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Privacy-Preserving Computations of Predictive Medical Models with Minimax Approximation and Non-Adjacent Form

  • Jung Hee Cheon
  • Jinhyuck Jeong
  • Joohee Lee
  • Keewoo LeeEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10323)

Abstract

In 2014, Bos et al. introduced a cloud service scenario to provide private predictive analyses on encrypted medical data, and gave a proof of concept implementation by utilizing homomorphic encryption (HE) scheme. In their implementation, they needed to approximate an analytic predictive model to a polynomial, using Taylor approximations. However, their approach could not reach a satisfactory compromise so that they just restricted the pool of data to guarantee suitable accuracy. In this paper, we suggest and implement a new efficient approach to provide the service using minimax approximation and Non-Adjacent Form (NAF) encoding. With our method, it is possible to remove the limitation of input range and reduce maximum errors, allowing faster analyses than the previous work. Moreover, we prove that the NAF encoding allows us to use more efficient parameters than the binary encoding used in the previous work or balaced base-B encoding. For comparison with the previous work, we present implementation results using HElib. Our implementation gives a prediction with 7-bit precision (of maximal error 0.0044) for having a heart attack, and makes the prediction in 0.5 s on a single laptop. We also implement the private healthcare service analyzing a Cox Proportional Hazard Model for the first time.

Keywords

Homomorphic encryption Healthcare Predictive analysis Minimax approximation Non-Adjacent Form Cloud service 

Notes

Acknowledgement

This work was supported by Institute for Information & communications Technology Promotion (IITP) grant funded by the Korea government (MSIP) (No. B0717-16-0098). The authors would like to thank Yong Soo Song, Kyoohyung Han, and the anonymous reviewers for valuable comments and suggestions.

References

  1. [Ach13]
    Achieser, N.I.: Theory of Approximation. Courier Corporation, Chelmsford (2013)zbMATHGoogle Scholar
  2. [Alb17]
    Albrecht, M.R.: On dual lattice attacks against small-secret LWE and parameter choices in HElib and SEAL. Cryptology ePrint Archive, Report 2017/047 (2017). http://eprint.iacr.org/2017/047
  3. [APS15]
    Albrecht, M.R., Player, R., Scott, S.: On the concrete hardness of learning with errors. J. Math. Cryptol. 9(3), 169–203 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [AYDA+14]
    Abadi, A., Yavari, P., Dehghani-Arani, M., Alavi-Majd, H., Ghasemi, E., Amanpour, F., Bajdik, C.: Cox models survival analysis based on breast cancer treatments. Iran. J. Cancer Prev. 7(3), 124 (2014)Google Scholar
  5. [BGV12]
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, pp. 309–325. ACM (2012)Google Scholar
  6. [BLLN13]
    Bos, J.W., Lauter, K., Loftus, J., Naehrig, M.: Improved security for a ring-based fully homomorphic encryption scheme. In: Stam, M. (ed.) IMACC 2013. LNCS, vol. 8308, pp. 45–64. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-45239-0_4 CrossRefGoogle Scholar
  7. [BLN14]
    Bos, J.W., Lauter, K., Naehrig, M.: Private predictive analysis on encrypted medical data. J. Biomed. Inform. 50, 234–243 (2014)CrossRefGoogle Scholar
  8. [Bra12]
    Brakerski, Z.: Fully homomorphic encryption without modulus switching from classical GapSVP. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 868–886. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32009-5_50 CrossRefGoogle Scholar
  9. [BRD+00]
    Biondo, S., Ramos, E., Deiros, M., Ragué, J.M., De Oca, J., Moreno, P., Farran, L., Jaurrieta, E.: Prognostic factors for mortality in left colonic peritonitis: a new scoring system. J. Am. Coll. Surg. 191(6), 635–642 (2000)CrossRefGoogle Scholar
  10. [BSJ+05]
    Boekholdt, S.M., Sacks, F.M., Jukema, J.W., Shepherd, J., Freeman, D.J., McMahon, A.D., Cambien, F., Nicaud, V., De Grooth, G.J., Talmud, P.J., et al.: Cholesteryl ester transfer protein TaqIB variant, high-density lipoprotein cholesterol levels, cardiovascular risk, and efficacy of pravastatin treatment individual patient meta-analysis of 13 677 subjects. Circulation 111(3), 278–287 (2005)CrossRefGoogle Scholar
  11. [BTC87]
    Boyd, C.R., Tolson, M.A., Copes, W.S.: Evaluating trauma care: the TRISS method. J. Trauma Acute Care Surg. 27(4), 370–378 (1987)CrossRefGoogle Scholar
  12. [BWA+05]
    Blankstein, R., Ward, R.P., Arnsdorf, M., Jones, B., Lou, Y.-B., Pine, M.: Female gender is an independent predictor of operative mortality after coronary artery bypass graft surgery contemporary analysis of 31 midwestern hospitals. Circulation 112(9 suppl), I–323 (2005)Google Scholar
  13. [CCK+13]
    Cheon, J.H., Coron, J.-S., Kim, J., Lee, M.S., Lepoint, T., Tibouchi, M., Yun, A.: Batch fully homomorphic encryption over the integers. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 315–335. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38348-9_20 CrossRefGoogle Scholar
  14. [CKLY15]
    Cheon, J.H., Kim, J., Lee, M.S., Yun, A.: CRT-based fully homomorphic encryption over the integers. Inf. Sci. 310, 149–162 (2015)MathSciNetCrossRefGoogle Scholar
  15. [CLT14]
    Coron, J.-S., Lepoint, T., Tibouchi, M.: Cryptanalysis of two candidate fixes of multilinear maps over the integers. IACR Cryptology ePrint Archive 2014, p. 975 (2014)Google Scholar
  16. [CMNT11]
    Coron, J.-S., Mandal, A., Naccache, D., Tibouchi, M.: Fully homomorphic encryption over the integers with shorter public keys. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 487–504. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22792-9_28 CrossRefGoogle Scholar
  17. [CNT12]
    Coron, J.-S., Naccache, D., Tibouchi, M.: Public key compression and modulus switching for fully homomorphic encryption over the integers. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 446–464. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-29011-4_27 CrossRefGoogle Scholar
  18. [CO84]
    Cox, D.R., Oakes, D.: Analysis of Survival Data, vol. 21. CRC Press, Boca Raton (1984)Google Scholar
  19. [Cox58]
    Cox, D.R.: The regression analysis of binary sequences. J. R. Stat. Soc. Ser. B (Methodol.) 20(2), 215–242 (1958). JSTOR. www.jstor.org/stable/2983890 MathSciNetzbMATHGoogle Scholar
  20. [Cox72]
    Cox, D.R.: Regression models and life-tables. J. R. Stat. Soc. Ser. B 34(2), 187–220 (1972)MathSciNetzbMATHGoogle Scholar
  21. [Cox92]
    Cox, D.R.: Regression models and life-tables. In: Kotz, S., Johnson, N.L. (eds.) Breakthroughs in Statistics. SSS, pp. 527–541. Springer, New York (1992).  https://doi.org/10.1007/978-1-4612-4380-9_37 CrossRefGoogle Scholar
  22. [CSVW]
    Costache, A., Smart, N.P., Vivek, S., Waller, A.: Fixed point arithmetic in SHE schemes. Technical report, Cryptology ePrint Archive, Report 2016/250 (2016). http://eprint.iacr.org/2016/250
  23. [DGBL+15]
    Dowlin, N., Gilad-Bachrach, R., Laine, K., Lauter, K., Naehrig, M., Wernsing, J.: Manual for using homomorphic encryption for bioinformatics. Microsoft Research (2015). http://research.microsoft.com/pubs/258435/ManualHEv2.pdf
  24. [DPMC13]
    D’Agostino, R.B., Pencina, M.J., Massaro, J.M., Coady, S.: Cardiovascular disease risk assessment: insights from Framingham. Glob. Heart 8(1), 11–23 (2013)CrossRefGoogle Scholar
  25. [DVP+08]
    D’Agostino, R.B., Vasan, R.S., Pencina, M.J., Wolf, P.A., Cobain, M., Massaro, J.M., Kannel, W.B.: General cardiovascular risk profile for use in primary care the Framingham heart study. Circulation 117(6), 743–753 (2008)CrossRefGoogle Scholar
  26. [FHS]
  27. [Fra65]
    Fraser, W.: A survey of methods of computing minimax and near-minimax polynomial approximations for functions of a single independent variable. J. ACM (JACM) 12(3), 295–314 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Gen09a]
    Gentry, C.: A fully homomorphic encryption scheme. PhD thesis, Stanford University (2009). https://crypto.stanford.edu/craig/
  29. [Gen09b]
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing-STOC 2009, pp. 169–169. ACM Press (2009)Google Scholar
  30. [GHS09]
    Gentry, C., Halevi, S., Smart, N.P.: Homomorphic evaluation of the AES circuit. Cryptology ePrint Archive, Report 2012/099 (2009). https://eprint.iacr.org/2012/099
  31. [GHS12]
    Gentry, C., Halevi, S., Smart, N.P.: Homomorphic evaluation of the AES circuit. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 850–867. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32009-5_49 CrossRefGoogle Scholar
  32. [HS13]
    Halevi, S., Shoup, V.: Design and implementation of a homomorphic-encryption library. IBM Research, Manuscript (2013)Google Scholar
  33. [HS14]
    Halevi, S., Shoup, V.: Algorithms in HElib. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 554–571. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-44371-2_31 CrossRefGoogle Scholar
  34. [HS15]
    Halevi, S., Shoup, V.: Bootstrapping for HElib. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 641–670. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46800-5_25 Google Scholar
  35. [KEAS00]
    Kologlu, M., Elker, D., Altun, H., Sayek, I.: Validation of MPI and PIA II in two different groups of patients with secondary peritonitis. Hepatogastroenterology 48(37), 147–151 (2000)Google Scholar
  36. [LRM]
  37. [MR08]
    Mattner, L., Roos, B.: Maximal probabilities of convolution powers of discrete uniform distributions. Stat. Probab. Lett. 78(17), 2992–2996 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  38. [NP51]
    Novodvorskii, E.P., Pinsker, I.S.: The process of equating maxima. Uspekhi Matematicheskikh Nauk 6(6), 174–181 (1951)MathSciNetGoogle Scholar
  39. [Rem34]
    Remez, E.Y.: Sur le calcul effectif des polynomes d’approximation de tschebyscheff. CR Acad. Sci. Paris 199, 337–340 (1934)Google Scholar
  40. [Riv90]
    Rivlin, T.-J.: Chebyshev Polynomials. Wiley, New York (1990)zbMATHGoogle Scholar
  41. [SV14]
    Smart, N.P., Vercauteren, F.: Fully homomorphic SIMD operations. Des. Codes Crypt. 71(1), 57–81 (2014)CrossRefzbMATHGoogle Scholar
  42. [TCK67]
    Truett, J., Cornfield, J., Kannel, W.: A multivariate analysis of the risk of coronary heart disease in Framingham. J. Chronic Dis. 20(7), 511–524 (1967)CrossRefGoogle Scholar
  43. [TH02]
    Tabaei, B.P., Herman, W.H.: A multivariate logistic regression equation to screen for diabetes development and validation. Diab. Care 25(11), 1999–2003 (2002)CrossRefGoogle Scholar
  44. [TS14]
    Tolosie, K., Sharma, M.K.: Application of Cox proportional hazards model in case of tuberculosis patients in selected Addis Ababa health centres, Ethiopia. Tuberc. Res. Treat. 2014, 11 p. (2014).  https://doi.org/10.1155/2014/536976. Article ID 536976
  45. [VDGHV10]
    van Dijk, M., Gentry, C., Halevi, S., Vaikuntanathan, V.: Fully homomorphic encryption over the integers. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 24–43. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13190-5_2 CrossRefGoogle Scholar
  46. [Vei60]
    Veidinger, L.: On the numerical determination of the best approximations in the Chebyshev sense. Numer. Math. 2(1), 99–105 (1960)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© International Financial Cryptography Association 2017

Authors and Affiliations

  • Jung Hee Cheon
    • 1
  • Jinhyuck Jeong
    • 1
  • Joohee Lee
    • 1
  • Keewoo Lee
    • 1
    Email author
  1. 1.Seoul National University (SNU)SeoulRepublic of Korea

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