Shamir's celebrated secret sharing scheme provides an efficient method for encoding a secret of arbitrary length @math among any @math players such that for a threshold parameter @math , (i) the knowledge of any @math shares does not reveal any information about the secret and, (ii) any choice of @math shares fully reveals the secret. It is known that any such threshold secret sharing scheme necessarily requires shares of length @math , and in this sense Shamir's scheme is optimal. The more general notion of ramp schemes requires the reconstruction of secret from any @math shares, for a positive integer gap parameter @math . Ramp secret sharing scheme necessarily requires shares of length @math . Other than the bound related to secret length @math , the share lengths of ramp schemes can not go below a quantity that depends only on the gap ratio @math . In this work, we study secret sharing in the extremal case of bit-long shares and arbitrarily small gap ratio @math , where standard ramp secret sharing becomes impossible. We show, however, that a slightly relaxed but equally effective notion of semantic security for the secret, and negligible reconstruction error probability, eliminate the impossibility. Moreover, we provide explicit constructions of such schemes. One of the consequences of our relaxation is that, unlike standard ramp schemes with perfect secrecy, adaptive and non-adaptive adversaries need different analysis and construction. For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any @math fraction of observed shares, and reconstruction from any @math fraction of shares, for any choices of @math . Our construction achieves secret length @math , which we show to be optimal. For adaptive adversaries, we construct explicit schemes attaining a secret length @math .
In @cite_23 @cite_17 , the secret is one bit. In @cite_20 , secrets of length equal to a fraction of @math (number of players) is considered. This time binary secret sharing with adaptive and non-adaptive adversaries similar to the model we consider in this work is defined. However the paper considers only a privacy threshold @math , and reconstruction is from the full share set ( @math always). Their goal is to achieve large secrets @math over binary shares with large privacy parameter @math , which is also similar to ours. They have an additional goal of keeping the computational complexity of the reconstruction algorithm within AC @math , which we do not consider in this work. Their large privacy parameter @math is with a @math much smaller than @math , which means that the relative threshold gap @math can not be arbitrarily small.
@cite_20 @cite_23 @cite_17
1146167248 2083952939 2805329953 2167765756
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. A tunable measure for information leakage called maximal a-leakage is introduced. This measure quantifies the maximal gain of an adversary in refining a tilted version of its prior belief of any (potentially random) function of a dataset conditioning on a disclosed dataset. The choice of @math determines the specific adversarial action ranging from refining a belief for @math to guessing the best posterior for @math , and for these extremal values this measure simplifies to mutual information (MI) and maximal leakage (MaxL), respectively. For all other @math this measure is shown to be the Arimoto channel capacity. Several properties of this measure are proven including: (i) quasi-convexity in the mapping between the original and disclosed datasets; (ii) data processing inequalities; and (iii) a composition property. A full version of this paper is in [1]. Privacy amplification allows two parties Alice and Bob knowing a partially secret string S to extract, by communication over a public channel, a shorter, highly secret string S'. Bennett, Brassard, Crepeau, and Maurer showed that the length of S' can be almost equal to the conditional Renyi entropy of S given an opponent Eve's knowledge. All previous results on privacy amplification assumed that Eve has access to the public channel but is passive or, equivalently, that messages inserted by Eve can be detected by Alice and Bob. In this paper we consider privacy amplification secure even against active opponents. First it is analyzed under what conditions information-theoretically secure authentication is possible even though the common key is only partially secret. This result is used to prove that privacy amplification can be secure against an active opponent and that the size of S' can be almost equal to Eve's min-entropy about S minus 2n 3 if 5 is an n-bit string. Moreover, it is shown that for sufficiently large n privacy amplification is possible when Eve's min-entropy about S exceeds only n 2 rather than 2n 3.