Secure Computation from Random Error Correcting Codes

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4515)


Secure computation consists of protocols for secure arithmetic: secret values are added and multiplied securely by networked processors. The striking feature of secure computation is that security is maintained even in the presence of an adversary who corrupts a quorum of the processors and who exercises full, malicious control over them. One of the fundamental primitives at the heart of secure computation is secret-sharing. Typically, the required secret-sharing techniques build on Shamir’s scheme, which can be viewed as a cryptographic twist on the Reed-Solomon error correcting code. In this work we further the connections between secure computation and error correcting codes. We demonstrate that threshold secure computation in the secure channels model can be based on arbitrary codes. For a network of size n, we then show a reduction in communication for secure computation amounting to a multiplicative logarithmic factor (in n) compared to classical methods for small, e.g., constant size fields, while tolerating \(t < ({1 \over 2} - {\epsilon}) {n} \) players to be corrupted, where ε> 0 can be arbitrarily small. For large networks this implies considerable savings in communication. Our results hold in the broadcast/negligible error model of Rabin and Ben-Or, and complement results from CRYPTO 2006 for the zero-error model of Ben-Or, Goldwasser and Wigderson (BGW). Our general theory can be extended so as to encompass those results from CRYPTO 2006 as well. We also present a new method for constructing high information rate ramp schemes based on arbitrary codes, and in particular we give a new construction based on algebraic geometry codes.


Linear Code Secure Computation Secret Sharing Scheme Arbitrary Code Algebraic Geometry Code 
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.


  1. 1.
    Beaver, D.: Efficient multiparty protocols using circuit randomization. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 420–432. Springer, Heidelberg (1992)Google Scholar
  2. 2.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation. In: Proceedings of STOC 1988, pp. 1–10. ACM Press, New York (1988)Google Scholar
  3. 3.
    Blakley, G.R.: Safeguarding cryptographic keys. In: Proceedings of National Computer Conference ’79. AFIPS Proceedings, vol. 48, pp. 313–317 (1979)Google Scholar
  4. 4.
    Blakley, G.R., Meadows, C.: Security of ramp schemes. In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 242–268. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  5. 5.
    Chaum, D., Crépeau, C., Damgaard, I.: Multi-party unconditionally secure protocols. In: Proceedings of STOC 1988, pp. 11–19. ACM Press, New York (1988)Google Scholar
  6. 6.
    Chen, H., Cramer, R.J.F.: Algebraic Geometric Secret Sharing Schemes and Secure Multi-Party Computations over Small Fields. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 521–536. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Cramer, R.J.F., Damgård, I.B., de Haan, R.: Atomic Secure Multi-party Multiplication with Low Communication. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 329–346. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Cramer, R., Damgaard, I., Dziembowski, S.: On the complexity of verifiable secret sharing and multi-party computation. In: Proceedings of STOC 2000, pp. 325–334. ACM Press, New York (2000)Google Scholar
  9. 9.
    Cramer, R.J.F., Damgård, I.B., Dziembowski, S., Hirt, M., Rabin, T.: Efficient Multiparty Computations Secure against an Adaptive Adversary. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 311–326. Springer, Heidelberg (1999)Google Scholar
  10. 10.
    Cramer, R.J.F., Damgård, I.B., Maurer, U.M.: General secure multi-party computation from any linear secret-sharing scheme. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 316–334. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. 11.
    Cramer, R.J.F., Daza, V., Gracia, I., Urroz, J.J., Leander, G., Martí-Farré, J., Padró, C.: On codes, matroids and secure multi-party computation from linear secret sharing schemes. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 327–343. Springer, Heidelberg (2005)Google Scholar
  12. 12.
    Cramer, R.J.F., Damgård, I.B., Fehr, S.: On the Cost of Reconstructing a Secret, or VSS with Optimal Reconstruction Phase. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 503–523. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Franklin, M., Yung, M.: Communication complexity of secure computation. In: Proceedings of STOC 1992, pp. 699–710. ACM Press, New York (1992)Google Scholar
  14. 14.
    Gaborit, P., Otmani, A.: Experimental constructions of self-dual codes. Manuscript (2002), Available from
  15. 15.
    García, A., Stichtenoth, H.: On the asymptotic behavior of some towers of function fields over finite fields. J. Number Theory 61, 248–273 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Goldreich, O., Micali, S., Wigderson, A.: How to Play Any Mental Game. In: Proceedings of STOC 1987, pp. 218–229. ACM Press, New York (1987)Google Scholar
  17. 17.
    Goppa, V.D.: Codes on algebraic curves. Soviet Math. Dokl. 24, 170–172 (1981)zbMATHGoogle Scholar
  18. 18.
    Karchmer, M., Wigderson, A.: On span programs. In: Proceedings of the Eight Annual Structure in Complexity Theory Conference, pp. 102–111. IEEE Computer Society Press, Los Alamitos (1993)CrossRefGoogle Scholar
  19. 19.
    Kurosawa, K., Okada, K., Sakano, K., Ogata, W., Tsujii, S.: Nonperfect Secret Sharing Schemes and Matroids. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 126–141. Springer, Heidelberg (1994)Google Scholar
  20. 20.
    Lang, S.: Algebra. Addison-Wesley, Reading (1997)Google Scholar
  21. 21.
    MacWilliams, F.J., Sloane, N.J.A., Thompson, J.G.: Good self-dual codes exist. Discrete Math. 3, 153–162 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Massey, J.L.: Minimal codewords and secret sharing. In: Proceedings of the 6-th Joint Swedish-Russian Workshop on Information Theory, Molle, Sweden, August 1993, pp. 269–279 (1993)Google Scholar
  23. 23.
    Massey, J.L.: Some applications of coding theory in cryptography. In: Codes and Ciphers: Cryptography and Coding IV, pp. 33–47 (1995)Google Scholar
  24. 24.
    Ogata, W., Kurosawa, K.: Some Basic Properties of General Nonperfect Secret Sharing Schemes. J. UCS 4(8), 690–704 (1998)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Ozarow, L.H., Wyner, A.D.: Wire-tap-channel II. AT&T Bell Labs Tech. J. 63, 2135–2157 (1984)zbMATHGoogle Scholar
  26. 26.
    Rabin, T., Ben-Or, M.: Verifiable secret sharing and multiparty protocols with honest majority. In: Proceedings of ACM STOC 1989, pp. 73–85 (1989)Google Scholar
  27. 27.
    Rains, E.M., Sloane, N.J.A.: Self-Dual Codes. A long survey article written for the Handbook of Coding Theory (1998), Available from
  28. 28.
    Shamir, A.: How to share a secret. Communications of the ACM 22(11), 612–613 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Thompson, J.G.: Weighted averages associated to some codes. Scripta Math. 29, 449–452 (1973)zbMATHMathSciNetGoogle Scholar
  30. 30.
    van Lint, J.H.: Introduction to Coding Theory. Graduate Texts in Mathematics. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  31. 31.
    Wei, V.K.: Generalized Hamming Weights for Linear Codes. IEEE Transactions on Information Theory 37(5), 1412–1418 (1991)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  1. 1.Department of Computing and Information Technology, School of Information Science and EngineeringFudan UniversityShanghaiChina
  2. 2.Mathematical InstituteLeiden UniversityThe Netherlands
  3. 3.Weizmann Institute of ScienceRehovotIsrael
  4. 4.CWIAmsterdamThe Netherlands
  5. 5.MITCambridgeUSA

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