Advertisement

Threshold Properties of Prime Power Subgroups with Application to Secure Integer Comparisons

  • Rhys Carlton
  • Aleksander Essex
  • Krzysztof Kapulkin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10808)

Abstract

We present a semantically secure somewhat homomorphic public-key cryptosystem working in sub-groups of \(\mathbb {Z}_{n}^{*}\) of prime power order. Our scheme introduces a novel threshold homomorphic property, which we use to build a two-party protocol for secure integer comparison. In contrast to related work which encrypts and acts on each bit of the input separately, our protocol compares multiple input bits simultaneously within a single ciphertext. Compared to the related protocol of Damgård et al. [9, 10] we present results showing this approach to be both several times faster in computation and lower in communication complexity.

Keywords

Public-key encryption Homomorphic encryption Homomorphic threshold Secure integer comparison 

References

  1. 1.
    Applebaum, B., Ishai, Y., Kushilevitz, E., Waters, B.: Encoding functions with constant online rate, or how to compress garbled circuit keys. SIAM J. Comput. 44(2), 433–466 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benaloh, J.: Dense probabilistic encryption. In: Workshop on Selected Areas of Cryptography (1994)Google Scholar
  3. 3.
    Benhamouda, F., Herranz, J., Joye, M., Libert, B.: Efficient cryptosystems from \(2^k\)-th power residue symbols. J. Cryptol. 30(2), 519–549 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blake, I.F., Kolesnikov, V.: Conditional encrypted mapping and comparing encrypted numbers. In: Di Crescenzo, G., Rubin, A. (eds.) FC 2006. LNCS, vol. 4107, pp. 206–220. Springer, Heidelberg (2006).  https://doi.org/10.1007/11889663_18 CrossRefGoogle Scholar
  5. 5.
    Boneh, D., Goh, E.-J., Nissim, K.: Evaluating 2-DNF formulas on ciphertexts. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 325–341. Springer, Heidelberg (2005).  https://doi.org/10.1007/978-3-540-30576-7_18 CrossRefGoogle Scholar
  6. 6.
    Bost, R., Popa, R.A., Tu, S., Goldwasser, S.: Machine learning classification over encrypted data. In: NDSS (2015)Google Scholar
  7. 7.
    Chou, T., Orlandi, C.: The simplest protocol for oblivious transfer. In: Lauter, K., Rodríguez-Henríquez, F. (eds.) LATINCRYPT 2015. LNCS, vol. 9230, pp. 40–58. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-22174-8_3 CrossRefGoogle Scholar
  8. 8.
    Coron, J.-S., Joux, A., Mandal, A., Naccache, D., Tibouchi, M.: Cryptanalysis of the RSA subgroup assumption from TCC 2005. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 147–155. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-19379-8_9 CrossRefGoogle Scholar
  9. 9.
    Damgård, I., Geisler, M., Krøigaard, M.: Efficient and secure comparison for on-line auctions. In: Pieprzyk, J., Ghodosi, H., Dawson, E. (eds.) ACISP 2007. LNCS, vol. 4586, pp. 416–430. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-73458-1_30 CrossRefGoogle Scholar
  10. 10.
    Damgård, I., Geisler, M., Krøigaard, M.: A correction to efficient and secure comparison for online auctions. Int. J. Appl. Cryptol. 1(4), 323–324 (2009)CrossRefzbMATHGoogle Scholar
  11. 11.
    Fischlin, M.: A cost-effective pay-per-multiplication comparison method for millionaires. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 457–471. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-45353-9_33 CrossRefGoogle Scholar
  12. 12.
    Garay, J., Schoenmakers, B., Villegas, J.: Practical and secure solutions for integer comparison. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 330–342. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-71677-8_22 CrossRefGoogle Scholar
  13. 13.
    Goldwasser, S., Micali, S.: Probabilistic encryption & how to play mental poker keeping secret all partial information. In: Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, STOC 1982, pp. 365–377 (1982)Google Scholar
  14. 14.
    Groth, J.: Cryptography in subgroups of \(z^*_n\). In: Proceedings of the Theory of Cryptography: Second Theory of Cryptography Conference, TCC 2005, Cambridge, MA, USA, 10–12 February 2005, pp. 50–65 (2005)Google Scholar
  15. 15.
    Joye, M., Libert, B.: Efficient cryptosystems from 2k-th power residue symbols. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 76–92. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38348-9_5 CrossRefGoogle Scholar
  16. 16.
    Kolesnikov, V., Sadeghi, A.-R., Schneider, T.: How to combine homomorphic encryption and garbled circuits. Sig. Process. Encrypted Domain 100, 2009 (2009)Google Scholar
  17. 17.
    Kolesnikov, V., Sadeghi, A.-R., Schneider, T.: Improved garbled circuit building blocks and applications to auctions and computing minima. In: Garay, J.A., Miyaji, A., Otsuka, A. (eds.) CANS 2009. LNCS, vol. 5888, pp. 1–20. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-10433-6_1 CrossRefGoogle Scholar
  18. 18.
    Lin, H.-Y., Tzeng, W.-G.: An efficient solution to the millionaires’ problem based on homomorphic encryption. In: Ioannidis, J., Keromytis, A., Yung, M. (eds.) ACNS 2005. LNCS, vol. 3531, pp. 456–466. Springer, Heidelberg (2005).  https://doi.org/10.1007/11496137_31 CrossRefGoogle Scholar
  19. 19.
    Lipmaa, H., Toft, T.: Secure equality and greater-than tests with sublinear online complexity. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7966, pp. 645–656. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39212-2_56 CrossRefGoogle Scholar
  20. 20.
    Mckee, J., Pinch, R.: Further attacks on server-aided RSA cryptosystems. (1998, Unpublished)Google Scholar
  21. 21.
    Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48910-X_16 Google Scholar
  22. 22.
    Schoenmakers, B., Tuyls, P.: Practical two-party computation based on the conditional gate. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 119–136. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-30539-2_10 CrossRefGoogle Scholar
  23. 23.
    Toft, T.: Sub-linear, secure comparison with two non-colluding parties. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 174–191. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-19379-8_11 CrossRefGoogle Scholar
  24. 24.
    Veugen, T.: Improving the DGK comparison protocol. In: 2012 IEEE International Workshop on Information Forensics and Security (WIFS), pp. 49–54. IEEE (2012)Google Scholar
  25. 25.
    Veugen, T.: Encrypted integer division and secure comparison. Int. J. Appl. Cryptol. 3(2), 166–180 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yao, A.C.-C.: How to generate and exchange secrets. In: 27th FOCS, pp. 162–167. IEEE Computer Society Press (1986)Google Scholar
  27. 27.
    Yu, C.-H., Yang, B.-Y.: Probabilistically correct secure arithmetic computation for modular conversion, zero test, comparison, MOD and exponentiation. In: Visconti, I., De Prisco, R. (eds.) SCN 2012. LNCS, vol. 7485, pp. 426–444. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32928-9_24 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Western UniversityLondonCanada

Personalised recommendations