Asymptotically Ideal Hierarchical Secret Sharing Based on CRT for Integer Ring
School of Mathematics and Big Data, Chaohu University（巢湖学院数学与大数据学院）；School of Mathematics and Statistics, Hefei University（合肥学院数学与统计学院）；School of Computer Science and Technology, University of Science and Technology of China（中国科学技术大学计算机科学与技术学院）
In Shamir's secret sharing scheme, all participants possess equal privileges. However, in many practical scenarios, it is often necessary to assign different levels of authority to different participants. To address this requirement, Hierarchical Secret Sharing (HSS) schemes were developed, which partitioned all participants into multiple subsets and assigned a distinct privilege level to each. Existing Chinese Remainder Theorem (CRT)-based HSS schemes benefit from flexible share sizes, but either exhibit security flaws or have an information rate less than $\frac{1}{2}$. In this work, we propose a disjunctive HSS scheme and a conjunctive HSS scheme by using the CRT for integer ring and one-way functions. Both schemes are asymptotically ideal and are proven to be secure.
School of Mathematics and Big Data, Chaohu University（巢湖大学数学与大数据学院）；School of Computer Science and Technology, University of Science and Technology of China（中国科学技术大学计算机科学与技术学院）；School of Artificial Intelligence and Big Data, Hefei University（合肥大学人工智能与大数据学院）
With the rapid development of MANET, secure and practical authentication is becoming increasingly important. The existing works perform the research from two aspects, i.e., (a)secure key division and distributed storage, (b)secure distributed authentication. But there still exist several unsolved problems. Specifically, it may suffer from cheating problems and fault authentication attack, which can result in authentication failure and DoS attack towards authentication service. Besides, most existing schemes are not with satisfactory efficiency due to exponential arithmetic based on Shamir's sc…
For conventional secret sharing, if cheaters can submit possibly forged shares after observing shares of the honest users in the reconstruction phase then they cannot only disturb the protocol but also only they may reconstruct the true secret. To overcome the problem, secret sharing scheme with properties of cheater-identification have been proposed. Existing protocols for cheater-identifiable secret sharing assumed non-rushing cheaters or honest majority. In this paper, we remove both conditions simultaneously, and give its universal construction from any secret sharing scheme. To resolve th…
Splitting a secret s between several participants, we generate (for each value of s) shares for all participants. The goal: authorized groups of participants should be able to reconstruct the secret but forbidden ones get no information about it. In this paper we introduce several notions of non- perfect secret sharing, where some small information leak is permitted. We study its relation to the Kolmogorov complexity version of secret sharing (establishing some connection in both directions) and the effects of changing the secret size (showing that we can decrease the size of the secret and th…
Sharing a secret efficiently amongst a group of participants is not easy since there is always an adversary / eavesdropper trying to retrieve the secret. In secret sharing schemes, every participant is given a unique share. When the desired group of participants come together and provide their shares, the secret is obtained. For other combinations of shares, a garbage value is returned. A threshold secret sharing scheme was proposed by Shamir and Blakley independently. In this (n,t) threshold secret sharing scheme, the secret can be obtained when at least t out of n participants contribute the…
It is well-known that the linear secret-sharing scheme (LSSS) can be constructed from linear error-correcting codes (Brickell [1], R.J. McEliece and D.V.Sarwate [2],Cramer, el.,[3]). The theory of linear codes from algebraic-geometric curves (algebraic-geometric (AG) codes or geometric Goppa code) has been well-developed since the work of V.Goppa and Tsfasman, Vladut, and Zink(see [17], [18] and [19]). In this paper the linear secret-sharing scheme from algebraic-geometric codes, which are non-threshold scheme for curves of genus greater than 0, are presented . We analysis the minimal access s…
Secret sharing schemes for classical secrets can be classified into classical secret sharing schemes and quantum secret sharing schemes. Classical secret sharing has been known to be able to distribute some shares before a given secret. On the other hand, quantum mechanics extends the capabilities of secret sharing beyond those of classical secret sharing. We propose quantum secret sharing with the capabilities in designing of access structures more flexibly and realizing higher efficiency beyond those of classical secret sharing, that can distribute some shares before a given secret.
A secret can be an encrypted message or a private key to decrypt the ciphertext. One of the main issues in cryptography is keeping this secret safe. Entrusting secret to one person or saving it in a computer can conclude betrayal of the person or destruction of that device. For solving this issue, secret sharing can be used between some individuals which a coalition of a specific number of them can only get access to the secret. In practical issues, some of the members have more power and by a coalition of fewer of them, they should know about the secret. In a bank, for example, president and …
We consider an extension of Massey's construction of secret sharing schemes using linear codes. We describe the access structure of the scheme and show its connection to the dual code. We use the $g$-fold weight enumerator and invariant theory to study the access structure.
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 $n$ shares, such that any set of at least $t_1$ shares suffice to decode the secret, and any set of at most $t_2 < t_1$ shares reveal no information about the secret. Assuming that each party holds a share and a user wishes to decode the secret by receiving information from a set of parties; the question we study is how to minimize the amount of communication between the user and the parties. We show that the necessary amount of communicat…
In this paper, we propose a class of threshold secret sharing schemes with repairing function between shares without the help of the dealer, that we called repairable threshold secret sharing schemes. Specifically, if a share fails, such as broken or lost, it will be repaired just by some other shares. A construction of such repairable threshold secret sharing schemes is designed by applying linearized polynomials and regenerating codes in distributed storage systems. In addition, a new repairing rate is introduced to characterize the performance and efficiency of the repairing function. Then …
Secret Sharing Schemes (SSS) are methods for distributing a secret among a set of participants. One of the first Secret Sharing Schemes was proposed by M. Mignotte, based on the Chinese remainder theorem over the ring of integers. In this article we extend the Mignotte's scheme to the ring of Gaussian Integers and study some of its properties. While doing this we aim to solve a gap in a previous construction of such extension. In addition we show that any access structure can be made through a SSS over $ \mathbb{Z}[i]$.
We improve the flexibility in designing access structures of quantum stabilizer-based secret sharing schemes for classical secrets, by introducing message randomization in their encoding procedures. We generalize the Gilbert-Varshamov bound for deterministic encoding to randomized encoding of classical secrets. We also provide an explicit example of a ramp secret sharing scheme with which multiple symbols in its classical secret are revealed to an intermediate set, and justify the necessity of incorporating strong security criterion of conventional secret sharing. Finally, we propose an explic…
We consider secret sharing schemes with a classical secret and quantum shares. One example of such schemes was recently reported whose access structure cannot be realized by any secret sharing schemes with classical shares. In this paper, we report further quantum secret sharing schemes whose access structures cannot be realized by any classical secret sharing schemes.
The Shamir secret sharing scheme requires a Maximum Distance Separable (MDS) code, and in its most common implementation, a Reed-Solomon (RS) code is used. In this paper, we observe that the encoding procedure can be made simpler and faster by dropping the MDS condition and specifying the possible symbols that can be shared. In particular, the process can be made even faster by using array codes based on XOR operations instead of RS codes.
The multiple assignment scheme is to assign one or more shares to single participant so that any kind of access structure can be realized by classical secret sharing schemes. We propose its quantum version including ramp secret sharing schemes. Then we propose an integer optimization approach to minimize the average share size.
We show a simple example of a secret sharing scheme encoding classical secret to quantum shares that can realize an access structure impossible by classical information processing with limitation on the size of each share. The example is based on quantum stabilizer codes.
The work presents a novel quantum secret sharing strategy based on GHZ product state sharing between three parties. The dealer, based on the classical information to be shared, toggles his qubit and shares the product state. The other parties make their Bell measurements and collude to reconstruct the secret. Unlike the other protocols, this protocol does not involve the entire initial state reconstruction, rather uses selective qubits to discard the redundant qubits at the time of reconstruction to decrypt the secret. The protocol also allows for security against malicious attacks by an adver…
Shamir's celebrated secret sharing scheme provides an efficient method for encoding a secret of arbitrary length $\ell$ among any $N \leq 2^\ell$ players such that for a threshold parameter $t$, (i) the knowledge of any $t$ shares does not reveal any information about the secret and, (ii) any choice of $t+1$ shares fully reveals the secret. It is known that any such threshold secret sharing scheme necessarily requires shares of length $\ell$, and in this sense Shamir's scheme is optimal. The more general notion of ramp schemes requires the reconstruction of secret from any $t+g$ shares, for a …
In this paper we consider to use the quantum stabilizer codes as secret sharing schemes for classical secrets. We give necessary and sufficient conditions for qualified and forbidden sets in terms of quantum stabilizers. Then we give a Gilbert-Varshamove-type sufficient condition for existence of secret sharing schemes with given parameters, and by using that sufficient condition, we show that roughly 19% of participants can be made forbidden independently of the size of classical secret, in particular when an $n$-bit classical secret is shared among $n$ participants having 1-qubit share each.…
We first explain the notion of secret sharing and also threshold schemes, which can be implemented with the Shamir's secret sharing. Subsequently, we review social secret sharing (NSG'10,NS'10) and its trust function. In a secret sharing scheme, a secret is shared among a group of players who can later recover the secret. We review the construction of a social secret sharing scheme and its application for resource management in cloud, as explained in NS'12. To clarify the social secret sharing scheme, we first review its trust function according to NL'06. In this scheme, a secret is maintained…
For a secret sharing scheme, two parameters $d_{min}$ and $d_{cheat}$ are defined in [12] and [13]. These two parameters measure the error-correcting capability and the secret-recovering capability of the secret sharing scheme against cheaters. Some general properties of the parameters have been studied in [12,[9] and [13]. The MDS secret-sharing scheme was defined in [12] and it is proved that MDS perfect secret sharing scheme can be constructed for any monotone access structure. The famous Shamir $(k,n)$ threshold secret sharing scheme is the MDS with $d_{min}=d_{cheat}=n-k+1$. In [3] we pro…
In the paper we propose general framework for Automatic Secret Generation and Sharing (ASGS) that should be independent of underlying secret sharing scheme. ASGS allows to prevent the dealer from knowing the secret or even to eliminate him at all. 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 pr…
This paper first reviews some basic properties of cryptographic hash function, secret sharing scheme, and Latin square. Then we discuss why Latin square or its critical set is a good choice for secret representation and its relationship with secret sharing scheme. Further we enumerate the limitations of Latin square in a secret sharing scheme. Finally we propose how to apply cryptographic hash functions, herding attack technique to a Latin square based secret sharing scheme to overcome these limitations.
Secret Sharing techniques are now the building blocks of several security protocols. A (t;n) threshold secret sharing scheme is one in which t or more participant can join together to retrieve the secret.Traditional single secret sharing schemes are modified and generalized to share multiple secrets.Use of elliptic curve and pairing in secret sharing is gaining more importance.In this paper we propose a threshold multi secret sharing scheme where more than one secret is shared.When the threshold number of participants collate, the multi secret can be retrieved. The scheme make use of elliptic …
The first construction of strongly secure quantum ramp secret sharing by Zhang and Matsumoto had an undesirable feature that the dimension of quantum shares must be larger than the number of shares. By using algebraic curves over finite fields, we propose a new construction in which the number of shares can become arbitrarily large for fixed dimension of shares.
In the paper we discuss how to share the secrets, that are graphs. So, far secret sharing schemes were designed to work with numbers. As the first step, we propose conditions for "graph to number" conversion methods. Hence, the existing schemes can be used, without weakening their properties. Next, we show how graph properties can be used to extend capabilities of secret sharing schemes. This leads to proposal of using such properties for number based secret sharing.
Secret sharing is a method of dividing a secret among n par- ticipants and allows only qualified subset to reconstruct the secret and hence provides better reliability and availability of secret data.In the generalized secret sharing scheme, a monotone access structure of the set of participants is considered. The access structure specifies a qualified subset of participants who can reconstruct the secret from their shares.Generalized secret sharing schemes can be efficiently implemented by using (n; n) scheme.We have developed an efficient (n; n) scheme using Permutation Ordered Binary (POB) …
Secret sharing schemes are widely used now a days in various applications, which need more security, trust and reliability. In secret sharing scheme, the secret is divided among the participants and only authorized set of participants can recover the secret by combining their shares. The authorized set of participants are called access structure of the scheme. In Multi-Secret Sharing Scheme (MSSS), k different secrets are distributed among the participants, each one according to an access structure. Multi-secret sharing schemes have been studied extensively by the cryptographic community. Numb…
Secret sharing is the well-known problem of splitting a secret into multiple shares, which are distributed to shareholders. When enough or the correct combination of shareholders work together the secret can be restored. We introduce two new types of shares to the secret sharing scheme of Shamir. Crucial shares are always needed for the reconstruction of the secret, whereas mutual redundant shares only help once in reconstructing the secret. Further, we extend the idea of crucial and redundant shares to a compartmented secret sharing scheme. The scheme, which is based on Shamir's, allows distr…
This note proposes a method of space efficient secret sharing in which k secrets are mapped into n shares (n>=k) of the same size. Since, n can be chosen to be equal to k, the method is space efficient. This method may be compared with conventional secret sharing schemes that divide a single secret into n shares.
Threshold secret sharing schemes do not prevent any malicious behavior of the dealer or shareholders and so we need verifiable secret sharing, to detect and identify the cheaters, to achieve fair reconstruction of a secret. The problem of verifiable secret sharing is to verify the shares distributed by the dealer. A novel approach for verifiable secret sharing is presented in this paper where both the dealer and shareholders are not assumed to be honest. In this paper, we extend the term verifiable secret sharing to verify the shares, distributed by a dealer as well as shares submitted by shar…
Verifiable secret sharing (VSS) is designed to allow parties to collaborate to keep secrets. We describe here a method of fabricating false secret shares that appear to other parties to be legitimate, which can prevent assembly of the decryption key. This vulnerability affects VSS schemes using verification commitments bounded to a finite field.
In his paper Stadler develops techniques for improving the security of existing secret sharing protocols by allowing to check whether the secret shares given out by the dealer are valid. In particular, the secret sharing is executed over abelian groups. In this paper we develop similar methods over non-abelian groups.
The secret sharing schemes are the important tools in cryptography that are used as building blocks in many secured protocols. It is a method used for distributing a secret among the participants in a manner that only the threshold number of participants together can recover the secret and the remaining set of participants cannot get any information about the secret. Secret sharing schemes are absolute for storing highly sensitive and important information. In a secret sharing scheme, a secret is divided into several shares. These shares are then distributed to the participants one each and th…
Secret image sharing deals with splitting confidential images into several shares and the original image can be reconstructed from the qualified subset of the shares. Secret sharing schemes are used in transmission and storage of private medical images and military secrets. Increased confidentiality and availability are the major achievements. We propose an efficient (2, 2) scheme and (2, 3) scheme for secret image sharing. The scheme is lossless and also the share size is same as the secret size. The sharing and revealing phase uses simple modular arithmetic which can be very easily implement…
A secret sharing scheme is a cryptographic protocol to distribute a secret state in an encoded form among a group of players such that only authorized subsets of the players can reconstruct the secret. Classically, efficient secret sharing schemes have been shown to be induced by matroids. Furthermore, access structures of such schemes can be characterized by an excluded minor relation. No such relations are known for quantum secret sharing schemes. In this paper we take the first steps toward a matroidal characterization of quantum secret sharing schemes. In addition to providing a new perspe…
A rational secret sharing scheme is a game in which each party responsible for reconstructing a secret tries to maximize his utility by obtaining the secret alone. Quantum secret sharing schemes, either derived from quantum teleportation or from quantum error correcting code, do not succeed when we assume rational participants. This is because all existing quantum secret sharing schemes consider that the secret is reconstructed by a party chosen by the dealer. In this paper, for the first time, we propose a quantum secret sharing scheme which is resistant to rational parties. The proposed sche…
Data security and availability for operational use are frequently seen as conflicting goals. Research on searchable encryption and homomorphic encryption are a start, but they typically build from encryption methods that, at best, provide protections based on problems assumed to be computationally hard. By contrast, data encoding methods such as secret sharing provide information-theoretic data protections. Archives that distribute data using secret sharing can provide data protections that are resilient to malicious insiders, compromised systems, and untrusted components. In this paper, we cr…
Reversible data hiding in encrypted domain (RDH-ED) schemes based on symmetric or public key encryption are mainly applied to the security of end-to-end communication. Aimed at providing reliable technical supports for multi-party security scenarios, a separable RDH-ED scheme for secret image sharing based on Chinese remainder theorem (CRT) is presented. In the application of (t, n) secret image sharing, the image is first shared into n different shares of ciphertext. Only when not less than t shares obtained, can the original image be reconstructed. In our scheme, additional data could be emb…
A computational secret-sharing scheme is a method that enables a dealer, that has a secret, to distribute this secret among a set of parties such that a "qualified" subset of parties can efficiently reconstruct the secret while any "unqualified" subset of parties cannot efficiently learn anything about the secret. The collection of "qualified" subsets is defined by a Boolean function. It has been a major open problem to understand which (monotone) functions can be realized by a computational secret-sharing schemes. Yao suggested a method for secret-sharing for any function that has a polynomia…
Secret sharing allows split/distributed control over the secret (e.g. master key). Verifiable secret sharing (VSS) is the secret sharing extended by verification capacity. Usually verification comes at the price. We propose "free lunch", the approach that allows to overcome this inconvenience.
We present a new secret sharing algorithm that provides the storage efficiency of an Information Dispersal Algorithm (IDA) while providing perfect secret sharing. We achieve this by mixing the input message with random bytes generated using Repeatable Random Sequence Generator (RRSG). We also use the data from the RRSG to provide random polynomial evaluation points and optionally compute the polynomials on random isomorphic fields rather than a single fixed field.
We propose an adaptive threshold multi secret sharing scheme based solely on cryptographically secure hash functions. We show that the proposed scheme is also: perfect, ideal, verifiable, and proactive. Moreover the proposed scheme has a low computational complexity comparing to the most common schemes operating over finite fields.
How to construct an ideal multi-secret sharing scheme for general access structures is difficult. In this paper, we solve an open problem proposed by Spiez et al.recently [Finite Fields and Their Application, 2011(17) 329-342], namely to design an algorithm of privileged coalitions of any length if such coalitions exist. Furthermore, in terms of privileged coalitions, we show that most of the existing multi-secret sharing schemes based on Shamir threshold secret sharing are not perfect by analyzing Yang et al.'s scheme and Pang et al.'s scheme. Finally, based on the algorithm mentioned above, …
Secret sharing is a new alternative for outsourcing data in a secure way.It avoids the need for time consuming encryption decryption process and also the complexity involved in key management.The data must also be protected from untrusted cloud service providers.Secret sharing based solution provides secure information dispersal by making shares of the original data and distribute them among different servers.Data from the threshold number of servers can be used to reconstruct the original data.It is often impractical to distribute data among large number of servers.We have to achieve a trade …
A resilient secret sharing scheme is supposed to generate the secret correctly even after some shares are damaged. In this paper, we show how quantum error correcting codes can be exploited to design a resilient quantum secret sharing scheme, where a quantum state is shared among more than one parties.
Multi-secret sharing is an extension of secret sharing technique where several secrets are shared between the participants, each according to a specified access structure. The secrets can be reconstructed according to the access structure by participants using their private shares.Each participant has to hold a single share, additional information are made available in a public bulletin board.The scheme is computationally efficient and also each participant can verify the shares of the other participants and also the reconstructed secret.The scheme does not need any secure channel also.
The rising need of secret image sharing with high security has led to much advancement in lucrative exchange of important images which contain vital and confidential information. Multi secret image sharing system (MSIS) is an efficient and robust method for transmitting one or more secret images securely. In recent research, n secret images are encrypted into n or n+ 1 shared images and stored in different database servers. The decoder has to receive all n or n+1 encrypted images to reproduce the secret image. One can recover partial secret information from n-1 or fewer shared images, which po…
Secret sharing schemes based on the idea of hidden multipliers in encryption are proposed. As a platform, one can use both multiplicative groups of finite fields and groups of invertible elements of commutative rings, in particular, multiplicative groups of residue rings. We propose two versions of the secret sharing scheme and a version of ($k,n$)-thrested scheme. For a given $n$, the dealer can choose any $k.$ The main feature of the proposed schemes is that the shares of secrets are distributed once and can be used multiple times. This property distinguishes the proposed schemes from the se…
Hiding a secret is needed in many situations. Secret sharing plays an important role in protecting information from getting lost, stolen, or destroyed and has been applicable in recent years. A secret sharing scheme is a cryptographic protocol in which a dealer divides the secret into several pieces of share and one share is given to each participant. To recover the secret, the dealer requires a subset of participants called access structure. In this paper, we present a multi-secret sharing scheme over a local ring based on linear complementary dual codes using Blakley's method. We take a larg…
In this report, I explained the problem of Secret Sharing Scheme. Then based on the definition of the problem, two old methods: Blakley's Secret Sharing Scheme and Shamir's Secret Sharing are introduced. However, we explained the details of the first one since it's the topic of this work. Blakley's method has an application in distributing a key between different parties and reconstructing the key based on each share. However, this method is not efficient enough because of too large space states. Also, we tried to simulate a scenario for spreading a key between some users and tried to reconstr…
