Lower bounds on the information rate of secret sharing schemes with homogeneous access structure

We present some new lower bounds on the optimal information rate and on the optimal average information rate of secret sharing schemes with homogeneous access structure. These bounds are found by using some covering constructions and a new parameter, the k-degree of a participant, that is introduced in this paper. Our bounds improve the previous ones in almost all cases.     

Strongly secure ramp secret sharing schemes for general access structures

Ramp secret sharing (SS) schemes can be classified into strong ramp SS schemes and weak ramp SS schemes. The strong ramp SS schemes do not leak out any part of a secret explicitly even in the case that some information about the secret leaks out from some set of shares, and hence, they are more desirable than the weak ramp SS schemes. In this paper, it is shown that for any feasible general access structure, a strong ramp SS scheme can be constructed from a partially decryptable ramp SS scheme, which can be considered as a kind of SS scheme with plural secrets. As a byproduct, it is pointed out that threshold ramp SS schemes based on Shamir's polynomial interpolation method are not always strong.     

Contrast-optimal k out of n secret sharing schemes in visual cryptography

Visual cryptography and (k,n)-visual secret sharing schemes were introduced by Naor and Shamir (Advances in Cryptology — Eurocrypt 94, Springer, Berlin, 1995, pp. 1–12). A sender wishing to transmit a secret message distributes n transparencies amongst n recipients, where the transparencies contain seemingly random pictures. A (k,n)-scheme achieves the following situation: If any k recipients stack their transparencies together, then a secret message is revealed visually. On the other hand, if only k−1 recipients stack their transparencies, or analyze them by any other means, they are not able to obtain any information about the secret message. The important parameters of a scheme are its contrast, i.e., the clarity with which the message becomes visible, and the number of subpixels needed to encode one pixel of the original picture. Naor and Shamir constructed (k,k)-schemes with contrast 2^−(k−1). By an intricate result from Linial (Combinatorica 10 (1990) 349–365), they were also able to prove the optimality of these schemes. They also proved that for all fixed kn, there are (k,n)-schemes with contrast . For k=2,3,4 the contrast is approximately and . In this paper, we show that by solving a simple linear program, one is able to compute exactly the best contrast achievable in any (k,n)-scheme. The solution of the linear program also provides a representation of a corresponding scheme. For small k as well as for k=n, we are able to analytically solve the linear program. For k=2,3,4, we obtain that the optimal contrast is at least and . For k=n, we obtain a very simple proof of the optimality of Naor's and Shamir's (k,k)-schemes. In the case k=2, we are able to use a different approach via coding theory which allows us to prove an optimal tradeoff between the contrast and the number of subpixels.     

Two secret sharing schemes based on Boolean operations

Traditional secret sharing schemes involve complex computation. A visual secret sharing (VSS) scheme decodes the secret without computation, but each shadow is m times as big as the original. Probabilistic VSS solved the computation complexity and space complexity problems at once. In this paper we propose a probabilistic (2,n) scheme for binary images and a deterministic (n,n) scheme for grayscale images. Both use simple Boolean operations and both have no pixel expansion. The (2,n) scheme provides a better contrast and significantly smaller recognized areas than other methods. The (n,n) scheme gives an exact reconstruction.     

Combinatorial lower bounds for secret sharing schemes

In a perfect secret sharing scheme, it holds that , where S denotes the secret and denotes the set of the share of user i. On the other hand, it is well known that if S is not uniformly distributed, where denotes the set of secrets. In this case, . Then, which is bigger, or We first prove that for any distribution on S by using a combinatorial argument. This is a more sharp lower bound on for not uniformly distributed S. Our proof makes it intuitively clear why must be so large. Next, we extend our technique to show that max_i for some access structure.     

On identification secret sharing schemes

Let be a set of participants sharing a secret from a set of secrets. A secret sharing scheme is a protocol such that any qualified subset of can determine the secret by pooling their shares, the messages which they receive, without error, whereas non-qualified subsets of cannot obtain any knowledge about the secret when they pool what they receive. In (optimal) schemes, the sizes of shared secrets depend on the sizes of shares given to the participants. Namely the former grow up exponentially as the latter increase exponentially. In this paper, instead of determining the secret, we require the qualified subsets of participants to identify the secret. This change would certainly make no difference from determining secret if no error for identification were allowed. So here we relax the requirement to identification such that an error may occur with a vanishing probability as the sizes of the secrets grow up. Under relaxed condition this changing allows us to share a set of secrets with double exponential size as the sizes of shares received by the participants exponentially grow. Thus much longer secret can be shared. On the other hand, by the continuity of Shannon entropy we have that the relaxation makes no difference for (ordinary) secret sharing schemes. We obtain the characterizations of relations of sizes of secrets and sizes of the shares for identification secret sharing schemes without and with public message. Our idea originates from Ahlswede–Dueck’s awarded work in 1989, where the identification codes via channels were introduced.     

Reduce shadow size in aspect ratio invariant visual secret sharing schemes using a square block-wise operation

An aspect ratio invariant visual secret sharing (ARIVSS) scheme is a perfectly secure method for sharing secret images. Due to the nature of the VSS encryption, each secret pixel is expanded to m sub-pixels in each of the generated shares. The advantage of ARIVSS is that the aspect ratio of the recovered secret image is fixed and thus there is no loss of information when the shape of the secret image is our information. For example, a secret image of a circle is compromised to an ellipse if m does not have a square value. Two ARIVSS schemes based on processing one and four pixel blocks, respectively, were previously proposed. In this paper, we have generalized the square block-wise approach to further reduce pixel expansion.     

超圈量子存取结构及其秘密共享方案

2001年,Anderson等人提出了一个可改进的完善量子秘密共享方案（简称IPQSS方案）,本文就该方案可以实现的量子存取结构进行了深入研究.首先在同构意义下给出了所有的超圈量子存取结构的形式;然后求出了其对应的最优限制存取结构,并在理论上证明了所有超圈量子存取结构可由IPQSS方案来实现的条件,且证明了可实现的这些方案均是理想的.     

A hierarchical threshold secret image sharing

In the traditional secret image sharing schemes, the shadow images are generated by embedding the secret data into the cover image such that a sufficient number of shadow images can cooperate to reconstruct the secret image. In the process of reconstruction, each shadow image plays an equivalent role. However, a general threshold access structure could have other useful properties for the application. In this paper, we consider the problem of secret shadow images with a hierarchical threshold structure, employing Tassa’s hierarchical secret sharing to propose a hierarchical threshold secret image sharing scheme. In our scheme, the shadow images are partitioned into several levels, and the threshold access structure is determined by a sequence of threshold requirements. If and only if the shadow images involved satisfy the threshold requirements, the secret image can be reconstructed without distortion.     

Visual secret sharing with cheating prevention revisited

Visual secret sharing (VSS) is a variant form of secret sharing, and is efficient since secret decoding only depends on the human vision system. However, cheating in VSS, first showed by Horng et al., is a significant issue like a limelight. Since then, plenty of studies for cheating activities and cheating prevention visual secret sharing (CPVSS) schemes have been introduced. In this paper, we revisit some well-known cheating activities and CPVSS schemes, and then categorize cheating activities into meaningful cheating, non-meaningful cheating, and meaningful deterministic cheating. Moreover, we analyze the research challenges in CPVSS, and propose a new cheating prevention scheme which is better than the previous schemes in the aspects of some security requirements.     

Progressive sharing for a secret image

Based on the wavelet transform, a new progressive sharing scheme is proposed to share a secret image into several shadow images using SPIHT encoding processes and Shamir’s threshold scheme. Quality refinement of the recovered image is achieved by the data consumed from the threshold number (r) of shadow images and each single shadow image reveals no information about the secret image. The size of each shadow image is smaller than 1/r of the secret image and any number of shadow images that is less than r reveals no information about the secret image. The proposed approach is secure for image sharing and provides excellent peak signal-to-noise ratio (PSNR) versus rate performance. Experimental results have demonstrated the promising performance of this method in progressive sharing.     

一类超图存取结构的秘密共享方案的信息率

针对参与者人数为6的一类超图存取结构的完善秘密共享方案及其最优信息率进行了研究。利用这些存取结构与超图之间的关系, 给出了其对应的95种超图存取结构。对其中的57种超图存取结构运用理想超星判定定理等计算了它们最优信息率的精确值, 并给出了达到此信息率的秘密共享方案的具体构造方法; 对余下的38种超图存取结构运用λ-分解方法等给出了它们最优信息率的上下界。同时证明了具有n个顶点且秩为r的超星和超路径, 其超边数至多为n-r+1条; 并从理论上证明了顶点可约超图的最优信息率为1。     

A novel secret image sharing scheme for true-color images with size constraint

Rapid development of telecommunication and service has made researchers think of intelligent tools to assist users in delivering critical data securely. When it comes to share digital images, owing to high frequent use of Mega pixel digital cameras or camera phones, true-color images become one common image type. In the last few years, several researches have been devoted to study of secret image sharing. What seems lacking is a scheme for sharing true-color secret images with size constraint. This paper proposes a new secret image sharing scheme for true-color secret images. Through combination of neural networks and variant visual secret sharing, the quality of the reconstructed secret image and camouflage images are visually the same as the corresponding original images. Compared with other schemes, the proposed one alone supports true-color secret image with size constraint on shares. Experimental results and comparisons demonstrate the feasibility of this scheme.     

Efficient visual secret sharing scheme for color images

A k-out-of-n visual secret sharing scheme (VSSS) resolves the visual variant of the k-out-of-n secret sharing problem where only k or more out of n participants can reveal the secret by human visual system without any cryptographic computation. The best pixel expansion of the general k-out-of-n VSSS for c-colored images was c×m by Yang and Laih [New colored visual secret sharing schemes, Des Codes Cryptogr. 24 (2000) 325-335] where m is the pixel expansion of an existing binary k-out-of-n VSSS. Regarding the c-colored n-out-of-n scheme, the best pixel expansion is (c-1)2^n-1-c+2 and c(c-1)2^n-2-c when n is odd and even, respectively, by Blundo et al. [Improved schemes for visual cryptography, Des Codes Cryptogr. 24 (2001) 255-278]. In this paper, we propose a new c-colored k-out-of-n VSSS by using a pixel expansion of that is more efficient than ever.     

A general (k, n) scalable secret image sharing scheme with the smooth scalability

A novel (k, n) scalable secret image sharing (SSIS) scheme was proposed to encrypt a secret image into n shadow images. One can gradually reconstruct a secret image by stacking k or more shadows, but he/she cannot conjecture any information from fewer than k shadows. The advantage of a (k, n)-SSIS scheme is that it provides the threshold property (i.e., k is a threshold value necessary to start in to reveal the secret) as well as the scalability (i.e., the information amount of a reconstructed secret is proportional to the number of shadows used in decryption). All previous (k, n)-SSIS schemes did not have the smooth scalability so that the information amount can be “smoothly” proportional to the number of shadows. In this paper, we consider the smooth scalability in (k, n)-SSIS scheme.     

A necessary and sufficient condition for the asymptotic idealness of the GRS threshold secret sharing scheme

The study of the asymptotic idealness of the Goldreich–Ron–Sudan (GRS, for short) threshold secret sharing scheme was the subject of several research papers, where sufficient conditions were provided. In this paper a necessary and sufficient condition is established; namely, it is shown that the GRS threshold secret sharing scheme is asymptotically ideal under the uniform distribution on the secret space if and only if it is based on 1-compact sequences of co-primes.     

Threshold cryptography based on Asmuth-Bloom secret sharing

In this paper, we investigate how threshold cryptography can be conducted with the Asmuth-Bloom secret sharing scheme and present three novel function sharing schemes for RSA, ElGamal and Paillier cryptosystems. To the best of our knowledge, these are the first provably secure threshold cryptosystems realized using the Asmuth-Bloom secret sharing. Proposed schemes are comparable in performance to earlier proposals in threshold cryptography.     

一种简单的可验证秘密共享方案

分析了两种有效的可验证秘密共享方案：Feldman's VSS方案和Pedersen's VSS方案。但是它们都是门限方案,当推广到一般接入结构时,效率都很低。为此,提出了一个一般接入结构上的可验证秘密共享方案。参与者的共享由秘密分发者随机生成,采用秘密信道发送。每个授权子集拥有一个的公开信息,通过公开的信息,参与者能够验证属于自己份额的共享的有效性。该方案具有两种形式：一种是计算安全的,另一种是无条件安全的。其安全性分别等同于Feldman's VSS方案和Pedersen's VSS方案,但在相同的安全级别下,新方案更有效。     

Bit-level based secret sharing for image encryption

A new secret sharing scheme capable of protecting image data coded with B bits per pixel is introduced and analyzed in this paper. The proposed input-agnostic encryption solution generates B-bit shares by combining bit-level decomposition/stacking with a {k,n}-threshold sharing strategy. Perfect reconstruction is achieved by performing decryption through simple logical operations in the decomposed bit-levels without the need for any postprocessing operations. The framework allows for cost-effective cryptographic image processing of B-bit images over the Internet.