Some Applications of Polynomials for the Design of Cryptographic Protocols

  • Eyal Kushilevitz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2576)


This paper surveys some recent work on applications of polynomials (over finite fields) to the design of various cryptographic protocols. It is based on a talk given at the 3rd Conference on Security in Communication Networks, 2002.


Cryptographic Protocol Polynomial Representation Oblivious Transfer Arithmetic Circuit Private Information Retrieval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Ambainis. Upper bound on the communication complexity of private information retrieval. In 24th ICALP, LNCS 1256, pp. 401–407, 1997.Google Scholar
  2. 2.
    J. Bar-Ilan and D. Beaver. Non-cryptographic fault-tolerant computing in a constant number of rounds. In Proc. 8th ACM PODC, pages 201–209. ACM, 1989.Google Scholar
  3. 3.
    D. Beaver, S. Micali, and P. Rogaway. The round complexity of secure protocols (extended abstract). In Proc. 22nd STOC, pages 503–513. ACM, 1990.Google Scholar
  4. 4.
    A. Beimel and Y. Ishai. Information-theoretic private information retrieval: A unified construction. In 28th ICALP, vol. 2076 of LNCS, pp. 912–926, 2001.Google Scholar
  5. 5.
    A. Beimel, Y. Ishai, E. Kushilevitz, and J. F. Raymond, “Breaking the O(n 1/(2 k -1) Barrier for Information-Theoretic Private Information Retrieval”, In Proc. of FOCS, 2002.Google Scholar
  6. 6.
    A. Beimel, Y. Ishai, and T. Malkin. Reducing the servers’ computation in private information retrieval: PIR with preprocessing. In CRYPTO 2000, vol. 1880 ofLNCS, pp. 56–74, 2000.Google Scholar
  7. 7.
    M. Ben-Or, S. Goldwasser, and A. Wigderson. Completeness Theorems for Noncryptographic Fault-Tolerant Distributed Computations. Proc. 20th STOC88, pp. 1–10.Google Scholar
  8. 8.
    C. Blundo, A. De Santis, A. Herzberg, S. Kutten, U. Vaccaro, M. Yung Perfectly-Secure Key Distribution for Dynamic Conferences. Proc. CRYPTO 1992, 471–486Google Scholar
  9. 9.
    R. Canetti, Y. Ishai, R. Kumar, M. K. Reiter, R. Rubinfeld, and R. N. Wright. Selective private function evaluation with applications to private statistics. In 20th PODC, pp. 293–304, 2001.Google Scholar
  10. 10.
    D. Chaum, C. Crepeau, and I. Damgard. Multiparty Unconditionally Secure Protocols. In Proc. 20th STOC88, pages 11–19.Google Scholar
  11. 11.
    B. Chor and N. Gilboa. Computationally private information retrieval. In 29th STOC, pp. 304–313, 1997.Google Scholar
  12. 12.
    B. Chor, O. Goldreich, E. Kushilevitz, and M. Sudan. Private information retrieval. J. of the ACM, 45:965–981, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    G. Di-Crescenzo, Y. Ishai, and R. Ostrovsky. Universal service-providers for private information retrieval. J. of Cryptology, 14(1):37–74, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    U. Feige, J. Kilian, and M. Naor. A minimal model for secure computation (extended abstract). In Proc. 26th STOC, pages 554–563. ACM, 1994.Google Scholar
  15. 15.
    J. Feigenbaum, Y. Ishai, T. Malkin, K. Nissim, M. J. Strauss, and R. N. Wright. Secure multiparty computation of approximations. In 28th ICALP, vol. 2076 of LNCS, pp. 927–938, 2001.Google Scholar
  16. 16.
    R. Gennaro, Y. Ishai, E. Kushilevitz, and T. Rabin. On 2-Round Secure Multiparty Computation. In Proc. of Crypto, 2002.Google Scholar
  17. 17.
    R. Gennaro, M. O. Rabin, and T. Rabin. Fact-track multiparty computations with applications to threshold cryptography. In Proc. of 17th PODC, pages 101–111, 1998.Google Scholar
  18. 18.
    Y. Gertner, S. Goldwasser, and T. Malkin. A random server model for private information retrieval. In RANDOM’ 98, vol. 1518 of LNCS, pp. 200–217, 1998.Google Scholar
  19. 19.
    Y. Gertner, Y. Ishai, E. Kushilevitz, and T. Malkin. Protecting data privacy in private information retrieval schemes. JCSS, 60(3):592–629, 2000.zbMATHMathSciNetGoogle Scholar
  20. 20.
    S.W. Golub, “Shift Register Sequences”, 1967.Google Scholar
  21. 21.
    O. Goldreich, S. Micali, and A. Wigderson. How to Play Any Mental Game. In Proc. 19th STOC, pages 218–229. ACM, 1987.Google Scholar
  22. 22.
    Y. Ishai and E. Kushilevitz. Private simultaneous messages protocols with applications. In ISTCS97, pages 174–184, 1997.Google Scholar
  23. 23.
    Y. Ishai and E. Kushilevitz. Improved upper bounds on information theoretic private information retrieval. 31st STOC, pp. 79–88, 1999.Google Scholar
  24. 24.
    Y. Ishai and E. Kushilevitz. Randomizing polynomials: A new representation with applications to round-efficient secure computation. In Proc. 41st FOCS, 2000.Google Scholar
  25. 25.
    Y. Ishai and E. Kushilevitz. Perfect Constant-Round Secure Computation via Perfect Randomizing Polynomials. In Proc. ICALP’ 02, pp. 244–256.Google Scholar
  26. 26.
    M. Ito, A. Saito, and T. Nishizeki. Secret sharing schemes realizing general access structures. In Proc. IEEE Global Telecommunication Conf., Globecom 87, pages 99–102, 1987.Google Scholar
  27. 27.
    T. Itoh. Efficient private information retrieval. IEICE Trans. Fund. of Electronics, Commun. and Comp. Sci., E82-A(1):11–20, 1999.Google Scholar
  28. 28.
    J. Katz and L. Trevisan. On the efficiency of local decoding procedures for errorcorrecting codes. In 32nd STOC, pp. 80–86, 2000.Google Scholar
  29. 29.
    A. Kiayias and M. Yung. Secure games with polynomial expressions. In 28th ICALP, vol. 2076 of LNCS, pp. 939–950, 2001.Google Scholar
  30. 30.
    A. Kiayias and M. Yung. Cryptographic Hardness Based on the Decoding of Reed-Solomon Codes. In 29th ICALP, pp. 232–243, 2002.Google Scholar
  31. 31.
    E. Kushilevitz and R. Ostrovsky. Replication is not needed: Single database, computationally-private information retrieval. In 38th FOCS, pp. 364–373, 1997.Google Scholar
  32. 32.
    E. Mann. Private access to distributed information. Master’s thesis, Technion, Haifa, 1998.Google Scholar
  33. 33.
    F.J. Macwilliams and N.J.A. Sloane, “The Theory of Error Correcting Codes”, 1977.Google Scholar
  34. 34.
    M. Naor and K. Nissim. Communication preserving protocols for secure function evaluation. In 33th STOC, 2001.Google Scholar
  35. 35.
    M. Naor and B. Pinkas. Oblivious transfer and polynomial evaluation. In 31st STOC, pp. 245–254, 1999.Google Scholar
  36. 36.
    R. Ostrovsky and V. Shoup. Private information storage. In 29th STOC, pp. 294–303, 1997.Google Scholar
  37. 37.
    G.B. Purdy, “A high Security Log-In Procedure”, CACM 17(8), pp. 442–445, 1974.MathSciNetGoogle Scholar
  38. 38.
    M. O. Rabin. Efficient dispersal of information for security, load balancing, and fault tolerance. J. ACM 38, 335–348 (1989).CrossRefMathSciNetGoogle Scholar
  39. 39.
    T. Rabin and M. Ben-Or. Verifiable Secret Sharing and Multiparty Protocols with Honest Majority. In Proc. 21st STOC, pages 73–85. ACM, 1989.Google Scholar
  40. 40.
    A. Shamir. How to share a secret. Commun. ACM, 22(6):612–613, June 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    A. C-C. Yao. How to Generate and Exchange Secrets. In Proc. 27th FOCS, pages 162–167. IEEE, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Eyal Kushilevitz
    • 1
  1. 1.Computer Science DepartmentTechnionIsrael

Personalised recommendations