Secret sharing schemes for ports of matroids of rank 3

summary:A secret sharing scheme is ideal if the size of each share is equal to the size of the secret. Brickell and Davenport showed that the access structure of an ideal secret sharing scheme is determined by a matroid. Namely, the minimal authorized subsets of an ideal secret sharing scheme are in correspondence with the circuits of a matroid containing a fixed point. In this case, we say that the access structure is a matroid port. It is known that, for an access structure, being a matroid port is not a sufficient condition to admit an ideal secret sharing scheme. In this work we present a linear secret sharing scheme construction for ports of matroids of rank 3 in which the size of each share is at most $n$ times the size of the secret. Using the previously known secret sharing constructions, the size of each share was $O(n^2/\log n)$ the size of the secret. Our construction is extended to ports of matroids of any rank $k\geq 2$, obtaining secret sharing schemes in which the size of each share is at most $n^{k-2}$ times the size of the secret. This work is complemented by presenting lower bounds: There exist matroid ports that require $(\mathbb{F}_q,\ell)$-linear secret schemes with total information ratio $\Omega(2^{n/2}/\ell n^{3/4}\sqrt{\log q})$

Secret sharing and duality

Secret sharing is an important building block in cryptography. All explicitly defined secret sharing schemes with known exact complexity bounds are multi-linear, thus are closely related to linear codes. The dual of such a linear scheme, in the sense of duality of linear codes, gives another scheme for the dual access structure. These schemes have the same complexity, namely the largest share size relative to the secret size is the same. It is a long-standing open problem whether this fact is true in general: the complexity of any access structure is the same as the complexity of its dual. We give an almost answer to this question. An almost perfect scheme allows negligible errors, both in the recovery and in the independence. There exists an almost perfect ideal scheme on 174 participants whose complexity is strictly smaller than that of its dual

Low-Bandwidth Recovery of Linear Functions of Reed-Solomon-Encoded Data

We study the problem of efficiently computing on encoded data. More specifically, we study the question of low-bandwidth computation of functions F:F^k ? F of some data ? ? F^k, given access to an encoding ? ? F? of ? under an error correcting code. In our model - relevant in distributed storage, distributed computation and secret sharing - each symbol of ? is held by a different party, and we aim to minimize the total amount of information downloaded from each party in order to compute F(?). Special cases of this problem have arisen in several domains, and we believe that it is fruitful to study this problem in generality. Our main result is a low-bandwidth scheme to compute linear functions for Reed-Solomon codes, even in the presence of erasures. More precisely, let ? > 0 and let ?: F^k ? F? be a full-length Reed-Solomon code of rate 1 - ? over a field F with constant characteristic. For any ? ? [0, ?), our scheme can compute any linear function F(?) given access to any (1 - ?)-fraction of the symbols of ?(?), with download bandwidth O(n/(? - ?)) bits. In contrast, the naive scheme that involves reconstructing the data ? and then computing F(?) uses ?(n log n) bits. Our scheme has applications in distributed storage, coded computation, and homomorphic secret sharing

Secret Sharing and Shared Information

Secret sharing is a cryptographic discipline in which the goal is to distribute information about a secret over a set of participants in such a way that only specific authorized combinations of participants together can reconstruct the secret. Thus, secret sharing schemes are systems of variables in which it is very clearly specified which subsets have information about the secret. As such, they provide perfect model systems for information decompositions. However, following this intuition too far leads to an information decomposition with negative partial information terms, which are difficult to interpret. One possible explanation is that the partial information lattice proposed by Williams and Beer is incomplete and has to be extended to incorporate terms corresponding to higher order redundancy. These results put bounds on information decompositions that follow the partial information framework, and they hint at where the partial information lattice needs to be improved.Comment: 9 pages, 1 figure. The material was presented at a Workshop on information decompositions at FIAS, Frankfurt, in 12/2016. The revision includes changes in the definition of combinations of secret sharing schemes. Section 3 and Appendix now discusses in how far existing measures satisfy the proposed properties. The concluding section is considerably revise

Quantum information transfer for qutrits

We propose a scheme for the transfer of quantum information among distant qutrits. We apply this scheme to the distribution of entanglement among distant nodes and to the generation of multipartite antisymmetric states. We also discuss applications to quantum secret sharing

Securing a Quantum Key Distribution Network Using Secret Sharing

We present a simple new technique to secure quantum key distribution relay networks using secret sharing. Previous techniques have relied on creating distinct physical paths in order to create the shares. We show, however, how this can be achieved on a single physical path by creating distinct logical channels. The technique utilizes a random 'drop-out' scheme to ensure that an attacker must compromise all of the relays on the channel in order to access the key

Quantum secure direct communication based on supervised teleportation

We present a quantum secure direct communication(QSDC) scheme as an extension for a proposed supervised secure entanglement sharing protocol. Starting with a quick review on the supervised entanglement sharing protocol -- the "Wuhan" protocol [Y. Li and Y. Liu, arXiv:0709.1449v2], we primarily focus on its further extend using for a QSDC task, in which the communication attendant Alice encodes the secret message directly onto a sequence of 2-level particles which then can be faithfully teleported to Bob using the shared maximal entanglement states obtained by the previous "Wuhan" protocol. We also evaluate the security of the QSDC scheme, where an individual self-attack performed by Alice and Bob -- the out of control attack(OCA) is introduced and the robustness of our scheme on the OCA is documented.Comment: 5 pages, 1 table, oral contribution in the Conference on Quantum Optics and Applications in Computing and Communications, Photonics Asia 2007, Proc. of SPI