New Bounds for Distributed Storage Systems with Secure Repair

Transcription

1 New Bouns for Distribute Storage Systems with Secure Repair Ravi Tanon 1 an Soheil Mohajer 1 Discovery Analytics Center & Department of Computer Science, Virginia Tech, Blacksburg, VA Department of Electrical an Computer Engineering, University of Minnesota, Twin Cities, MN Abstract In this paper, we consier exact-repair istribute storage systems. Characterizing the optimal storage-vs-repair banwith traeoff for such systems remains an open problem, in general with results available in the literature for very specific instances. We characterize the optimal traeoff between storage an repair banwith if in aition to exact-repair requirements, an aitional requirement of pair-wise symmetry on the repair process is impose. The optimal traeoff surprisingly consists of only one efficient point, namely the minimum banwith regenerating (MBR point. These results are also extene to the case in which the store ata (or repair ata must be secure from an external wiretapper, an the corresponingly optimal secure traeoffs are also characterize. The main technical tool use in the converse proofs is a use of Han s inequality for sub-sets of ranom variables. Finally, we also present results in which the pair-wise symmetry constraint is relaxe an a new converse boun is obtaine for the case of exact repair in which an external aversary can access the repair ata of any one noe. This boun improves upon the existing best known results for the secure an exact repair problem. I. INTRODUCTION Contemporary istribute storage systems store massive amounts of ata over a set of istribute noes. Besies the traitional goals of achieving reliability by introucing reunancy, new aspects such as efficient repair of faile storage noes make their esign even more challenging. To overcome these issues, the concept of regenerating coes for istribute storage systems was introuce by Dimakis et al. 3]. A istribute storage system (DSS consists of n storage noes each with a storage capacity of α units of ata such that the entire file of size B can be recovere by accessing any k < n noes. This is calle as the reconstruction property of the DSS. Whenever a noe fails, noes (where k n 1 participate in the repair process by sening β units of ata each. This proceure is terme as the regeneration of a faile noe an β is referre to as the per-noe repair banwith. In 3], by using the concepts of network coing ], the authors show that the parameters of a DSS must satisfy B min (α, ( iβ. (1 i=0 Thus, in orer to store a file of size B, there exists a funamental traeoff between α (storage an β (total repair banwith. However, this traeoff is in general achievable tanonr@vt.eu, soheil@umn.eu only for the functional-repair case 3]. In functional repair, a faile noe is replace by a new noe such that the resulting DSS has the same reconstruction an regeneration capabilities as before. In particular, the contents of the repaire noe may not necessarily be ientical to the faile noe even though the esirable properties of the DSS are preserve. In contrast to functional repair, exact repair regeneration requires the repair process to replace a faile noe with an ientical new noe. Exact repair is a practically appealing property specially when it is esirable that the store contents remain intact over time. Furthermore, the file recovery process is also easier in this case as the reconstruction proceure nee not change whenever a faile noe is replace. While characterizing the storage-vs-banwith traeoff for the case of exact repair remains a challenging open problem in general, two extreme points of this traeoff (epening on whether α or β is minimize first namely, the minimum storage regenerating case (MSR an the minimum banwith regenerating (MBR case have been stuie extensively (see 5], 6] an references therein. Beyon these points, the optimal exact-repair traeoff for the (, 3, 3-DSS was characterize in 7] where it has been shown that the optimal traeoffs for functional an exact repair are ifferent. There have been several recent works in this regar, namely new outer bouns (converse results beyon (, 3, 3 in 1], as well as novel achievable schemes in 15] an 16], an new results for secure an exact repair in 17] 19]. Even though these recent avances have brought forth new insights on the funamental problem of exact repair, the general problem of characterizing the optimal traeoff for the exact repair problem remains open till ate. Besies the requirements of exact repair, another important esign concern for such systems is that of ata security. Pawar et.al 9] introuce the notion of information theoretic security for such systems where various moels for the aversary are introuce an investigate. In one of those moels (which we refer to as Type-I moel, the aversary can access the ata store in any l < k noes. In another moel (referre to as the Type-II moel, the aversary is more powerful an can access not only the content of the noes but the repair ata of any l < k noes sent by other noes to repair. The focus of this paper is on the exact repair problem with such security constraints an the main contributions of this paper are two fol: For the istribute storage system with an external

2 aversary, we obtain new outer bouns for the exact an secure repair problem. In particular, we consier the general (n, k, DSS operating in the presence of an eavesropper which can rea the repair ata of any l noes. The goal is to fin the maximum size of ata which can be store while being information theoretically secure from such aversary. An upper boun on the amissible file size for this general problem was obtaine in 9] together with some optimality results; however the general problem still remains open. We evelop a new boun for this problem for general parameters (n, k, an for l = 1 an show that it outperforms the bouns of 9]. We also focus on the exact repair problem with an aitional conition of pair-wise symmetric repair. The pair-wise symmetric repair conition refers to the following: consier any two storage noes, for example, noes i an j an a set of other ( 1 helper noes which we enote collectively as H. Now, consier the repair of noe i from noe j an the fixe ( 1 helper noes an enote the repair ata sen by noe j as S j i. Similarly, consiering the repair of noe j from noe i an the same set of fixe helper noes, we efine the repair ata sent from noe i to noe j as. Then, the pair-wise symmetric repair conition enforces that = S j i. In other wors, it means that given the fixe set of ( 1 helper noes, the ata which noe i sens to repair noe j must be the same as the ata which noe j must sen to repair noe i. Through a novel converse proof, we characterize the optimal traeoff of the exact repair problem along with the pair-wise symmetry conition. Interestingly, the traeoff only has one efficient point (namely the minimum banwith regenerating (MBR point, the achievability for which has been establishe in 11]. The pair-wise symmetry restriction also presents an operation interpretation of the MBR point an shows that there exists no other (more efficient exact repair point satisfying such a conition. Notation: We enote the set {1,,..., m} by m] for m Z +. For any set A n] we efine W A = {W i : i A}. For sets A, B n], we efine S A B = { : i A, j B}. Finally we use S A B to enote the union of repair ata in both irections, i.e, (S A B, S B A. II. SYSTEM MODEL A (n, k,, l DSS consists of n storage noes that store a file F of size B across n noes, with each noe storing up to α units of ata. A ata collector connects to any k < n noes in orer to reconstruct the file F. This is known as the MDS property of the DSS 3]. We focus on single noe failures in which at any given point only one noe in the system coul fail. For the repair of a faile noe, any out of the remaining (n 1 alive noes sen β α units of ata in orer to ai the repair process. The parameter β is referre to as the total repair banwith. From an information theoretic perspective, the goal is to store a file F, whose entropy is B, i.e., H(F = B. Let W i enote the storage content at noe i, for i = 1,..., n. Hence, H(W i α, i = 1,,..., n. ( Due to the MDS property we also have H (F W K = 0, K n] with K = k. (3 Let enote the ata sent by noe i to repair noe j. Due to the repair banwith constraint, we have H( β, i, j n] ( an for exact repair of noe j from noes, we also have H(W j S D j = 0, j / D n], D =. (5 Since is a function of the ata store in noe i, we have H( W i = 0. Secrecy Constraints: We also have another parameter l < k, which signifies the number of noes an aversary can wiretap. In particular, we consier two types of aversaries: Type-I: aversary can wiretap only the store ata on any l noes. For any l < k noes to be secure, we require I (F ; W L = 0, L n] with L l (6 where W L is the ata store on noes whose inex belong to L. Type-II: aversary can wiretap the repair ata of any l noes. For the repair of any l < k noes to be secure, we require I (F ; S n1, S n..., S nl = 0, (7 where S ni is the repair ata ownloae from any other noes to repair noe n i. Note that the noes repairing n i might be ifferent from those helping to repair n j. We first note that the Type-I aversary is in general weaker than Type-II aversary since the constraint (7 implies (6 but not the other way aroun. Finally, we note that by setting l = 0, we recover the original problem with no security constraints. III. MAIN RESULTS & DISCUSSION Theorem 1 The secrecy capacity of an (n, k, storage system with Type-II secrecy against an eavesropper who observes the repair ata for any l = 1 noe is upper boune by B S k 1 α + (k 1(3 k β. (8 The best existing results for the Type-II secrecy problem with general parameters are in 9] which shows that B S min (α, ( iβ. (9 i=l The result of Theorem 1 improves upon the above boun in general. We next illustrate our boun in (8 together with the

3 (n, k, =(5,, ` =1 1/3 5/1 1/5 New boun (8 = B S = B S Pair-wise Symmetric Repair (10 Theorem The optimal (α, β trae-off for an (n, k,, l istribute storage system with exact an pair-wise symmetric repair is given by B S (k l (( k for both Type-I an Type-II secrecy. ( ] l ( α min, β (11 1/6 1/3 8/1 /5 Pawar et.al s boun (9 1/ /3 = (MBR Fig. 1: Bouns for (n, k, = (5,, an l = 1. boun in (9 for the case of (n, k, = (5,, an l = 1 in Figure 1. We strongly believe that even being tighter, the boun in Theorem 1 is not achievable. We in fact conjecture that the secrecy capacity for a general (n, k,, l system with strong secrecy constraint is upper boune by B S (k l (( k ( l ] min ( α, β (10 In particular, this conjecture suggests that the MBR point in the only efficient point on the optimum trae-off for the strong secrecy capacity of DSSs. Specific scenarios for which the Type-II secrecy problem has been completely solve are in 17] 19], an correspon to the following cases: All (n, k,, l, 3 (n, k, = (n, n 1, n 1 an l = n. For all these cases, this conjecture has been prove to be true. Even though we are unable to prove this conjecture uner the general setting, we next consier a constraine version of this problem with a pair-wise symmetric repair conition an prove that the optimal trae-off for this restricte problem is inee given by the above boun. The pair-wise symmetric repair conition refers to the following: consier any two storage noes, for example, noes i an j an a set of other ( 1 helper noes which we enote collectively as H. Now, consier the repair of noe i from noe j an noes in H an enote the repair ata sen by noe j as S j i. Similarly, consiering the repair of noe j from noe i an the same set of fixe helper noes H, we efine the repair ata sent from noe i to noe j as. Then, the pair wise symmetric repair conition enforces that = S j i. In other wors, it means that given the fixe set of ( 1 helper noes, the ata which noe i sens to repair noe j must be the same as the ata which noe j must sen to repair noe i an hence the name pair-wise symmetric repair. This result shows that with the symmetric pair-wise repair constraint, the only efficient point in the storage-banwith traeoff is the MBR point. This also provies another operational interpretation of the MBR point. The proofs for Theorems 1 an are presente next. IV. PROOF OF THEOREM 1 In this section we focus on the case of Type-II secrecy with l = 1, an present the proof of Theorem 1. Let A + 1] be an arbitrary set with A = k 1, an i + 1] \ A be an inex not in A. We have B S = H(F = H(F S +1]\{i} i = H(F, S +1] i H(S +1] i (1 H(F, W A, S +1] i H(S +1] i = H(W A, S +1] i H(S +1] i = H(W A + H(S +1] i W A H(S +1] i H(W A + H(S +1] i S A i H(S +1] i = H(W A H(S A i. (13 where in (1 we just ae S i i to what the eavesropper observes, an set S i i to be a ummy variable with zero entropy. In orer to boun H(S A i we can write H(S A i = ( 1 k 1 A +1]\{i} A = 1 ( A +1]\{i} A = H(S A i (1 H(S A i (15 k 1 H(W i. (16 where (1 follows from the symmetry of the repair variables, that is, H(S A i is invariant w.r.t. the choice of A + 1] \ {i} provie that A = k 1; (15 follows from Han s inequality 13]; an inequality in (16 hols since the content of noe i can be reconstructe from repair ata ownloae from noes in A with A =. In orer to upper boun H(W A, we can write H(W A = H(W i W i 1] = H(Wi I(W i ; W i 1] ], (17

4 an boun the mutual information above by I(W i ; W i 1] I(S i i 1] ; S i 1] i = I(S i i] ; S i] i (18 = H(S i i] + H(S i] i H(S i i] (19 where in (18 we just ae S i i which is a ummy ranom variable, with zero entropy. Next we boun terms in (19 iniviually. First note that H(S i i] + H(S k:+1] 1:] j=1 j=1 i=j = H i +1 j=1 i=j = H +1 j=1 i=j j=1 W j (0 H(W A. (1 Note that in orer to get (0, we can argue that W 1 can be recovere from +1 j=1 S j 1. Once W 1 is available, we can construct S 1, which together with +1 j= S j can reconstruct W. Now, from W 1 an W, we fin S 1 3 an S 3, respectively, an then with the help of +1 j=3 S j 3 we recover W 3. Continuing this proceure, all the content of all iscs in k 1] can be recovere. Similarly, H(S i] i + H(S k:+1] 1:] j=1 = H i S j i, j=1 i=j+1 j=1 i=j+1 S j i, S j i, +1 j=k S j i W i H(W A. ( Finally we can boun the last term in (19 using the following lemma. The proof of lemma is presente in Appenix I. Lemma 1 For the (n, k, istribute storage system with exact repair requirements, we have H(S i i] H(S +1] i H(S +1] ] H(S +1] i H(W A. (3 Plugging (19, (1, (, an (3 in (17, we get H(W A H(W i + H(S k:+1] 1:] + which together with (16 imply H(S +1] i ( B S 1 H(W i + 1 H(S k:+1] 1:] ( k 1 + k 1 H(S +1] i k 1 ( (k 1( + k (k 1( α + + β = k 1 (k 1(3 k α + β, which is the esire boun. Note that this boun is a vali one for. For <, the problem has been completely solve from the results in 17] 19]. V. PROOF OF THEOREM We prove the boun for Type-I secrecy. Valiity of the boun for Type-II secrecy is just an immeiate consequence of the fact that Type-II secrecy is stronger than Type-I. We boun the entropy of the file, F as follows B S = H(F = H(F W 1,..., W l (5 H(F, W l+1,..., W k W 1,..., W l = H(W l+1,..., W k W 1,..., W l + H(F W 1,..., W l, W l+1,..., W k = H(W l+1,..., W k W 1,..., W l (6 k = H(W i W 1,..., W i 1 (7 where (5 follows from the Type-I (weaker secrecy constraint, an (6 follows from the property that the file F must be recoverable from the ata store in any k noes. Next, the rest of the proof is evote to upper bouning the terms appearing in the summation. In particular, there are two ways to upper boun the terms. The first one is rather

5 straightforwar an is as follows: B S k k H(W i W 1,..., W i 1 H(S +1] i S i 1] i k ( i + 1β (8 to obtain one of the terms in Theorem an has the epenence on the repair banwith, β. We next prove the other boun (which epens on the storage parameter α. To this en, we rewrite the summation in (7 as k B S H(W i W 1,..., W i 1 (9 = k ] H(W i I(W i ; W i 1] (30 In orer to further upper boun the above expression, we state the following Lemma which is central to the proof. We present the proof of Lemma in Appenix II. Intuitively, this lemma states that with exact an pair-wise symmetric repair constraint, the mutual information between the content on a noe W s+1 an the ata store at s other noes is lower boune as a fraction of the entropy of the noe W s+1. Lemma For the (n, k, istribute storage system with exact an pair-wise symmetric repair requirements, we have ( s I(W s+1 ; W 1,..., W s H(W s+1 (31 for any 1 s. Using Lemma, we further upper boun (30 as follows: B S = = k k k k ] H(W i I(W i ; W i 1] H(W i ( i + 1 ( i + 1 ( i 1 H(W i Hence, from (8 an (3, we arrive at k ( α B S ( i + 1 min, β = (k l (( k ] H(W i α (3 ( ] l ( α min, β (33 thus completing the proof of Theorem. VI. CONCLUSIONS We consiere the istribute storage problem with exact repair an information theoretic secrecy constraints. We evelope a novel outer boun on the storage an repair banwith traeoff. Our results for the general problem improve upon the best known outer bouns for this problem for all (n, k, parameters when the eavesropper can access the repair ata of any one noe. Furthermore, uner a more restrictive scenario of pair-wise symmetric repair, we completely characterize the corresponing traeoff an show that it correspons to the MBR point. The key technical element in the proofs of these results involves a novel use of Han s inequality. Future work inclues extensions of this technique for tightening of the bouns for l 1. APPENDIX I PROOF OF LEMMA 1 We can prove a stronger claim, that is, for any τ, we have τ τ H(S +1] i H(S i i] + H(S +1] τ]. (3 This can be prove by inuction on τ. It is clear that for τ = 1, the summation in the RHS reuces to the entropy of a ummy variable S 1 1, which is zero. The remaining terms are just ientical, an so the inequality hols with equality. Now, assume the claim hols for τ 1. Then we have τ H(S +1] i = H(S +1] i + H(S +1] τ ] H(S i i] + H(S +1] ] + H(S +1] τ = H(S i i] + H(S +1] ], S ] τ (35 + H(S +1] τ, S τ ] (36 = H(S i i] + H(S +1] ], S τ τ] + H(S +1] τ, S τ τ] (37 H(S i i] + H(S τ τ] + H(S +1] ], S +1] τ, S τ τ] (38 τ H(S i i] + H(S +1] τ]. (39 Note that (35 hols because of the inuction assumption for τ 1. in (36 we use the fact that for any 1 j τ, having repair ata from ( + 1 (from noes excluing itself one can recover W j, an hence S j i, for any i. In particular, S i i 1] is a function of the repair ata;

6 in (37 we just use the efinition of S τ τ], an the fact that S τ τ is a ummy variable; an (38 follows from inequality H(XZ + H(Y Z H(XY Z + H(Z. This completes the proof. First note that APPENDIX II PROOF OF LEMMA H(S s+1 s] W s+1 = 0 (0 ue to the fact that S s+1 i is the repair ata sent by the noe s + 1 in the repair of noe i, for i = 1,..., s. Furthermore, we also note that H(S s] s+1 W s] = 0 (1 since S i s+1 is the repair ata sent by noe i in the repair of noe s+1. We can thus lower boun the mutual information I(W s+1 ; W s] as I(W s+1 ; W s] I(S s+1 s] ; S s] s+1 = I(S s] s+1 ; S s] s+1 ( = H(S s] s+1. (3 In (, we invoke the pair-wise symmetric repair constraint, i.e., = S j i an replace S s+1 i by S i s+1 for i = 1,..., s. To complete the proof of the lemma, we next show the following H(S s] s+1 s H(W s+1. ( To prove (, we can write H(S s] s+1 = ( 1 s s 1 ( A +1]\{s+1} A =s A +1]\{s+1} A = H(S A s+1 (5 H(S A s+1 (6 s H(W s+1 (7 where (5 follows from the fact that the system is symmetric, an H(S A s+1 is the same for all A with A = s; in (6 we have use Han s inequality 13] for ranom variables X i = S i s+1, for i + 1] \ {s + 1}, which is 1 ( s A: A =s H(X(A s 1 ( A : A = H(X(A. Finally, (7 follows from the fact that W s+1 must be recoverable from the repair ata coming in from noes. Substituting (7 in (3 we get the esire boun. ] K. V. Rashmi, N. B. Shah an P. V. Kumar, Enabling noe repair in any erasure coe for istribute storage, in arxiv: , Jun ] A. G. Dimakis, P. B. Gofrey, Y. Wu, M. Wainwright an K. Ramchanran, Network coing for istribute storage systems, IEEE Trans. Inf. Theory, vol. 56, no. 9, pp , Sept ] R. Ahlswee, N. Cai, S.-Y. R. Li an R. W. Yeung, Network information flow, IEEE Trans. Inf. Theory, vol. 6, pp , Jul ] N. B. Shah, K. V. Rashmi, P. V. Kumar an K. Ramchanran, Distribute storage coes with repair-by-transfer an non-achievability of interior points on the storage-banwith traeoff, IEEE Trans. Inf. Theory, vol. 58, no. 3, pp , Mar ] V. Caambe, S. Jafar, H. Maleki, K. Ramchanran an C. Suh, Asymptotic interference alignment for optimal repair of MDS coes in istribute storage, IEEE Trans Inf. Theory, vol. 59, no. 5, pp , May ] C. Tian, Rate region of the (,3,3 exact-repair regenerating coes, in Proc. Intern. Symp. Inf Theory, ISIT, Istanbul, Turkey, Jun ] A. S. Rawat, O. O. Koyluoglu, N. Silberstein, an S. Vishwanath, Optimal locally repairable an secure coes for istribute storage systems, in arxiv: , Aug ] S. Pawar, S. El Rouayheb, an K. Ramchanran, Securing ynamic istribute storage systems against eavesropping an aversarial attacks, IEEE Trans. Inf. Theory, vol. 58, pp , Mar ] O. O. Koyluoglu, A. S. Rawat, an S. Vishwanath, Secure Cooperative Regenerating Coes for Distribute Storage Systems,, in arxiv: , Oct ] N. B. Shah, K. V. Rashmi, an P. V. Kumar, Information-theoretically secure regenerating coes for istribute storage, in Proc. IEEE Global Commun. Conf., GLOBECOM, Houston, TX, Dec ] S. Goparaju, S. El Rouayheb, R. Calerbank an H. Vincent Poor, Data Secrecy in Distribute Storage Systems uner Exact Repair, in Proc. IEEE International Symposium on Network Coing, NETCOD, Calgary, Canaa, Jun ] T. M. Cover an J. A. Thomas, Elements of Information Theory, New York: Wiley. 1] B. Sasiharan, K. Senthoor, P. V. Kumar, An Improve Outer Boun on the Storage-Repair-Banwith Traeoff of Exact-Repair Regenerating Coes, in arxiv: , Dec ] C. Tian, B. Sasiharan, V. Aggarwal, V. A. Vaishampayan, P. V. Kumar, Layere, Exact-Repair Regenerating Coes Via Embee Error Correction an Block Designs, in arxiv: , Aug ] S. Goparaju, S. El Rouayheb, R. Calerbank, New Coes an Inner Bouns for Exact Repair in Distribute Storage Systems, in arxiv:10.33, Feb ] R. Tanon, S. Amuru, T. C. Clancy an R. M. Buehrer, Towars Optimal Secure Distribute Storage Systems with Exact Repair, in arxiv: , Oct ] R. Tanon, S. Amuru, T. C. Clancy an R. M. Buehrer, On Secure Distribute Storage Systems with Exact Repair, in Proc. IEEE International Conference on Communications (ICC, Syney, Australia, June ] R. Tanon, S. Amuru, T. C. Clancy an R. M. Buehrer, Distribute Storage Systems with Secure an Exact Repair - New Results, in Proc. Information Theory an Applications Workshop (ITA, San Diego, CA, Feb. 01. REFERENCES 1] K. V. Rashmi, N. B. Shah, D. Gu, H. Kuang, D. Borthakur an K. Ramchanran, A Solution to the Network Challenges of Data Recovery in Erasure-coe Distribute Storage Systems: A Stuy on the Facebook Warehouse Cluster, in arxiv: , Sept. 013.