A distributed secret sharing system is considered that consists of a dealer, @math storage nodes, and @math users. Each user is given access to a certain subset of storage nodes where it can download the data. The dealer wants to securely convey a specific secret s j to user @math via storage nodes, for @math = 1, 2,…, m, in such a way that no user gets any information about other users' secrets in an information-theoretic sense. To this end, we propose to study protocols where the dealer encodes secrets into several secret shares and loads them into the storage nodes. Given a certain number of storage nodes we find the maximum number of users that can be served in such protocols and construct schemes that achieve this. We further define two major properties for such distributed secret sharing systems; communication complexity is defined as the total amount of data that needs to be downloaded by users in order to reconstruct their secrets; and storage overhead is defined as the total amount of data loaded by the dealer into the storage nodes normalized by the total size of secrets. Lower bounds on minimum communication complexity and storage overhead are characterized given any @math and @math . Furthermore, we construct distributed secret sharing protocols, under certain conditions on the system parameters, that attain these lower bounds thereby providing schemes that are optimal in terms of both the communication complexity and storage overhead.
The @math secret sharing scheme provided by Shamir in @cite_20 , is as follows. Given a secret @math it produces @math @math in such a way that the secret @math can be reconstructed given any @math or more of the secret shares, and knowledge of any @math or fewer secret shares does not reveal any information about @math , in the information-theoretic sense. To this end, a @math -degree polynomial @math is constructed as follows: shamir polynomial P(x)=s+ i=1 ^ t-1 p_i x^i where @math 's are i.i.d and are selected uniformly at random from @math . Let @math denote @math distinct non-zero elements from @math . The secret shares are produced by evaluating @math at @math 's, i.e., @math We refer to the encoder @math that produces @math according to this procedure given the input @math as @math Shamir's secret encoder.
@cite_20
1641218312 2090352935 1146167248 2083952939
A secret sharing scheme is a method to store information securely and reliably. Particularly, in a threshold secret sharing scheme , a secret is encoded into @math shares, such that any set of at least @math shares suffice to decode the secret, and any set of at most @math d @math t_ 1 d n$ . The scheme is based on a generalization of Shamir’s secret sharing scheme and preserves its simplicity and efficiency. In addition, we consider the setting of secure distributed storage where the proposed communication efficient secret sharing schemes not only improve decoding bandwidth but further improve disk access complexity during decoding. In the secret sharing schemes already published, the security of the scheme is mostly established upon difficult mathematical problems such as the solution of discrete logarithm and the factoring of large number. However, such schemes take up lots of resources while result in low speed. In this paper we propose a new secret-sharing scheme, which is based on shamir secret sharing scheme and use the NTRU and one-way hashing function to verify the validity of data. The security of our scheme is built on quick and efficient NTRU algorithm. It can use the public channel to issue information, detect effectively a variety of cheating or forgery behaviors, and ensure the credibility of resumption of secret. A new multi-secret sharing (t, n) threshold scheme is proposed in this paper. The scheme uses the Lagrange interpolating polynomial to split and reconstruct the secrets based on Shamir secret sharing scheme, and verifies the legality of data by NTRU algorithm and one-way hashing function. Compared with other public key cryptosystems such as elliptic curve cryptography, the proposed is simpler in design, which requires less calculation and fewer storage spaces. It can detect effectively a variety of cheating and forgery behaviors, which guarantee that the reconstruction of secret is the secure and trustworthy. We consider the problem of threshold secret sharing in groups with hierarchical structure. In such settings, the secret is shared among a group of participants that is partitioned into levels. The access structure is then determined by a sequence of threshold requirements: a subset of participants is authorized if it has at least k0 0 members from the highest level, as well as at least k1 > k0 members from the two highest levels and so forth. Such problems may occur in settings where the participants differ in their authority or level of confidence and the presence of higher level participants is imperative to allow the recovery of the common secret. Even though secret sharing in hierarchical groups has been studied extensively in the past, none of the existing solutions addresses the simple setting where, say, a bank transfer should be signed by three employees, at least one of whom must be a department manager. We present a perfect secret sharing scheme for this problem that, unlike most secret sharing schemes that are suitable for hierarchical structures, is ideal. As in Shamir's scheme, the secret is represented as the free coefficient of some polynomial. The novelty of our scheme is the usage of polynomial derivatives in order to generate lesser shares for participants of lower levels. Consequently, our scheme uses Birkhoff interpolation, i.e., the construction of a polynomial according to an unstructured set of point and derivative values. A substantial part of our discussion is dedicated to the question of how to assign identities to the participants from the underlying finite field so that the resulting Birkhoff interpolation problem will be well posed. In addition, we devise an ideal and efficient secret sharing scheme for the closely related hierarchical threshold access structures that were studied by Simmons and Brickell.