On Secret Sharing Schemes based on Regenerating Codes

Transcription

1 On Secret Sharing Schemes base on Regenerating Coes Masazumi Kurihara (University of Electro-Communications) 0 IEICE General Conference Okayama Japan 0- March 0. (Proc. of 0 IEICE General Conference AS-- pp.s7-s8)

2 Data file Distribute Storage System (Secret Sharing Scheme?) Network encoing Data Collector (Reconstruction) Helper noe Helper noe Helper noe Faile noe (Repair the faile noe) (Regeneration)

3 Regenerating Coes A.G.Dimakis P.B.Gofrey Y.Wu M.J.Wainwright K.Ramchanran Network Coing for Distribute Storage Systems IEEE IT Trans. Vol.6 no [Dimakis et al.00] Repair Problem ( --- symmetry --- ) -- Repairing a faile noe -- Regenerating ata that was store in the faile noe

4 Contents. Concept of regenerating coes. Relation between network-coing an istribute-storage (The results of Dimakis et al.). Several current regenerating coes. Secret sharing scheme base on regenerating coes (Secure regenerating coes)

5 Keywors an their relation Coes for Distribute Storage Reconstruction(Recovery) Erasure coes (MDS coes RS coes etc.) Reconstruction an Regeneration Regenerating coes Network Coing (for multicast non-multicast) Liner network Coing Ranom linear network Coing Secret Sharing MDS-coe-base Secret Sharing Regenerating-coe-base Secret Sharing (Secure Regenerating Coes)

6 Distribute Storage using ( n k ) MDS coes ( n ) MDS coe ( n k ) MDS coe Message (one symbol) Noe Noe Share DC Message ( k symbols) Noe Noe Share k DC Noe k Noe n at most n- faile noes Noe n Traeoff between Reunancy an Reliability at most n-k faile noes Performance metric 6

7 Regenerating Coes Collection of six parameters ( n k α B ) Reconstruction property A ata collector can reconstruct the original ata by using the ata ownloae from any k active noes. Regeneration property A faile noe can regenerate the ata which was store in itself by using the ata ownloae from any active noes. Erasure coes Regenerating coes 7

8 message B= 6 (Message size) Noe u u u 6 encoing Message Encoing an Shares (Share size = storage ) Noe c c Noe Noe Noe Share α= c c c c c c c c ( n k α B ) n= (Total number of noes in a network) 8

9 message B= 6 (message size) u u u u u 6 encoing n= Reconstruction Property( k ) Share α= (Share size = storage ) c c c c c c c c c c ( n k α B ) c c c c c c k= k shares provie is the minimum ata for reconstructing the original message. DC Reconstruction u u u 6 Original message 9

10 Maintaining the reliable istribute storage system (Repair a faile noe i.e. regenerate the share of it) cut Share α= message B= 6 c c u u u 6 c c c c Faile noe c c c c 0

11 Regeneration by reconstruction message B= 6 u u u 6 encoing cut Share α= c c c c c c c c c c c Share c c c c c k= The original message Repair-Banwith kα=x=6(=b) Reconstruction u u u 6 c c encoing

12 message B= 6 u u u 6 encoing Point ( k ) Regeneration Property ( an ) cut Share α= (Share size = storage ) c c c c c c c c c c ( n k α B ) = (piece size ) = Repair-Banwith =x= (<6=B) Repair the faile noe -vector = ( ) = (piece-vector size ) Regeneration Noe c c ( Exact-regeneration )

13 message B= 6 u u u u u 6 encoing cut Share α= (Share size = storage ) c c c c c c c c Why k? If <k then = (piece size ) ( =<=k ) = Repair the faile noes Noe Noe Noe Repair-Banwith x k = x c c c c c c skip c c k shares provie is the minimum ata for reconstructing the message.

14 . Reucing the repair-banwith ( Fixe k an ). Traeoff between Storage α an Repair-Banwith message B u u u 6 encoing cut Share α (Share size = Storage ) c c c c c c c c c c Trae-off (piece size ) Repair-Banwith Repair the faile noe -vector = ( ) (piece-vector size ) Regeneration Noe c c

15 Traeoff between Storage α an Repair-Banwith Banwith to repair one noe / / /9 O Minimum Storage Regenerating(MSR) point (α) point / /8 /9 Storage per noe α ( B= k= = ) Minimum Banwith Regenerating(MBR) point α

16 Contents. Concept of regenerating coes. Relation between network-coing an istribute-storage. Several current regenerating coes. Secret sharing scheme base on regenerating coes (Secure regenerating coes) 6

17 (Linear) Network Coing for multicast t Single source s terminal t i terminal Capacity for network coe for multicast mim{maxflow( s t i ) i =... m} t m terminal 7

18 Max-flow min-cut theorem Min-cut t s isjoint path t Max-flow from source s to terminal t_{} maxflow( s t ) = Information flow graph 8

19 Repairable istribute storage system ( Network coing for multicast )(/) Repaire noes k Message B DC k DC n- n Keep n active noes in a network 9

20 Repairable istribute storage system ( Network coing for multicast )(/) k Message B DC k DC n- n 0

21 Repairable istribute storage system ( Network coing for multicast )(/) k Message B Multicast DC k DC n- n

22 Capacity for (nkαb)regenerating Coes(/) isjoint path k Message B DC k DC n- n

23 Capacity for (nkαb)regenerating Coes(/) k Message B DC - α α in out - in α Zoom in out in α out k DC

24 Capacity for (nkαb)regenerating Coes(/) Message B - in α out in - α out in α out k DC isjoint path Flow from Source to DC

25 Capacity for (nkαb)regenerating Coes(/) Message B - in α out in α out k - in α out DC isjoint path

26 Capacity for (nkαb)regenerating Coes(/) Message B in α min{ α } - out in α out k isjoint path min{ (-) α } - in α min{ (-) α } out DC The Capacity C for (nkαb)regenerating Coes Min-cut C = k i= min{( i + ) α} B Information flow graph 6

27 MSR an MBR points (α) Capacity for (nkαb) regenerating coes Minimum Storage Regenerating(MSR) point Minimum Banwith Regenerating(MBR) point 7 = + = k i i C } ) min{( α + = = ) ( k k B α α + = = ) ( k k B α

28 Traeoff between Storage α an Repair-Banwith Banwith to repair one noe / / /9 O Minimum Storage Regenerating(MSR) point (α) point / /8 /9 Storage per noe α ( B= k= = ) Minimum Banwith Regenerating(MBR) point α 8

29 Contents. Concept of regenerating coes. Relation between network-coing an istribute-storage. Several current regenerating coes. Secret sharing scheme base on regenerating coes (Secure regenerating coes) 9

30 Exact vs. Functional Regeneration [Rashmi et al.0] Functional-Regeneration(Functional-Repair) The resulting network of n noes continues to possess reconstruction an regeneration properties Exact-Regeneration(Exact-Repair) The repaire noe stores exactly the same ata as was store in it prior to failure. Functional-Repair Partial Exact-Repair Exact-Repair (Creit : Fig. in [Suh et al. 00]) 0

31 Coes for regenerations Functional-Regeneration Network Coing for multicast (Multicasting Problem on information flow graph) [Dimakis et al.007][wu et al.007] [Dimakis et al.00] Partial Exact-Regeneration Interference alignment (MSR coes =n-)[shah et al.00] ( MSR coes with systematic-noes an parity-noes ) Exact-Regeneration Interference alignment (MSR coes) [Suh et al.0] [Shah et al.0b] Complete graph an MDS coe (MBR coes) [Shah et al. 0a] Prouct-Matrix framework [Rashmi et al.0] (nk) MBR coes for all values of (nk) (nk) MSR coes for all values of (nk) >=k- (the first- general- explicit- exact-constructions of MSR an MBR coes)

32 MSR coes by Interference alignment (Fig. in [Suh et al.00]) ( n= k=α= = = B= ) MSR coe example Share (size α=) Systematic noe a a Message B= a a b b Parity noe a a a a b b + b + b + b + b = Regeneration b + b a + + a + b b a + a + b + b piece (size =) piece (size =) piece (size =) Noe a a

33 MBR an MSR coes on Prouct-Matrix framework [Rashmi et al.0] Using the property of symmetric matrices such that if a matrix M is a symmetric matrix then where M A ( t t AM ) = MA : x : x symmetric matrix matrix AB ( AB) t BA example BA t since t t t t M A = ( A M ) = ( A M ) M = M t

34 ( n k )MBR coes n by complete graph an ( B )MDS coe[shah et al.0](/) k (=α=n- = B= (n-)k - ) example (n= k==α= = B=6) MBR coe Share message u u 6 c c c c c c c c c 6 c c c c c c c 6 c c 6 Complete graph with n= noes an n eges

35 MBR coes by Complete graph an MDS coe (/) example (n= k==α= = B=6) MBR coe Share message c c c c c c Reconstruction DC u u 6 c c c c c c6 c c c c c6 c c c 6 c Noe Regeneration c c c

36 Contents. Concept of regenerating coes. Relation between network-coing an istribute-storage. Several current regenerating coes. Secret sharing scheme base on regenerating coes (Secure regenerating coes) 6

37 Information flow Share -vector message ( size B ) u u u 6 Share Share c c ( size α ) DC c c c c c c c c c c ( size ) c c Noe Share Reconstruction Regeneration -vector = ( ) c c (size ) 7

38 Secure Regenerating Coes (Secret Sharing base on Regenerating Coes) Share -vector message ( size B ) S secret R Ranom symbols Share Share c c ( size α ) DC c c c c c c c c c c ( size ) c c Noe Share Reconstruction Regeneration -vector = ( ) c c (size ) 8

39 Regeneration by reconstruction message B S S R R R R encoing cut Share α c c c c c c c c c c c Share c c c c c k Entire message c c Repair-Banwith kα=b Reconstruction S S R R R R encoing 9

40 Regeneration by regenerating coes cut Share α message B S S R R R R encoing c c c c c c c c c c Noe Repair the faile noe -vector = ( ) (piece-vector size ) Regeneration c c Repair-Banwith <B) ( Exact-regeneration ) 0

41 Secure Regenerating Coes (Secret sharing base on Regenerating coes) Exact-Regeneration(MSR an MBR coes) Informationally secure regenerating coes Secure MBR coes Complete graph an MDS coe : [Pawar et al. 0] Prouct-Matrix framework : Secure MSR coes [Kurihara an Kuwakao0]( non-linear ramp scheme) [Shah et al. 0] Prouct-Matrix framework : [Kurihara an Kuwakao0ab] [Kuwakao an Kurihara0] Computationally secure regenerating coes [Kuwakao an Kurihara0] [Shah et al. 0]

42 Computationally-Secure Regenerating(CS-R) coes message [Kuwakao an Kurihara0] Ranom symbols secret Regenerating coe Informationally-Secure Regenerating(IS-R) coe share share Creit: Kuwakao an Kurihara Computationally-Secure Regenerating coe CSS0

43 Computationally-Secure Regenerating(CS-R) coes message Ranom symbols Key Regenerating coe [Kuwakao an Kurihara0] Improve Coing Rate Informational security (IS-R coe) secret encripting ciphertext Regenerating coe Computational security share share Creit: Kuwakao an Kurihara Computationally-Secure Regenerating coe CSS0 share

44 Recall ( n k )MBR coes n by complete graph an ( B )MDS coe[shah et al.0a] (=α=n- = B= (n-)k - k ) example (n= k==α= = B=6) MBR coe Share message u u 6 c c c c c c c c c 6 c c c c c c c 6 c c 6 Complete graph with n= noes an n eges

45 Secure ( n k )MBR coes [Pawar et al. 0] by Complete graph an MDS coe example (n= k==α= = l= ) secure MBR coe Message B=6 u u 6 c c c Share c c c c c c 6 Setting : S :Secret R R... R : Ranom symbols Perfect secrecy for at most two shares Encoing : S R R R R R c c c 6 (Fig. in [Pawar et al. 0]) c c = R for i =... i i 6 = S + R + R R

46 Secure Regenerating Coes (Secret Sharing base on Regenerating Coes) Share -vector message ( size B ) S secret R Ranom symbols Share Share c c ( size α ) DC c c c c c c c c c c ( size ) c c Noe Share Reconstruction Regeneration -vector = ( ) c c (size ) 6

47 Secrecy Capacity for secure regenerating coe [Pawar et al. 0](/) Ranom variables - Secret : - Share : - : - -Vector : C D D ( i f h Reconstruction Property S For any k shares i f D Regeneration Property ( faile noe ) For any pieces for repair the faile noe that is piece-vector ( = n) f = ( D {... n} h {... n}\{ f }) f C C k f H ( S C ) = C k ) ( f 0 D f D f D f H ( C f D f ) = 0 {... n}) f f 7

48 Secrecy Capacity for secure regenerating coe [Pawar et al. 0](/) Regeneration Property ( faile noe ) For any pieces for repair the faile noe D f D f D f that is piece-vector C f H ( C f D f ) = 0 (The share is uniquely etermine from the piece-vector.) H ( S D ) H ( S C f In general f i.e. f since the Markov chain S D f C f. Thus if I( S; ) = 0 then I( S; ) = 0 because D f However it is not always true that H ( D f C f ) 0. ) H ( S) = H ( S D ) H ( S C ) H ( S) f C f f I( S; D ) I( S; C f f ) f D f 8

49 Secrecy Capacity for secure regenerating coe [Pawar et al. 0](/) Secrecy property For m piece-vector D for any m repaire noes f H ( S D D ) H ( S) Secrecy capacity for (n k α B ; m ) secure regeneration coe : C S = C S sup H ( S) H ( S C Ck ) = 0 H ( S D D ) = H ( S) The upper boun of the secrecy capacity : f D fm m f fm = m Reconstruction Property C S k i= m+ min{( i + ) α} Information flow graph 9

50 Conclusions Regenerating coes for istribute storage [Dimakis et al.00] Reconstruction an Regeneration(repair faile noe) Reucing banwith to repair Traeoff between Storage an Repair-Banwith MSR an MBR coes Network coing for istribute storage [Dimakis et al.00] Capacity for regenerating coes Linear network coing for multicast Exact-regeneration vs. Functional-regeneration Secure Regenerating Coes Secret sharing base on (exact-)regenerating coes Share piece an piece-vector 0

51 References(/) [Dimakis et al.00] Dimakis A.G.; Gofrey P.B.; Yunnan Wu; Wainwright M.J.; Ramchanran K.; Network Coing for Distribute Storage Systems Information Theory IEEE Transactions on Volume: 6 Issue: 9 Publication Year: 00 Page(s): 9 [Dimakis et al.007]a. G. Dimakis P. B. Gofrey Y. Wu M. Wainwright an K. Ramchanran "Network coing for istribute storage systems" Proc. IEEE INFOCOM 007 [Wu et al.007]y. Wu A. G. Dimakis an K. Ramchanran "Deterministic regenerating coes for istribute storage" Proc. Allerton Conf. Control Computing an Communication 007 [Rashmi et al.0] Rashmi K.V.; Shah N.B.; Kumar P.V.; Optimal Exact-Regenerating Coes for Distribute Storage at the MSR an MBR Points via a Prouct-Matrix Construction Information Theory IEEE Transactions on Volume: 7 Issue: 8 Publication Year: 0 Page(s): 7-9 [Suh et al.0] Changho Suh; Ramchanran K.; Exact-Repair MDS Coe Construction Using Interference Alignment Information Theory IEEE Transactions on Volume: 7 Issue: Publication Year: 0 Page(s): -

52 References(/) [Shah et al.00]n. B. Shah K. V. Rashmi P. V. Kumar an K. Ramchanran Explicit coes minimizing repair banwith for istribute storage in Proc. IEEE Information Theory Workshop Cairo Egypt Jan.00. [Shah et al.0] Shah N.B.; Rashmi K.V.; Kumar P.V.; Information-Theoretically Secure Regenerating Coes for Distribute Storage Global Telecommunications Conference (GLOBECOM 0) 0 IEEE Publication Year: 0 Page(s): [Shah et al.0a] Shah N.B.; Rashmi K.V.; Vijay Kumar P.; Ramchanran K.; Distribute Storage Coes With Repair-by-Transfer an Nonachievability of Interior Points on the Storage- Banwith Traeoff Information Theory IEEE Transactions on Volume: 8 Issue: 0 Page(s): 87-8 [Shah et al.0b] Shah N. B.; Rashmi K. V.; Kumar P. V.; Ramchanran K.; Interference Alignment in Regenerating Coes for Distribute Storage: Necessity an Coe Constructions Information Theory IEEE Transactions on Volume: 8 Issue: 0 Page(s): - 8 [Pawaer et al.0] Pawar S.; El Rouayheb S.; Ramchanran K.; Securing Dynamic Distribute Storage Systems Against Eavesropping an Aversarial Attacks Information Theory IEEE Transactions on Volume: 7 Issue: 0 0 Page(s): 67-67

53 References(/) [Kurihara an Kuwakao0a]M.Kurihara an H.Kuwakao ``On regenerating coes an secret sharing for istribute storage'' IEICE Technical Report(in Japanese) IT00-6(0-0) pp.-8(in Japanese) Jan. 0. [Kurihara an Kuwakao0b]M.Kurihara an H.Kuwakao `On an extene version of Rashmi-Shah-Kumar regenerating coes an secret sharing for istribute storage'' IEICE Technical Report IT00-(0-0) pp.0-0(in Japanese) Mar. 0. [Kurihara an Kuwakao0c]M.Kurihara an H.Kuwakao ``On ramp secret sharing schemes for istribute storage systems uner repair ynamics'' IEICE Technical Report IT0-7(0-7) pp.-6(in Japanese) July 0. [Kurihara an Kuwakao0]M.Kurihara an H.Kuwakao ``Ramp secret sharing schemes base on MBR coes'' IEICE Technical Report ISEC0-(0-) pp.6-68(in Japanese) Nov. 0. [Kuwakao an Kurihara an0]h.kuwakao an M.Kurihara ``Computationally-Secure Regenerating Coe'' Proceeings of Computer Security Symposium 0 pp Oct. 0.

54 The last slie Delete 6 pages in which recent results ware written. [Kurihara an Kuwakao 0] (See the full version) // 時 分