Some Basic Properties of General Nonperfect Secret Sharing Schemes

Oriol Farràs

Mitsugu Iwamoto, Junji Shikata

Fundamental results on secret sharing schemes (SSSs) are discussed in the setting where security and share size are measured by (conditional) min-entropies. We first formalize a unified framework of SSSs based on (conditional) Rέnyi entropies, which includes SSSs based on Shannon and min entropies etc. as special cases. By deriving the lower bound of share sizes in terms of Rέnyi entropies based on the technique introduced by Iwamoto-Shikata, we obtain the lower bounds of share sizes measured by min entropies as well as by Shannon entropies in a unified manner. As the main contributions of this paper, we show two existential results of non-perfect SSSs based on min-entropies under several important settings. We first show that there exists a nonperfect SSS for arbitrary binary secret information and arbitrary monotone access structure. In addition, for every integers k and n (k ≤ n), we prove that the ideal non-perfect (k, n)-threshold scheme exists even if the distribution of the secret is not uniformly distributed.

I-Ching Yu, Feng-Li Lin, Ching-Yu Huang

We construct general schemes for multipartite quantum secret sharing using multilevel systems, and find that the consistency conditions for valid measurements can be summarized in two simple algebraic conditions. The scheme using very high-dimensional mutually unbiased bases can in principle achieve perfect security against intercept-resend attack; and for the scheme using mutually biased bases, it reaches the optimal but nonperfect security at the four-level system. We also address the security issue against general attacks in the context of our multilevel schemes. In particular, we propose a protocol to enhance both the efficiency and the security against an entanglement-assisted participant's attack by incorporating quantum-key-distribution and measurement-basis-encrypted schemes so that its security is as robust as quantum-key distribution.

Ernest F. Brickell, Daniel M. Davenport

Ernest F. Brickell, Daniel M. Davenport

Kamil Kulesza, Zbigniew Kotulski

In the paper we propose a general, abstract framework for Automatic Secret Generation and Sharing (ASGS) that should be independent of underlying Secret Sharing Scheme (SSS). ASGS allows to prevent the Dealer from knowing the secret. The Basic Property Conjecture (BPC) forms the base of the framework. Due to the level of abstraction, results are portable into the realm of quantum computing. Two situations are discussed. First concerns simultaneous generation and sharing of the random, prior nonexistent secret. Such a secret remains unknown until it is reconstructed. Next, we propose the framework for automatic sharing of a known secret. In this case the Dealer does not know the secret and the secret Owner does not know the shares. We present opportunities for joining ASGS with other extended capabilities, with special emphasis on PVSS and pre-positioned secret sharing. Finally, we illustrate framework with practical implementation.