) MBR code
2k
Masazumi Kurihara(UEC, Tokyo, Japan)
example
Secure ( n, k )MBR codes [Pawar, et al., 2011] by Complete graph and MDS code
45
321 ,, ccc
541 ,, ccc
642 ,, ccc
653 ,, ccc
1
2
3
4
Share
Message B=6
,,,,,,
654
321
uuuuuu
(n=4, k=d=α=3, β=1, l= 2) secure MBR code
symbols Random : ,...,,,Secret :
521 RRRS
5216 ...,5,...,2,1for RRRSc
iRc ii
++++===
Setting :
Encoding :
Perfect secrecy for at most two shares
(Fig.5 in [Pawar, et al., 2011]) Masazumi Kurihara(UEC, Tokyo, Japan)
example
543
21
,,,,,
RRRRRS
Secure Regenerating Codes (Secret Sharing based on Regenerating Codes)
46
1
2
3
4
5
message ( size B )
1,3d
Regeneration
Piece
( size β )
2,3dPiece
4,3dPiece
5,3dPiece
),,,( 5,34,32,31,33 ddddd =Piece-vector
2,31,3 ,ccNode 3
( size α ) DC 2,11,1 ,cc
2,41,4 ,cc
Share
Share
Reconstruction
(size dβ )
Share
R
Ssecret
Random symbols
Masazumi Kurihara(UEC, Tokyo, Japan)
Share Piece
Piece-vector
2,11,1 ,cc
2,21,2 ,cc
2,31,3 ,cc
2,41,4 ,cc
2,51,5 ,cc
Secrecy Capacity for secure regenerating code [Pawar, et al., 2011](1/3)
• Random variables
• Reconstruction Property
For any k shares
• Regeneration Property ( failed node ) For any d pieces for repair the failed node ,
that is, piece-vector ,
47
}),...,1{( , ),,(
}){\},...,1{ },,...,1{( ),,2,1(
,1,
,
nfDDDfnhnfD
niCS
dfff
hf
i
∈=
∈∈=
- Secret :
- Piece : - Piece-Vector :
- Share :
0),,|( 1 =kCCSH
,,,1 kCC
dff DD ,1, ,,fD0)|( =ff DCH
ff
Masazumi Kurihara(UEC, Tokyo, Japan)
Secrecy Capacity for secure regenerating code [Pawar, et al., 2011](2/3)
• Regeneration Property ( failed node ) For any d pieces for repair the failed node ,
that is, piece-vector ,
(The share is uniquely determined from the piece-vector .)
• In general, , i.e., ,
since the Markov chain .
Thus, if then , because
• However, it is not always true that .
48
dff DD ,1, ,,fD0)|( =ff DCH
ff
)|()|( ff CSHDSH ≤
ff CDS →→
0)|( ≠ff CDH
fDfC
)()|()|()( SHCSHDSHSH ff ≤≤=
0);( =fDSI 0);( =fCSI
Masazumi Kurihara(UEC, Tokyo, Japan)
);();( ff CSIDSI ≥
Secrecy Capacity for secure regenerating code [Pawar, et al., 2011](3/3)
• Secrecy property
For m piece-vector for any m repaired nodes ,
• Secrecy capacity for (n, k, d, α, β, B ; m ) secure regeneration code :
• The upper bound of the secrecy capacity :
49
)( sup SHCS =SC
)(),,|(0),,|(
1
1
SHDDSHCCSH
m
k
==
fmf DD ,,1 mff ,,1
)(),,|( 1 SHDDSH fmf =
∑+=
+−≤k
miS idC
1},)1min{( αβ
Masazumi Kurihara(UEC, Tokyo, Japan)
Information flow graph
Reconstruction Property
Conclusions • Regenerating codes for distributed storage [Dimakis, et al.,2010]
– Reconstruction and Regeneration(repair failed node)
– Reducing bandwidth to repair
– Tradeoff between Storage and Repair-Bandwidth
• MSR and MBR codes
• Network coding for distributed storage [Dimakis, et al.,2010] – Capacity for regenerating codes
– Linear network coding for multicast
• Exact-regeneration vs. Functional-regeneration
• Secure Regenerating Codes – Secret sharing based on (exact-)regenerating codes
– Share, piece, and piece-vector
50 Masazumi Kurihara(UEC, Tokyo, Japan)
References(1/3) • [Dimakis, et al.,2010] Dimakis, A.G.; Godfrey, P.B.; Yunnan Wu; Wainwright, M.J.; Ramchandran, K.;
Network Coding for Distributed Storage Systems
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• [Dimakis, et al.,2007]A. G. Dimakis , P. B. Godfrey , Y. Wu , M. Wainwright and K. Ramchandran
"Network coding for distributed storage systems", Proc. IEEE INFOCOM, 2007
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Optimal Exact-Regenerating Codes for Distributed Storage at the MSR and MBR Points via a Product-Matrix Construction
Information Theory, IEEE Transactions on , Volume: 57 , Issue: 8, Publication Year: 2011 , Page(s): 5227 - 5239
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Exact-Repair MDS Code Construction Using Interference Alignment
Information Theory, IEEE Transactions on ,Volume: 57 , Issue: 3, Publication Year: 2011 , Page(s): 1425 - 1442
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References(2/3) • [Shah, et al.,2010]N. B. Shah, K. V. Rashmi, P. V. Kumar, and K. Ramchandran,
“Explicit codes minimizing repair bandwidth for distributed storage,”
in Proc. IEEE Information Theory Workshop, Cairo, Egypt, Jan.2010.
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Global Telecommunications Conference (GLOBECOM 2011), 2011 IEEE , Publication Year: 2011 , Page(s): 1 – 5
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Interference Alignment in Regenerating Codes for Distributed Storage: Necessity and Code Constructions
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Securing Dynamic Distributed Storage Systems Against Eavesdropping and Adversarial Attacks
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Masazumi Kurihara(UEC, Tokyo, Japan) 52
References(3/3) • [Kurihara and Kuwakado,2011a]M.Kurihara and H.Kuwakado,
``On regenerating codes and secret sharing for distributed storage,''
IEICE Technical Report(in Japanese), IT2010-56(2011-01), pp.13-18(in Japanese), Jan. 2011.
• [Kurihara and Kuwakado,2011b]M.Kurihara and H.Kuwakado,
`On an extended version of Rashmi-Shah-Kumar regenerating codes and secret sharing for distributed storage,''
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``On ramp secret sharing schemes for distributed storage systems under repair dynamics,''
IEICE Technical Report, IT2011-17(2011-7), pp.41-46(in Japanese), July 2011.
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Masazumi Kurihara(UEC, Tokyo, Japan) 53