Abstract
In this work we study relations between secret sharing and perfect zero knowledge in the non-interactive model. Both secret sharing schemes and non-interactive zero knowledge are important cryptographic primitives with several applications in the management of cryptographic keys, in multi-party secure protocols, and many other areas. Secret sharing schemes are very well-studied objects while non-interactive perfect zero-knowledge proofs seem to be very elusive. In fact, since the introduction of the non-interactive model for zero knowledge, the only perfect zero-knowledge proof known was for quadratic non residues.
In this work, we show that a large class of languages related to quadratic residuosity admits non-interactive perfect zero-knowledge proofs. More precisely, we give a protocol for proving non-interactively and in perfect zero knowledge the veridicity of any “threshold” statement where atoms are statements about the quadratic character of input elements. We show that our technique is very general and extend this result to any secret sharing scheme (of which threshold schemes are just an example).
Work supported by MURST and CNR
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References
M. Bellare and M. Yung, Certifying Cryptographic Tools: The case of Trapdoor Permutations, in CRYPTO 92.
M. Ben-Or, O. Goldreich, S. Goldwasser, J. Hastad, S. Micali, and P. Rogaway, Everything Provable is Provable in Zero Knowledge, in “Advances in Cryptology — CRYPTO 88”, vol. 403 of “Lecture Notes in Computer Science”, Springer Verlag, pp. 37–56.
G. R. Blackley, Safeguarding Chryptographic Keys, Proceedings AFIPS 1979 National Computer Conference, pp. 313–317, June 1979.
J. Boyar, K. Friedl, and C. Lund, Practical Zero-Knowledge Proofs: Giving Hints and Using Deficiencies, Journal of Cryptology, n. 4, pp. 185–206, 1991.
M. Blum, A. De Santis, S. Micali, and G. Persiano, Non-Interactive Zero-Knowledge, SIAM Journal of Computing, vol. 20, no. 6, Dec 1991, pp. 1084–1118.
M. Blum, P. Feldman, and S. Micali, Non-Interactive Zero-Knowledge and Applications, Proceedings of the 20th Annual ACM Symposium on Theory of Computing, 1988, pp. 103–112.
R. Boppana, J. Hastad, and S. Zachos, Does co-NP has Short Interactive Proofs?, Inf. Proc. Lett., vol. 25, May 1987, pp. 127–132.
A. De Santis, S. Micali, and G. Persiano, Non-Interactive Zero-Knowledge Proof-Systems, in “Advances in Cryptology — CRYPTO 87”, vol. 293 of “Lecture Notes in Computer Science”, Springer Verlag, pp. 52–72.
A. De Santis, G. Persiano, and M. Yung, Perfect Zero-Knowledge Proofs for Graph Isomorphism Languages. manuscript.
U. Feige, D. Lapidot, and A. Shamir, Multiple Non-Interactive Zero-Knowledge Proofs Based on a Single Random String, in Proceedings of 22nd Annual Symposium on the Theory of Computing, 1990, pp. 308–317.
L. Fortnow, The Complexity of Perfect Zero-Knowledge, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, pp. 204–209.
O. Goldreich and E. Kushilevitz, A Perfect Zero Knowledge Proof for a Decision Problem Equivalent to Discrete Logarithm, in “Advances in Cryptology-CRYPTO 88”, Ed. S. Goldwasser, vol. 403 of “Lecture Notes in Computer Science”, Springei-Verlag, pp. 57–70.
O. Goldreich, S. Micali, and A. Wigderson, Proofs that Yield Nothing but their Validity and a Methodology of Cryptographic Design, Proceedings of 27th Annual Symposium on Foundations of Computer Science, 1986, pp. 174–187.
O. Goldreich, S. Micali, and A. Wigderson, How to Play Any Mental Game, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pp. 218–229.
S. Goldwasser, S. Micali, and C. Rackoff, The Knowledge Complexity of Interactive Proof-Systems, SIAM Journal on Computing, vol. 18, n. 1, February 1989.
R. Impagliazzo and M. Yung, Direct Minimum Knowledge Computations “Advances in Cryptology — CRYPTO 87”, vol. 293 of “Lecture Notes in Computer Science”, Springer Verlag pp. 40–51.
I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, John Wiley and Sons, 1960, New York.
A. Shamir, How to share a secret, Communication of the ACM, vol. 22, n. 11, November 1979, pp. 612–613.
G. J. Simmons, An Introduction to Shared Secret and/or Shared Control Schemes and Their Application, Contemporary Cryptology, IEEE Press, pp. 441–497, 1991.
D. R. Stinson, An Explication of Secret Sharing Schemes, Design, Codes and Cryptography, Vol. 2, pp. 357–390, 1992.
M. Tompa and H. Woll, Random Self-Reducibility and Zero-Knowledge Interactive Proofs of Possession of Information, Proc. 28th Symposium on Foundations of Computer Science, 1987, pp. 472–482.
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© 1994 Springer-Verlag Berlin Heidelberg
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De Santis, A., Di Crescenzo, G., Persiano, G. (1994). Secret Sharing and Perfect Zero Knowledge. In: Stinson, D.R. (eds) Advances in Cryptology — CRYPTO’ 93. CRYPTO 1993. Lecture Notes in Computer Science, vol 773. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48329-2_7
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DOI: https://doi.org/10.1007/3-540-48329-2_7
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