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Key Recovery Using Noised Secret Sharing with Discounts Over Large Clouds Sushil Jajodia 1, W. Litwin 2 & Th. Schwarz 3 1 George Mason University, Fairfax, VA {jajodia@gmu.edu} 2 Université Paris Dauphine, Lamsade {witold.litwin@dauphine.fr} 3 Thomas Schwarz, UCU, Montevideo {tschwarz@ucu.edu.uy}jajodia@gmu.eduwitold.litwin@dauphine.frtschwarz@ucu.edu.uy 1

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What ? New schemes for an encryption key recovery Lost key is always recoverable in practical time – From a specifically encrypted backup Using a Noised Secret Sharing Scheme (NS) – We proposed such schemes in 2012 2

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What ? The schemes guaranteed 1.Recoverability in practical time of any key backed up using an NS- scheme E.g. in 10 min max But only provided a large cloud 1K – 1M nodes 3

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What ? Such a cloud is a scarce & usually external resource Possibly expensive & hard to use illegally o Money trail, traces in many logs… The constraint should make backup disclosure attempt rare But not impossible 4

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What ? After all, there is also no guarantee that someone does not pass a red light right on you Never mind, you still move around, don’t you ? 2. The schemes guaranteed also the absence of any other scheme recovering the key faster than the NS-scheme on the same cloud 5

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What ? We now enhance the proposed schemes: Key owner escrowing the key backup may retain a secret discount (code) Discounted recovery may be orders of magnitude less complex than the full- cost (brute force) recovery – The only possibility for 2012 schemes E.g. 256 times less complex for 1B code 6

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What ? Less complex means: – Less computations all together – Faster recovery on the same cloud E.g. 256 times for 1B code – Smaller cloud for the same time E.g. 256 times for 1B code In every case, cheaper in cloud cost – E.g. up to 256 times for 1B code 7

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What ? We present schemes for discounted recovery These naturally allow also the full- cost recovery But even if this case, these are not the same schemes as our earlier ones 8

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Why Currently, it’s better to encrypt the outsourced data Using client-side encryption – See news Especially Big Data – If you count on server-side encryption: good luck – Also see news 9

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Why You may also enjoy songs about privacy – By renowned CS Prof Th. Imielinski (Rutgers U) – http://www.system-crash.com/mp3/09- System%20Crash%20_%20Privacy.mp3 http://www.system-crash.com/mp3/09- System%20Crash%20_%20Privacy.mp3 – http://www.system- crash.com/mp3/HackersRevenge/05_SystemCrash_HackersRevenge_We WillForgiveYourFuture.mp3 http://www.system- crash.com/mp3/HackersRevenge/05_SystemCrash_HackersRevenge_We WillForgiveYourFuture.mp3 10

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Why Key loss danger is Achilles’ heel of modern crypto – Makes many folks refraining of any encryption With occasionally known disclosure consequences – Other loose many tears if unthinkable happens 11

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Why Frequent causes of key loss – Corrupted or lost memory & no backup – Sad loss of the key owner with all her/his secrets – Disgruntled employee departed with the key & backups 12

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Why Frequent causes of key disclosure – Hacking of one among many (simple) backup copies at owner’s site – Sad loss of the key owner with all her/his secrets – Disgruntled employee departed with the key & backups 13

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Why Frequent causes of key disclosure – Dishonest super-user / system administrator M andatory escrow of any key copy in many companies – Insider attack within an escrow service 14

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Why Frequent consequences of key loss – Valuable company data Possibly years of work by many people – U npublished valuable manuscripts Personally known case – Possibly critical personal health data Time-series of CAT Scans, ECGs, Blood analysis, med. treatments… 15

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Why Frequent consequences of key disclosure – Listen to news 16

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How ? Backup Encryption Client (Key owner) X backs up encryption key K X estimates 1-node inhibitive recovery time D – Say 70 days D measures trust to Escrow – Lesser trust ? Choose 700 days 17

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Client Side Encryption D determines minimal cloud size N for future recovery in any acceptable calculus time R – Chosen by recovery requestor E.g. 10 min – X expects N > D / R but also N D / R E.g. N 10K for D = 70 days – N 100K for D = 700 days 18

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Client Side Encryption X creates a classical shared secret for K – Sees K as a large integer, E.g., 256b long for AES – Basically, X creates a 2-share secret – Share s 0 is a random integer – Share s 1 is s 1 = s 0 XOR K Common knowledge: – K = s 0 XOR s 1 19

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Client Side Encryption X transforms the shared secret into a noised one – X makes s 0 a noised share : Chooses a 1-way hash H – E.g. SHA 256 Computes the hint h = H (s 0 ) – Chooses the noise space I = 0,1…,m,…M-1 – For some large M determined as we explain soon 20

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Client-Side Encryption – Each noise m and s 0 define a noise share s In a way we show soon as well – There are M different pseudo random noise shares All but one are different from s 0 But it is not known which one is s 0 – The only way to find for any s whether s = s 0 is to attempt the match H (s) ?= h 21

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Noised Secret Sharing 22

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Client Side Encryption X estimates the 1-node throughput T – # of match attempts H (s) ?= h per time unit 1 Sec by default X sets M to M = Int (DT). – M may be expected 2 40 ÷ 2 50 in practice M determines the worst-case and the average complexity of the recovery Respectively O(M) and O(M / 2) As we will show 23

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Client Side Encryption X forms base noise share f = p | 0 Hides the noised share as s 0 n = (f, M, h) in its noise share space [f, f + M[ Sends the noised shared secret S’ = (s 0 n, s 1 ) to Escrow as the key backup 24

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Client Side Encryption : Discount Code X may define some discount code d Basically, d is the m – bit suffix of s 0 In practice, we expect m = 8, 16. Represented in ASCII or Unicode d is then simply one or two digits, e.g., ‘j’. X retains d secretly for future recovery request(or) 25

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Discount Code Action 26 o 2 m times reduced noise space I ; M’ = M / 2 m o 2 m times reduced noise share space S’

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Escrow-Side Recovery (Backup Decryption) Escrow E receives legitimate request of S recovery in time R at most E sends data S” = (s 0 n, R) to some cloud node C with request for processing accordingly – Keeps s 1 out of the cloud C will choose between static or scalable recovery schemes 27

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Recovery Processing Parameters Node load L n : # of noises among M assigned to node n for match attempts Throughput T n : # of match attempts node n can process / sec Bucket (node) capacity B n : # of match attempts node n can process / time R – B n = R T n Load factor n = L n / B n 28

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Node Load 29 L B T 1 α L B T L B T OverloadNormal loadOptimal load 1 R t

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Recovery Processing Parameters Notice the data storage oriented vocabulary Node n respects R iff n ≤ 1 – Assuming T constant during the processing The cloud respects R if for every n we have n ≤ 1 This is our goal – For both static and scalable schemes we now present 30

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Static Scheme 31 Intended for a homogenous Cloud – All nodes provide the same throughput

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Static Scheme Full-Cost Recovery Init Phase Node C that got S” from E becomes coordinator Calculates a (M) = M / B (C) – Usually (M) >> 1 Defines N as a (M) – Implicitly considers the cloud as homogenous E.g., N = 10K or N = 100K in our ex. 32

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Static Scheme Full-Cost Recovery Map Phase C asks for allocation of N-1 nodes Associates logical address n = 1, 2…N-1 with each new node & 0 with itself Sends out to every node n data (n, a 0, P) – a 0 is its own physical address, e.g., IP – P specifies Reduce phase 33

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Static Scheme Full-Cost Recovery Reduce Phase P requests node n to attempt matches for every noise share s = (f + m) such that n = m mod N In practice, e.g., while m < M: – Node 0 loops over noise m = 0, N, 2N… So over the noise shares f, f + N, f + 2N… – Node 1 loops over noise m = 1, N+1, 2N+1… – ….. – Node N – 1 loops over m = (your guess here) 34

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Static Scheme : Node Load for Full Cost Recovery 35 T 1 α 0 1 2 N - 1 1 R, B t, L …….. f + 2N f + N f …….. f + 2N + 1 f + N + 1 f + 1 …….. f + 2N + 2 f + N + 2 f + 2 …….. f + 3N - 1 f + 2N - 1 f + N - 1 M

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Static Scheme : Termination Node n that gets the successful match sends s to C Otherwise node n locally terminate C asks every node to terminate – Details depend on actual cloud C forwards s as s 0 to E 36

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Static Scheme : Termination E discloses the secret S and sends S to Requestor – Bill included (we guess) E.g., up to 400$ on CloudLayer for – D = 70 days – R = 10 min – Both implied N = 10K with private option 37

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Static Scheme Observe that N ≥ D / R and N D / R – If the initial estimate of T by S owner holds Observe also that for every node n, we have (n) ≤ 1 – Provided the initial estimate of T by C holds in time 38

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Static Scheme Under above assumptions maximal recovery time is indeed R – See the papers for details Average recovery time is R / 2 – Every noise share is indeed equally likely to be the lucky one 39

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Static Scheme : Discounted Recovery 40 m < M’ = M / 2 m : m is bit length of d M’

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Static Scheme : Discounted Recovery If full-cost recovery needs 10K nodes for R = 10 min max. recovery calculus time and m = 8 then: The 10 min max. recovery requires only 40 – node wide cloud The 10K – node wide cloud suffices for R = 2.3 sec 1K cloud suffices for the recovery time up to 25 sec Etc 41

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Static Scheme See papers for – Details, – Numerical examples – Proof of correctness The scheme really partitions I Whatever is N and s 0, one and only one node finds s 0 42

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Static Scheme Safety – No disclosure method can in practice offer the (1-node) calculus faster than the scheme – Dictionary attack, inverted file of hints… Other properties 43

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Scalable Scheme Heterogeneous cloud – Throughput may differ among the nodes 44

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Scalable Scheme Intended for heterogenous clouds – Different node throughputs – Basically only locally known E.g. – Private or hybrid cloud – Public cloud without so-called private node option 45

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Scalable Scheme Each node n : n = 0,1… recursively splits its load through hash partitioning Until its load factor n decreases to n ≤ 1 See our previous papers for details 46

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Scalable Scheme Init phase similar up to (M) for full cost or (M’) for discounted recovery calculus – Basically (M) and (M’) >> 1 – Also we note it now 0 If > 1 we say that node overflows Node 0 sets then its level j to j = 0 and splits – Requests node 2 j = 1 – Sets j to j = 1 – Sends to node 1, (S”, j, a 0 ) 47

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Scalable Scheme As result – There are N = 2 nodes – Both have j = 1 – Node 0 and node 1 should each process M / 2 match attempts We show precisely how on next slides – Iff both 0 and 1 are no more than 1 Usually it should not be the case The splitting should continue as follows 48

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Scalable Scheme Recursive rule – Each node n splits until n ≤ 1 – Each split increases node level j n to j n + 1 – Each split creates new node n’ = n + 2 j n – Each node n’ gets j n’ = j n initially Node 0 splits thus perhaps into nodes 1,2,4… Until 0 ≤ 1 Node 1 starts with j= 1 and splits into nodes 3,5,9… Until 1 ≤ 1 49

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Scalable Scheme Node 2 starts with j = 2 and splits into 6,10,18… Until 2 ≤ 1 Your general rule here Node with smaller T splits more times and vice versa 50

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Scalable Scheme : Splitting 51 5 B α = 0.5 3 0 1 2 4 8 16 1 2 4 α = 0.8 α = 0.7 5 Node 0 split 5 times. Other nodes did not split. j T T T

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Scalable Scheme If cloud is homogenous, the address space is contiguous Otherwise, it is not – No problem – Unlike for a extensible or linear hash data structure 52

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Scalable Scheme : Reduce phase Every node n attempts matches for every noise k [0, M-1] such that n = k mod 2 j n. If node 0 splits three times, in Reduce phase it attempts to match noised shares (f + k) with k = 0, 8, 16… If node 1 splits three times, it attempts to match noised shares (f + k) with k = 1, 17, 33… Etc. 53

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Scalable Scheme : Reduce Phase 54 3 B α = 0.5 0 1 4 j T …. f + 16 f + 8 f …. f + 33 f + 17 f + 1 L

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Scalable Scheme N ≥ D / R – If S owner initial estimate holds For homogeneous cloud, N is 30% greater on the average and twice as big at worst / static scheme Cloud cost may still be cheaper – No need for private option Versatility may still make it preferable besides 55

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Scalable Scheme Max recovery time is up to R – Depends on homogeneity of the cloud Average recovery time is up to R /2 See again the papers for – Examples – Correctness – Safety – … – Detailed perf. analysis remains future work 56

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Multi-share noised secret sharing 57 For max recovery time R arbitrarily close to the average one : R = R ave * (k + 1) / k for M = D T / k ; M’ = M / 2 m

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Multi-share noised secret sharing 58 We now have – R = R ave * (k + 1) / k – Welcome capability for some Drawback: discount code becomes randomly valued d = (d 0 …d k-1 ) E.g. d = (j,…,a) at the figure Each d i is m – bit wide Perhaps harder to retain for k > 4 See the paper for paths to solutions See previous papers for scheme analysis

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Related Work RE scheme for outsourced LH* files CSCP scheme for outsourced LH* records sharing Crypto puzzles One way (hash) functions with trapdoors 30-year old excitement around Clipper chip Botnets Snowden’s Leaks (so-called) 59

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Conclusion NS-schemes with discounts effectively relieve the fear of encryption key loss/disclosure – Achilles’ heel of cryptography Key is always recoverable from its backup Especially easily through a discounted recovery 60

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Conclusion The backup disclosure is hard at will So can be also arbitrarily unlikely But remains always a possibility Like a driver going through a red light right on us Despite all precautions But we still drive our cars, don’t we ? 61

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Conclusion To our best knowledge, our technique is the only with the discussed properties at present It appears of interest to many apps It should especially help potential BigData users Deterred up to now for many By potentially catastrophic consequences of outsourced data loss / disclosure 62

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Conclusion NS sharing is a generalization of the well- known secret sharing An arbitrary additional cost C still protects the secret, even if one gets all the shares – Lowered at will by the discounts The “classical” scheme amounts to C = 0 NS sharing appears of general interest for the cryptography – Beyond our current use 63

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Conclusion Consider a company secret split into shares among some insiders With NS sharing, the secret is still arbitrarily protected even if a collusion / hacking among the insiders happens – So that someone discloses all the shares Not with the classical algorithm 64

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Conclusion The scalable partitioning scheme appears also a general purpose tool Could be called Scalable Map-Reduce Apparently failed dream of Amazon’s cloud service – Under the name of Elastic Map-Reduce 65

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Future work Deeper formal analysis Proof of concept implementation – Started using the popular open source Vagrant VM for the scalable scheme Experiments show 2.7s on the average for a node to (re)start Vagrant on another node so it is ready to process data sent in. 66

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Future work 67 One should also test the static scheme on some easily available Map-Reduce implementation E.g. Amazon EMR in commercial world

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Future work 68 Nevertheless, all things considered, the static scheme with discounts appears ready for use All ‘bricks’ are already out there Ready to be assembled

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Thanks for Your Attention 69 Witold LITWIN & al

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