Abstract
A secret sharing scheme (SSS) is homomorphic, if the products of shares of secrets are shares of the product of secrets. For a finite abelian group G, an access structure \({\mathcal A}\) is G-ideal homomorphic, if there exists an ideal homomorphic SSS realizing the access structure \({\mathcal A}\) over the secret domain G. An access structure \({\mathcal A}\) is universally ideal homomorphic, if for any non-trivial finite abelian group G, \({\mathcal A}\) is G-ideal homomorphic.
A black-box SSS is a special type of homomorphic SSS, which works over any non-trivial finite abelian group. In such a scheme, participants only have black-box access to the group operation and random group elements. A black-box SSS is ideal, if the size of the secret sharing matrix is the same as the number of participants. An access structure \({\mathcal A}\) is black-box ideal, if there exists an ideal black-box SSS realizing \({\mathcal A}\).
In this paper, we study universally ideal homomorphic and black-box ideal access structures, and prove that an access structure \({\mathcal A}\) is universally ideal homomorphic (black-box ideal) if and only if there is a regular matroid appropriate for \({\mathcal A}\).
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References
Beimel, A., Chor, B.: Universally ideal secret sharing schemes. IEEE Transaction on Information Theory IT-40(3), 786–794 (1994)
Benaloh, J.C.: Secret sharing homomorphisms: Keeping shares of a secret secret. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 251–260. Springer, Heidelberg (1987)
Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-Cryptographic fault tolerant distributed computation. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC 1988), pp. 1–10 (1988)
Blakley, G.R.: Safeguarding cryptographic keys. Proc. AFIPS 1979 Nat. Computer Conf. 48, 313–317 (1979)
Brickell, E.F.: Some ideal secret sharing schemes. Journal of Combinatorial Mathematics and Combinatorial Computing 6, 105–113 (1989)
Brickell, E.F., Davenport, D.M.: On the classification of ideal secret sharing schemes. Journal of Cryptology 4(2), 123–134 (1991)
Cramer, R., Fehr, S.: Optimal black-box secret sharing over arbitrary abelian groups. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, p. 272. Springer, Heidelberg (2002)
Desmedt, Y., Frankel, Y.: Threshold cryptosystems. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 307–315. Springer, Heidelberg (1990)
Desmedt, Y., Frankel, Y.: Homomorphic zero-knowledge threshold schemes over any finite abelian group. SIAM Journal on Discrete Mathematics 7(4), 667–679 (1994)
van Dijk, M.: A linear construction of perfect secret sharing schemes. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 23–36. Springer, Heidelberg (1995)
Frankel, Y., Desmedt, Y.: Classification of ideal homomorphic threshold schemes over finite abelian groups. In: Rueppel, R.A. (ed.) EUROCRYPT 1992. LNCS, vol. 658, pp. 25–34. Springer, Heidelberg (1993)
Frankel, Y., Desmedt, Y., Burmester, M.: Non-existence of homomorphic general sharing schemes for some key spaces. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 549–556. Springer, Heidelberg (1993)
Ito, M., Saito, A., Nishizeki, T.: Secret sharing schemes realizing general access structure. Proc. IEEE Global Telecommunication Conf. Globecom 87, 99–102 (1987)
Jackson, W.-A., Martin, K.: Geometric secret sharing schemes and their duals. Designs, Codes and Cryptography 4(1), 83–95 (1994)
Karnin, E.D., Greene, J.W., Hellman, M.E.: On secret sharing systems. IEEE Transaction on Information Theory IT-29(1), 35–41 (1983)
Liu, M., Zhou, Z.: Ideal homomorphic secret sharing schemes over cyclic groups. Science in China, Ser. EÂ 41(6) (1998)
Simmons, G.S., Jackson, W.-A., Martin, K.: The geometry of shared secret schemes. Bulletin of the Institute of Combinatorics and its Applications, 71–88 (1991)
Shamir, A.: How to share a secret. Communication of the ACM 22(11), 612–613 (1979)
Shoup, V.: Practical threshold signatures. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 207–220. Springer, Heidelberg (2000)
Truemper, K.: Matroid decomposition. Academic, Boston (1992)
Welsh, D.J.A.: Matroid theory. Academic, London (1976)
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Zhou, Z. (2005). Classification of Universally Ideal Homomorphic Secret Sharing Schemes and Ideal Black-Box Secret Sharing Schemes. In: Feng, D., Lin, D., Yung, M. (eds) Information Security and Cryptology. CISC 2005. Lecture Notes in Computer Science, vol 3822. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11599548_32
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DOI: https://doi.org/10.1007/11599548_32
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