Secure Distributed Computation of the Square Root and Applications

  • Manuel Liedel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7232)


The square root is an important mathematical primitive whose secure, efficient, distributed computation has so far not been possible. We present a solution to this problem based on Goldschmidt’s algorithm. The starting point is computed by linear approximation of the normalized input using carefully chosen coefficients. The whole algorithm is presented in the fixed-point arithmetic framework of Catrina/Saxena for secure computation. Experimental results demonstrate the feasibility of our algorithm and we show applicability by using our protocol as a building block for a secure QR-Decomposition of a rational-valued matrix.


Square Root Fixed-Point Arithmetic Secure Computation QR-Decomposition 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beaver, D., Micali, S., Rogaway, P.: The round complexity of secure protocols. In: Proceedings of the Twenty-second Annual ACM Symposium on Theory of Computing, STOC 1990, pp. 503–513. ACM, New York (1990)CrossRefGoogle Scholar
  2. 2.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC 1988, pp. 1–10. ACM, New York (1988)CrossRefGoogle Scholar
  3. 3.
    Catrina, O., de Hoogh, S.: Improved Primitives for Secure Multiparty Integer Computation. In: Garay, J.A., De Prisco, R. (eds.) SCN 2010. LNCS, vol. 6280, pp. 182–199. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Catrina, O., de Hoogh, S.: Secure Multiparty Linear Programming Using Fixed-Point Arithmetic. In: Gritzalis, D., Preneel, B., Theoharidou, M. (eds.) ESORICS 2010. LNCS, vol. 6345, pp. 134–150. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Catrina, O., Saxena, A.: Secure Computation with Fixed-Point Numbers. In: Sion, R. (ed.) FC 2010. LNCS, vol. 6052, pp. 35–50. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC 1988, pp. 11–19. ACM, New York (1988)CrossRefGoogle Scholar
  7. 7.
    Cramer, R., Damgård, I.: Secure Distributed Linear Algebra in a Constant Number of Rounds. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 119–136. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Cramer, R., Damgård, I., Ishai, Y.: Share Conversion, Pseudorandom Secret-Sharing and Applications to Secure Computation. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 342–362. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game. In: Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, STOC 1987, pp. 218–229. ACM, New York (1987)CrossRefGoogle Scholar
  10. 10.
    Goldschmidt, R.E.: Applications of division by convergence. Master’s thesis, M.I.T. (1964)Google Scholar
  11. 11.
    Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press (1996)Google Scholar
  12. 12.
    Ito, M., Takagi, N., Yajima, S.: Efficient initial approximation for multiplicative division and square root by a multiplication with operand modification. IEEE Transactions on Computers 46, 495–498 (1997)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Markstein, P.: Software division and square root using goldschmidt’s algorithms. In: 6th Conference on Real Numbers and Computers, pp. 146–157 (2004)Google Scholar
  14. 14.
    Shamir, A.: How to share a secret. Commun. ACM 22, 612–613 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Stoer, J., Bulirsch, R.: Introduction to numerical analysis. Texts in applied mathematics. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  16. 16.
    Yao, A.C.: Protocols for secure computations. In: Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, SFCS 1982, pp. 160–164. IEEE Computer Society, Washington, DC, USA (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Manuel Liedel
    • 1
  1. 1.Fakultät für WirtschaftswissenschaftenUniversity of RegensburgGermany

Personalised recommendations