Secret Sharing Schemes with Algebraic Properties and Applications

  • Ignacio Cascudo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)


Secret sharing concerns the distribution of some secret information among a number of parties and is among the most well known tools in cryptography. Secret sharing schemes with certain additional algebraic properties, known as linearity and multiplicativity, have important applications in the area of secure multiparty computation and other areas such as zero knowledge proofs. Secret sharing also has a strong relationship with coding theory and motivates new problems in that field. I will survey several of the recent results in the area and some of their applications.


Secret Sharing Linear Code Secret Share Scheme Secure Multiparty Computation Knowledge Proof 
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.
    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 (1988)Google Scholar
  2. 2.
    Cascudo, I., Chen, H., Cramer, R., Xing, C.: Asymptotically good ideal linear secret sharing with strong multiplication over any fixed finite field. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 466–486. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Cascudo, I., Cramer, R., Mirandola, D., Zemor, G.: Squares of random linear codes. IEEE Trans. Inf. Theor. 61(3), 1159–1173 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cascudo, I., Cramer, R., Xing, C.: The torsion-limit for algebraic function fields and its application to arithmetic secret sharing. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 685–705. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Cascudo, I., Cramer, R., Xing, C.: The arithmetic codex. In: Proceedings of IEEE Information Theory Workshop (ITW 2012), pp. 75–79 (2012)Google Scholar
  6. 6.
    Cascudo, I., Cramer, R., Xing, C.: Bounds on the threshold gap in secret sharing and its applications. IEEE Trans. Inf. Theor. 59(9), 5600–5612 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cascudo, I., Cramer, R., Xing, C.: Torsion limits and Riemann-Roch systems for function fields and applications. IEEE Trans. Inf. Theor. 60(7), 3871–3888 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cascudo, I., Damgård, I., David, B., Giacomelli, I., Nielsen, J.B., Trifiletti, R.: Additively homomorphic UC commitments with optimal amortized overhead. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 495–515. Springer, Heidelberg (2015)Google Scholar
  9. 9.
    Chaum, D., Crépeau, C., Damgård, I.: Multi-party unconditionally secure protocols. In: Proceedings of STOC 1988, pp. 11–19. ACM Press (1988)Google Scholar
  10. 10.
    Chen, H., Cramer, R.: 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
  11. 11.
    Chen, H., Cramer, R., Goldwasser, S., de Haan, R., Vaikuntanathan, V.: Secure computation from random error correcting codes. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 291–310. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Cramer, R., 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
  13. 13.
    Cramer, R., Damgård, I., Nielsen, J.B.: Secure Multiparty Computation and Secret Sharing - An Information Theoretic Approach. Cambridge University PressGoogle Scholar
  14. 14.
    Cramer, R., Damgård, I., Pastro, V.: On the amortized complexity of zero knowledge protocols for multiplicative relations. In: Smith, A. (ed.) ICITS 2012. LNCS, vol. 7412, pp. 62–79. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Cramer, R., Xing, C., Yuan, C.: On Multi-point Local Decoding of Reed-Muller Codes. Manuscript (2016).
  16. 16.
    Damgård, I., David, B., Giacomelli, I., Nielsen, J.B.: Compact VSS and efficient homomorphic UC commitments. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014, Part II. LNCS, vol. 8874, pp. 213–232. Springer, Heidelberg (2014)Google Scholar
  17. 17.
    Damgård, I., Zakarias, S.: Constant-overhead secure computation of Boolean circuits using preprocessing. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 621–641. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  18. 18.
    Frederiksen, T.K., Jakobsen, T.P., Nielsen, J.B., Trifiletti, R.: On the complexity of additively homomorphic UC commitments. In: Kushilevitz, E., et al. (eds.) TCC 2016-A. LNCS, vol. 9562, pp. 542–565. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49096-9_23 CrossRefGoogle Scholar
  19. 19.
    Garcia, A., Stichtenoth, H.: A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vlǎduţ bound. Inventiones Math. 121, 211–222 (1995)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Harnik, D., Ishai, Y., Kushilevitz, E., Nielsen, J.B.: OT-combiners via secure computation. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 393–411. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  21. 21.
    Ishai, Y., Kushilevitz, E., Ostrovsky, R., Prabhakaran, M., Sahai, A., Wullschleger, J.: Constant-rate oblivious transfer from noisy channels. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 667–684. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  22. 22.
    Ishai, Y., Kushilevitz, E., Ostrovsky, R., Sahai, A.: Zero-knowledge from secure multiparty computation. In: Proceedings of 39th STOC, San Diego, CA, USA, pp. 21–30 (2007)Google Scholar
  23. 23.
    Ishai, Y., Kushilevitz, E., Ostrovsky, R., Sahai, A.: Extracting correlations. In: Proceedings of 50th IEEE FOCS, pp. 261–270 (2009)Google Scholar
  24. 24.
    Ishai, Y., Prabhakaran, M., Sahai, A.: Founding cryptography on oblivious transfer – efficiently. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 572–591. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  25. 25.
    Massey., J.: Minimal codewords and secret sharing. In: Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory (1993)Google Scholar
  26. 26.
    Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsAalborg UniversityAalborgDenmark

Personalised recommendations