In this thesis we discussed secret haring for infinite secret domains. The first part of the thesis is devoted to the result by Chor and Kushilevitz. We proved an impossibility of secret haring over countable set of secrets and shares. Our result is slightly more general the the one by Chor and Kushilevitz. Also we gave several counterexamples which show an importance of assumptions in our theorem. Later we discussed possible definition of secret sharing over continuum set of secrets. Particularly, using the idea by Gabor Tardos, we constructed a scheme that shows that the definition by Chor and Kushilevitz is ”intuitively incorrect”. Namely, it is possible that an unqualified set of players can reconstruct the secret. We suggested a modified definition and showed that the positive result by Chor and Kushilevitz holds for the new definition. Finally we gave an introduction to Lebesgue-Rokhlin probability spaces and showed that a secret from such spaces can be shared. We suggest a further research in this direction. It seems that the converse is true. Namely, if a secret sharing scheme exists for a secret picked up from a probability space, then this space is Lebesgue-Rokhlin. As a short digression we presented a nice scheme for the situation when a secret is a branch of a binary tree(possible infinite). This scheme seems to be useful as it allows to compute several primitives on a secret without any communication between participants. In the last part of the thesis we presented our concept of perfect uniform non-probabilistic secret sharing, in which we avoid the idea of probability and concentrate on cardinalities. We showed that there exists a perfect uniform non-probabilistic secret sharing scheme for any secret domain and any access structure(possibly infinite).In this thesis we discussed secret haring for infinite secret domains. The first part of the thesis is devoted to the result by Chor and Kushilevitz. We proved an impossibility of secret haring over countable set of secrets and shares. Our result is slightly more general the the one by Chor and Kushilevitz. Also we gave several counterexamples which show an importance of assumptions in our theorem. Later we discussed possible definition of secret sharing over continuum set of secrets. Particularly, using the idea by Gabor Tardos, we constructed a scheme that shows that the definition by Chor and Kushilevitz is ”intuitively incorrect”. Namely, it is possible that an unqualified set of players can reconstruct the secret. We suggested a modified definition and showed that the positive result by Chor and Kushilevitz holds for the new definition. Finally we gave an introduction to Lebesgue-Rokhlin probability spaces and showed that a secret from such spaces can be shared. We suggest a further research in this direction. It seems that the converse is true. Namely, if a secret sharing scheme exists for a secret picked up from a probability space, then this space is Lebesgue-Rokhlin. As a short digression we presented a nice scheme for the situation when a secret is a branch of a binary tree(possible infinite). This scheme seems to be useful as it allows to compute several primitives on a secret without any communication between participants. In the last part of the thesis we presented our concept of perfect uniform non-probabilistic secret sharing, in which we avoid the idea of probability and concentrate on cardinalities. We showed that there exists a perfect uniform non-probabilistic secret sharing scheme for any secret domain and any access structure(possibly infinite).

Jan 1, 20113 citations

Analysis of a kind of quantum cryptographic schemes based on secret sharing

Zhang Xing

Recently, Yang et al. proposed a kind of quantum cryptographic schemes based on secret sharing. The main idea is drawn from the case, where any n participants who share a secret K can co-operate as K does. This process can be applied to encryption, authentication, signature and so on. Unfortunately, since there is no identity authentication of the share’s holder, these schemes inherit the limitation of secret sharing in practice. If some participants do not follow the protocol, the protocol would be a failure at last. Moreover, these schemes encounter another serious attack. Any two participants can co-operate to defeat the schemes by sending different shares to the parties unknown to the distributor of the secret K, such that these parties can execute the protocol as well. The attack proves that it is not enough for quantum cryptographic schemes that only the secret key is secure.Recently, Yang et al. proposed a kind of quantum cryptographic schemes based on secret sharing. The main idea is drawn from the case, where any n participants who share a secret K can co-operate as K does. This process can be applied to encryption, authentication, signature and so on. Unfortunately, since there is no identity authentication of the share’s holder, these schemes inherit the limitation of secret sharing in practice. If some participants do not follow the protocol, the protocol would be a failure at last. Moreover, these schemes encounter another serious attack. Any two participants can co-operate to defeat the schemes by sending different shares to the parties unknown to the distributor of the secret K, such that these parties can execute the protocol as well. The attack proves that it is not enough for quantum cryptographic schemes that only the secret key is secure.

Science in China. Series G, Physics, mechanics & astronomy/Science in China. Series G, Physics, Mechanics & AstronomyJan 1, 20090 citations

Multiparty quantum secret sharing

Zhanjun Zhang, Yong Li, Zhong‐Xiao Man

Based on a quantum secure direct communication (QSDC) protocol [Phys. Rev. A 69 052319 (2004)], we propose a $(n,n)$ -threshold scheme of multiparty quantum secret sharing of classical messages (QSSCM) using only single photons. We take advantage of this multiparty QSSCM scheme to establish a scheme of multiparty secret sharing of quantum information (SSQI), in which only all quantum information receivers collaborate can the original qubit be reconstructed. A general idea is also proposed for constructing multiparty SSQI schemes from any QSSCM scheme.Based on a quantum secure direct communication (QSDC) protocol [Phys. Rev. A 69 052319 (2004)], we propose a $(n,n)$-threshold scheme of multiparty quantum secret sharing of classical messages (QSSCM) using only single photons. We take advantage of this multiparty QSSCM scheme to establish a scheme of multiparty secret sharing of quantum information (SSQI), in which only all quantum information receivers collaborate can the original qubit be reconstructed. A general idea is also proposed for constructing multiparty SSQI schemes from any QSSCM scheme.

arXiv:quant-ph/0412203Physical Review AApr 11, 2005470 citations

Based on a quantum secure direct communication (QSDC) protocol [Phys. Rev. A 69 052319 (2004)], we propose a

(n,n)

-threshold scheme of multiparty quantum secret sharing of classical messages (QSSCM) using only single photons. We take advantage of this multiparty QSSCM scheme to establish a scheme of multiparty secret sharing of quantum information (SSQI), in which only all quantum information receivers collaborate can the original qubit be reconstructed. A general idea is also proposed for constructing multiparty SSQI schemes from any QSSCM scheme.