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- 1. A Locality Sensitive Hashing Filter for Encrypted Vector Databases Junpei Kawamoto (University of Tsukuba, Japan) This work is partly supported by The Nakajima Foundation
- 2. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 2Vector databases• A kind of databases consists of vectors and values. • eg. a picture database feature vector picture (value) (129, 251, 94, …. )T (98, 112, 49, …. )T
- 3. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 3Vector databases• A kind of databases consist of vectors and values.• Simply assume the scheme is (k, v) • k: key vector attribute (feature vector, etc.) • v: value attribute (do not care about the data type)• Queries • Only over the key vector attribute. • Find tuples having key vectors k s.t. sim(k, q) ≧ α • q: query vector • α: threshold We employ cosine similarity and assume all vectors are normalised.
- 4. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 4Vector databases• A query example. Query vector: feature vector picture (value) (129, 251, 90, …. )T Threshold: 0.8 (129, 251, 94, …. )T (129, 251, 94, …. )T (98, 112, 49, …. )T
- 5. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 5Cloud sourced vector databases• A database owner wants to deploy it on a cloud service • To share data easily VDB deploy access VDB Colleagues Database owner (Database user) • The owner does not have to manage any servers.
- 6. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 6Privacy and security concerns• Can the owner and the users trust the cloud services? Malicious services can read Malicious services can read data in the VDB. queries from users. VDB deploy access VDB Database user Database owner • VDB might have sensitive information. • Queries (i.e. query vectors) also might be sensitive information.
- 7. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 7Encrypted vector databases• All tuples are encrypted before deploying them.• Queries are also encrypted. Malicious services cannot Malicious services cannot read any data in the EVDB.read any queries from users. EVDB deploy access VDB Colleagues Database owner (Database user) • Many approaches are proposed. • We use those methods as basic protocols.
- 8. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 8Encrypted vector databases• All tuples are encrypted before deploying them. • Enck: An algorithm to encrypt key vectors • Encv: An algorithm to encrypt values • A plain tuple (k, v) is encrypted to (Enck(k), Encv(v))• Queries are also encrypted. • Encq: An algorithm to encrypt query vector • A query vector q is encrypted to Encq(q)• An important property of those encryption algorithm • Invariance of similarity • k・q = Enck(k)・Encq(q) (cosine similarities are same after encryption)
- 9. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 9Encrypted vector databases• Decryption algorithm are also shared in owner and users. • Deck: An decryption algorithm for key vectors • Decv: An decryption algorithm for values • Decryption algorithms for query vectors are not necessary.• All encryption/decryption algorithms are • defined by each existing protocol, • secret for servers. We do not define those algorithms in this work.
- 10. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 10Encrypted vector databases• Malicious cloud services cannot read any data. Cannot decrypt any data. EVDB find tuples s.t. (Enck(k), Encv(v))Enck(k)・Encq(q) ≧ α VDB Database user Database owner• Cloud services also cannot optimise query processes. • Must compute similarities for all tuples.
- 11. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 11The Problem of existing protocols• Cloud services (servers) must check all tuples.• Because of encryptions • Structures of vectors are not same after encryption • Structure based indexing such as R-tree cannot work well • Server also cannot cache query results, since cannot know which queries are same.• We introduce a filtering method based on LSH. • We focus the fact that even after encryption, the similarities are not changed. • LSH is a compressed data structure to estimate similarities of vectors.
- 12. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 12Locality sensitive hashing (LSH)• Approximate similarities with small data• LSH consists of m functions: hi (I = 1, 2, …, m) 1; v・bi ≧ 0 hi(v) = (bi is the base vector of function hi) 0; otherwise• LSH value of a vector v • lsh(v) = (h1(v), h2(v), …, hm(v))• Property cos(u, v ) cos( (1 Pr[ lsh (u ) lsh ( v )])) • Pr[lsh(u)=lsh(v)]: how many hash values of the two vectors u and v have same values i.e. hi(u) = hi(v)
- 13. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 13Locality sensitive hashing (LSH)• eg. b1 • lsh(u) = (1, 1, 0) u • lsh(v) = (1, 1, 1) v • Pr[lsh(u) = lsh(v)] = 2/3 b2 • cos(u, v) 〜 cos(π(1 – 2/3)) = 1/2 b3• The accuracy of the approximation depends on • the number of base vectors m • the distribution of target vectors
- 14. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 14Locality sensitive hashing (LSH)• If the distribution of encrypted vectors is lopsided, LSH cannot distinguish those vectors efficiently b1 To distinguish v1-v3, additional b1 b4 v1 base vectors are needed. v1 v2 v2 b 5 v3 v3 b2 b2 b3 b3 • In worst case, the number of base vectors m = the number of tuples• We employ whitening transformation to reduce skew of the vector space.
- 15. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 15Whitening transformation• A technique to remove correlations from vectors • At first, compute the average vector μ and covariance matrix Σ. S = E((v - m )(v - m ) ) T • Then, decompose Σ. S = FLF-1 • The whitening matrix Wk is Wk = FL-1/2
- 16. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 16Whitening transformation• For any vector v, the whitened vector vw is v w = W (v - m ) k T• The covariance matrix of whitened vectors is E(v w vT ) w = E(WkT (v - m )(v - m )T Wk ) = E(L -1/2FT SFL -1/2 ) = I • there are no correlations between the whitened vectors.
- 17. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 17Applying Whitening• Original protocol (typical EVDB protocols) • encrypted vector of k: Enck(k) • query condition of q: find Enck(k) s.t. Enck(k)・Encq(q)≧α• Our proposal protocol Whitening • encrypted vector of k: WkT(Enck(k) – μ) • query condition of q: find WkT(Enck(k)–μ) s.t. WkT(Enck(k)–μ)・Wk-1Encq(q)≧α–μ・Encq(q) Counter whitening• The following two conditions are same • find Enck(k) s.t. Enck(k)・Encq(q)≧α • find WkT(Enck(k)–μ) s.t. WkT(Enck(k)–μ)・Wk-1Encq(q)≧α–μ・Encq(q)
- 18. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 18Applying Whitening• Define wrapped algorithms: • Enck*(k) = WkT(Enck(k) – μ) • Encq*(q) = Wk-1Encq(q) • Deck*(ke) = Deck((WkT)-1ke + μ)• These algorithms are shared between owner and users.
- 19. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 19 Preparing the LSH filter • At first, servers add LSH values to all tuples converted by the serverdeploy by the owner (Enck*(k), Encv(v)) VDB (lsh(Enck*(k)), Enck*(k), Encv(v)) server Database owner
- 20. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 20Preparing the LSH filter• Make groups by LSH values. LSH value tuple (1, 0, ……, 0) ((1, 0, ….., 0), Enck*(k1), Encv(v1)) ((1, 0, ….., 0), Enck*(k2), Encv(v2)) (1, 1, ……, 0) ((1, 1, ….., 0), Enck*(k1), Encv(v1))
- 21. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 21 Filtering • After receiving queries, server computes lsh of quey vector Compute lsh(Encq*(q))find Enck*(k) s.t. Enck*(k)・Encq*(q)≧α* LSH value tuple (1, 0, ……, 0) ((1, 0, ….., 0), Enck*(k1), Encv(v1)) ((1, 0, ….., 0), Enck*(k2), Encv(v2)) Database user (1, 1, ……, 0) ((1, 1, ….., 0), Enck*(k1), Encv(v1))where α* = α–μ・Encq(q)
- 22. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 22 Filtering • After receiving queries, server computes lsh of quey vector Estimate similarity between Compute lsh(Encq*(q)) Encq*(q) and this group by Pr[(1,0,…,0)=lsh(Encq*(q))]find Enck*(k) s.t. Enck*(k)・Encq*(q)≧α* LSH value tuple (1, 0, ……, 0) ((1, 0, ….., 0), Enck*(k1), Encv(v1)) ((1, 0, ….., 0), Enck*(k2), Encv(v2)) Database user (1, 1, ……, 0) ((1, 1, ….., 0), Enck*(k1), Encv(v1)) where α* = α–μ・Encq(q)
- 23. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 23 Filtering • After receiving queries, server computes lsh of quey vector Estimate similarity between Compute lsh(Encq*(q)) Encq*(q) and this group by Pr[(1,0,…,0)=lsh(Encq*(q))]find Enck*(k) s.t. Enck*(k)・Encq*(q)≧α* LSH value tuple (1, 0, ……, 0) ((1, 0, ….., 0), Enck*(k1), Encv(v1)) If the estimated similarity <α*, ((1, 0, ….., 0), Enck*(k2), Encv(v2)) skip this group Database user (1, 1, ……, 0) ((1, 1, ….., 0), Enck*(k1), Encv(v1)) where α* = α–μ・Encq(q)
- 24. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 24 Filtering • After receiving queries, server computes lsh of quey vector Estimate similarity between Compute lsh(Encq*(q)) Encq*(q) and this group by Pr[(1,1,…,0)=lsh(Encq*(q))]find Enck*(k) s.t. Enck*(k)・Encq*(q)≧α* LSH value tuple (1, 0, ……, 0) ((1, 0, ….., 0), Enck*(k1), Encv(v1)) ((1, 0, ….., 0), Enck*(k2), Encv(v2)) Database user (1, 1, ……, 0) ((1, 1, ….., 0), Enck*(k1), Encv(v1)) where α* = α–μ・Encq(q)
- 25. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 25 Filtering • After receiving queries, server computes lsh of quey vector Estimate similarity between Compute lsh(Encq*(q)) Encq*(q) and this group by Pr[(1,1,…,0)=lsh(Encq*(q))]find Enck*(k) s.t. Enck*(k)・Encq*(q)≧α* LSH value tuple (1, 0, ……, 0) ((1, 0, ….., 0), Enck*(k1), Encv(v1)) If the estimated similarity ≧α*, ((1, 0, ….., 0), Enck*(k2), Encv(v2)) check the actual query condition for all tuples in this group Database user (1, 1, ……, 0) ((1, 1, ….., 0), Enck*(k1), Encv(v1)) where α* = α–μ・Encq(q)
- 26. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 26 Filtering • After receiving queries, server computes lsh of quey vector Estimate similarity between Compute lsh(Encq*(q)) Encq*(q) and this group by Pr[(1,1,…,0)=lsh(Encq*(q))]find Enck*(k) s.t. Enck*(k)・Encq*(q)≧α* LSH value tuple (1, 0, ……, 0) ((1, 0, ….., 0), Enck*(k1), Encv(v1)) If the estimated similarity ≧α*, ((1, 0, ….., 0), Compute), Encv(v2)) Enck*(k2 check the actual query condition for all tuples in this group Enck*(k)・Encq*(q) Database user (1, 1, ……, 0) ((1, 1, ….., 0), Enck*(k1), Encv(v1)) We can omit to computing similarity for less similar vectors where α* = α–μ・Encq(q)
- 27. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 27Summary of our methodology• Client side • Use Enck*(k), Encq*(q), and Deck*(ke) • instead of original algorithms defined by the associated protocol. • Use query conditions Enck*(k)・Encq*(q) ≧ α – μ・Encq(q)• Server side • Add LSH values all tuples • Filter to less similar vectors using LSH values.
- 28. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 28Experimental evaluations• Effectiveness of whitening transformation.• Recall of query results. • Our filter uses approximation of LSH • So that query results have errors.• Query processing time.
- 29. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 29Effectiveness of whitening transformation• Comparing • how many different LSH values exist. (size) • how many vectors has same LSH values. (min, max) (the number of tuples = 10000) with whitening transformation without whitening transformation
- 30. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 30Effectiveness of whitening transformation• Comparing • how many different LSH values exist. (size) • how many vectors has same LSH values. (min, max) (the number of tuples = 100000) with whitening transformation without whitening transformation LSH filter can distinguish There is only one LSH value, key vectors minutely. which means LSH filter doesn’t work.
- 31. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 31Effectiveness of whitening transformation• Comparing • how many different LSH values exist. (size) • how many vectors has same LSH values. (min, max) In all cases, min. = 1 (the number of tuples = 100000) with whitening transformation without whitening transformation bigger m provides well distinguishability. almost vectors has the same LSH value.
- 32. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 32Recall of query results• Recalls depend on the number of base vectors• Much base vectors achieves higher recalls. (the number of tuples = 10000)
- 33. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 33 Query processing time • Calculate query processing times on an IPP EVDB. • IPP EVDB is a encrypted vector database†. • We omit the detail of IPP EVDB and the x-axis of the following fig. time (sec) (log scale) (the number of tuples = 100000)†J. Kawamoto, M. Yoshikawa: Private Range Query by Perturbation and Matrix Based Encryption. InProc. of the 6th IEEE International Conf. on Digital Information Management, pp. 211–216. (2011)
- 34. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 34 Query processing time • Calculate query processing times on an IPP EVDB. • IPP EVDB is a encrypted vector database†. • We omit the detail of IPP EVDB and the x-axis of the following fig. We can reduce query processing time time (sec) (log scale) m = 128 (recall = 0.6) (the number of tuples = 100000)†J. Kawamoto, M. Yoshikawa: Private Range Query by Perturbation and Matrix Based Encryption. InProc. of the 6th IEEE International Conf. on Digital Information Management, pp. 211–216. (2011)
- 35. Dec. 4, 2012 A Locality Sensitive Hashing Filter for Encrypted Vector Databases 35Conclusion and future work• Introduce a filtering methodology for EVDBs based on • locality sensitive hashing (LSH) • whitening transformation• Our filter uses an approximation • Query results may have false negative errors • Applicable when users aren’t expecting perfect query results • We will modify our filter to increase the accuracy of query results Thank you!

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