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  1. Order Preserving Encryption for Numeric DataRakesh AgrawalJerry KiernanRamakrishnan SrikantYirong XuIBM Almaden Research Center

  2. Outline • Motivation and Introduction • OPES encryption • Modeling the distribution • Experimental evaluation

  3. Motivation • Encryption is rapidly becoming a requirement in a myriad of business settings (e.g., health care, financial, retail, government), driven by legislations (e.g. SB1386, HIPAA) • Encrypting databases unleashes a host of problems: • Performance slowdown • Incompatibility with standard database features • E.g. comparison predicates and the use of indexes • Changes to applications for encryption • Encryption functions now appear in queries

  4. Order Preserving Encryption Function E is an order preserving encryption function, and p1 and p2 are two plaintext values, and c1 = E(p1) c2 = E(p2) if (p1 < p2) then (c1 < c2)

  5. Threat Model • The storage system used by the DBMS is untrusted, i.e. vulnerable to compromise • The DBMS software is trusted • Ciphertext only attack • The adversary has access to all (but only) encrypted values • Guard against percentile exposure • An adversary should not be able to get even an estimate of true values

  6. Design Goals • Query results from OPES will be sound and complete • Comparison operations will be performed without decrypting the operands • Standard database indexes can be used over encrypted data • Tolerate updates

  7. Integration of Encryption and Query Processing Users have a plaintext view of an encrypted database We hereafter strictly focus on the OPES algorithms Comparison operators are directly applied over encrypted columns Queries Plaintext queries are translated into equivalent queries over encrypted data Select name from Emp where sal > 100000 Translation layer Select decrypt (“xsxx”) from “cwlxss” where “xescs” > OPESencrypt(100000) DBMS Tables are encrypted using standard as well as order preserving encryption Encrypted data And metadata

  8. Outline • Motivation and Introduction • OPES encryption • Modeling the distribution • Experimental evaluation

  9. Approach • Plaintext data has unknown distribution • User selects the target (ciphertext) distribution • Ciphertext values exhibit the target distribution

  10. Encrypted Original Target Effect of OPES Encryption on Plaintext Distributions Input: Gaussian, Target: Zipf Input: Uniform, Target: Zipf

  11. OPES Key Generation Sample of source values from the plaintext distribution Sample of target values from the ciphertext distribution OPES Key Generation OPES Key

  12. OPES Keys Target to uniform Target Source to uniform Uniform Uniform Source

  13. Two Step Encryption • Source (plaintext) to uniform • Uniform to target (ciphertext)

  14. OPES Encryption Step II Step I Target Uniform Uniform Source Step II Step I Encrypt Decrypt

  15. Outline • Motivation and Introduction • OPES encryption • Modeling the distribution • Experimental evaluation

  16. Modeling the Distribution • Histograms • Equi-depth, equi-width, wavelets • Number of buckets required unreasonably large • Over fitting the model • Parametric • Poor estimation for irregular distributions • Hybrid [Konig and Weikum 99] • Query result size estimation • Approach • Partition the data into buckets • Model the distribution within a bucket as a spline • Fixed number of buckets

  17. Our Approach • Hybrid [Konig and Weikum 99] • Partition the data into buckets • Model the distribution within each bucket as a linear spline • The number of buckets is not fixed • We use MDL to determine the number of bucket boundaries

  18. MDL • The best model for encoding data minimizes the sum of the cost of • Describing the model • Describing data in terms of the model

  19. Model Costs • Data Cost • Using a mapping M from [pl,ph) to [fl,fh), the cost of encoding pi is • C(pi)=log(fi-E(i)) • DC(pl,ph) = C(pl)+C(pl+1)+…+C(ph-1) • Incremental Model Cost • Fixed cost for each additional bucket • Boundary value • Boundary parameters • Slope • Scale factor

  20. Computing Boundaries • Growth phase • [pl,ph) with h-l-1 sorted points {pl+1,pl+2,…,ph-1} • Compute spline for [pl,ph) • Compute [fl,fh) using the spline • Find further split point ps with fs having the maximum deviation from the expected value • Prune phase • LB(pl,ph)=DC(pl,ph)-DC(pl,ps)-DC(ps,ph)-IMC • GB(pl,ph)=LB(pl,ph)+GB(pl,ps)+GB(ps,ph) • if (GB > 0), the split at ps is retained

  21. Scaling Number of values in a bucket may be disproportional to the size of the bucket Uniform x x x x x Source x x x x x b b+1 b-1

  22. Updates • The scale factor ensures that each distinct plaintext value maps to distinct ciphertext values • Encrypted values need not be recomputed unless the distribution of plaintext values changes

  23. Quality of Encryption • KS Statistical Test • Can we disprove, to a certain required level of significance, the null hypothesis that two data sets are drawn from the same distribution function? • If not, then the ciphertext distribution cannot be distinguished from the specified target distribution

  24. Duplicates • Assumptions • A large number of duplicates may leak information about the distribution of values • Alternatively, • Map duplicates to distinct values • if (f = M(p), f’ = M(p+1)) • [f,f’) = M(p) • Equality expressed as a range • Equi-joins can no longer be expressed • However, many numeric attributes (e.g., salary) may rarely be used in joins

  25. Outline • Motivation and Introduction • OPES encryption • Modeling the distribution • Experimental evaluation

  26. Experimental Evaluation • Percentile exposure • Updatability • Key size • Time overhead

  27. Datasets • Census • UCI KDD archive, PUMS census data (30,000) records • Gaussian • Zipf • Uniform Default Source: Gaussian Target: Zipf

  28. Percentile Exposure

  29. Time to the Build Model

  30. Insertion Overhead

  31. Cost of Additional Insertion

  32. Retrieval Overhead

  33. Retrieval Time

  34. Related Work • Polynomial functions • Ignores the distribution of plaintext/ciphertext values • Database as a service • Requires post processing of query results • Privacy homomorphisms • Comparison operations not investigated • Keyword searches on encrypted data • Designed for keyword retrieval • Range queries not supported • Smartcard-based schemes • Infeasible for large ranges • Order-preserving hashing • Protecting the hash values from cryptanalysis is not a concern, nor is deciphering plaintext values from hash values • Designed for static collections

  35. Closing Remarks • Ensuring safety without impeding the flow of information is a hard problem • Current choices • Plaintext database • Encrypted databases with loss of functionality or performance • Our approach focused on the trade-off between security and efficiency • We developed an algorithm which could easily be integrated with current systems

  36. Backup

  37. Encode Encode(p) = z(sp2+p) p c [0,ph), s = q/(2r), z > 0 distribution has density function qp + r p is the source (target) value s is the quadratic coefficient z is the scale factor

  38. Decode z ! z2 + 4zsf Decode (f) = 2zs fc [0, fh), s = q/(2r), z > 0 f is the flattened value s is the quadratic coefficient z is the scale factor

  39. Order Preserving Encryption Ciphertext is the index value • Effectively hides the distribution of plaintext values • The key size is proportional to the number of distinct attribute values • Any updates require recomputing the key and ciphertext values Compute distinct attribute values in ascending order

  40. Target Distribution Requirement • Why isn’t the source-to-uniform transformation sufficient for order preserving encryption? • It is, but • The target distribution may cause an adversary to make incorrect assumptions about the source distribution • The organization of the source distribution cannot be inferred from the target

  41. Quadratic Coefficient x x x x x x x x x x … v = b1 b2 i1 j1 i2 j2 j2 – i2 j1 – i1 - vj2 – vi2 vj1 – vi1 q q = s = vb1 – vb2 j1 – i1 2 vj1 – vi1

  42. Scale Factor Constraints for all p c [0,w) : M(p+1) – M(p) o 2 Ensures that there is a distinct mapped value for each input value wf = Kn The width of a bucket in the mapped space is a function of the number of elements n in the bucket K is the minimum width needed across buckets

  43. Scale Factor The scale factor will stretch short buckets to the width of the largest bucket, further increasing the dimension of a bucket by a factor of the number of elements in the bucket Kn z = sw2 + w K = max [x(swi2+w)], i = 1, …, m, 2, s o 0 2/(1 + s(2w – 1)), s < 0 x =

  44. Slope The values within a single bucket are unevenly distributed within the bucket b-1 b