Abstract
We present two universally composable and practical protocols by which a dealer can, verifiably and non-interactively, secret-share an integer among a set of players. Moreover, at small extra cost and using a distributed verifier proof, it can be shown in zero-knowledge that three shared integers a,b,c satisfy ab = c. This implies by known reductions non-interactive zero-knowledge proofs that a shared integer is in a given interval, or that one secret integer is larger than another. Such primitives are useful, e.g., for supplying inputs to a multiparty computation protocol, such as an auction or an election. The protocols use various set-up assumptions, but do not require the random oracle model.
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
Abe, M., Cramer, R.J.F., Fehr, S.: Non-interactive Distributed-Verifier Proofs and Proving Relations among Commitments. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 206–223. Springer, Heidelberg (2002)
Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for Non-Cryptographic Fault-Tolerant Distributed Computation. In: Proc. ACM STOC ’88, pp. 1–10 (1988)
Boneh, D., Franklin, M.K.: Identity-Based Encryption from the Weil Pairing. SIAM J. Comput. 32(3), 586–615 (2003)
Bogetoft, P., Damgård, I.B., Jakobsen, T., Nielsen, K., Pagter, J., Toft, T.: A Practical Implementation of Secure Auctions Based on Multiparty Integer Computation. In: Di Crescenzo, G., Rubin, A. (eds.) FC 2006. LNCS, vol. 4107, pp. 142–147. Springer, Heidelberg (2006)
Boudot, F.: Efficient Proofs that a Committed Number Lies in an Interval. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 431–444. Springer, Heidelberg (2000)
Canetti, R.: Universally Composable Security: A New Paradigm for Cryptographic Protocols. In: Proc. of FOCS 2001, pp. 136–145 (2001), See also updated version on the Eprint archive, http://www.iacr.org
Canetti, R., Halevi, S., Katz, J.: Chosen-Ciphertext Security from Identity-Based Encryption. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 207–222. Springer, Heidelberg (2004)
Chaum, D., Crépeau, C., Damgård, I.: Multi-Party Unconditionally Secure Protocols. In: Proc. of ACM STOC ’88, pp. 11–19 (1988)
Chor, B., Goldwasser, S., Micali, S., Awerbuch, B.: Verifiable Secret Sharing and Achieving Simultaneity in the Presence of Faults (extended abstract). In: 26th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos (1985)
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)
Cramer, R.J.F., Fehr, S., Stam, M.: Black-Box Secret Sharing from Primitive Sets in Algebraic Number Fields. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 344–360. Springer, Heidelberg (2005)
Damgård, I., Fitzi, M., Kiltz, E., Nielsen, J.B., Toft, T.: Unconditionally Secure Constant-Rounds Multi-party Computation for Equality, Comparison, Bits and Exponentiation. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 285–304. Springer, Heidelberg (2006)
Damgård, I.B., Thorbek, R.: Linear Integer Secret Sharing and Distributed Exponentiation. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 75–90. Springer, Heidelberg (2006)
Damgård, I., Thorbek, R.: Non-Interactive Proofs for Integer Multiplication (full version). The Eprint archive (eprint.iacr.org/2007/086), http://www.iacr.org
Fujisaki, E., Okamoto, T.: A Practical and Provably Secure Scheme for Publicly Verifiable Secret Sharing and Its Applications. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 32–46. Springer, Heidelberg (1998)
Fujisaki, E., Okamoto, T.: Statistical Zero Knowledge Protocols to Prove Modular Polynomial Relations. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 16–30. Springer, Heidelberg (1997)
Gennaro, R., Rabin, M., Rabin, T.: Simplified VSS and Fast-Track Multiparty Computations with Applications to Threshold Cryptography. In: Proc. of ACM PODC’98 (1998)
Goldreich, O., Micali, S., Wigderson, A.: How to Play Any Mental Game or a Completeness Theorem for Protocols with Honest Majority. In: Proc. of ACM STOC ’87, pp. 218–229 (1987)
Hirt, M., Maurer, U.: Player Simulation and General Adversary Structures in Perfect Multiparty Computation. Journal of Cryptology: the journal of the International Association for Cryptologic Research 13, 31–60 (2000)
Ito, M., Saito, A., Nishizeki, T.: Secret sharing schemes realizing general access structures. In: Proc. IEEE Global Telecommunication Conf., Globecom 87, pp. 99–102 (1987)
Karchmer, M., Wigderson, A.: On Span Programs. In: Proc. of 8th IEEE Structure in Complexity Theory, pp. 102–111 (1993)
Pedersen, T.P.: Non-interactive and Information-Theoretic Secure Verifiable Secret Sharing. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 129–140. Springer, Heidelberg (1992)
Shamir, A.: How to share a secret. Communication of the Association for Computing Machinery 22(11) (1979)
Waters, B.: Efficient Identity-Based Encryption Without Random Oracles. In: Cramer, R.J.F. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 114–127. Springer, Heidelberg (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Damgård, I., Thorbek, R. (2007). Non-interactive Proofs for Integer Multiplication. In: Naor, M. (eds) Advances in Cryptology - EUROCRYPT 2007. EUROCRYPT 2007. Lecture Notes in Computer Science, vol 4515. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72540-4_24
Download citation
DOI: https://doi.org/10.1007/978-3-540-72540-4_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72539-8
Online ISBN: 978-3-540-72540-4
eBook Packages: Computer ScienceComputer Science (R0)