![worsening the security properties; a new computational problem, EPRP, supposed
to be harder than the DLP, has been introduced, but the derived verification
schemes, missing the homomorphic property, are not extensible to additional
shareholders, and the dealer must be a trusted entity, since any malicious behavior
of this party cannot be detected. Further research should be carried out on the
possibility of modifying the proposed problem in order to augment it with the
homomorphic property, so that a resulting VSS scheme would present shareholder
extensibility, and to investigate if this kind of problem can be also exploited in
interactive proofs for authenticating the dealer’s integrity and in publickey based
cryptosystems. Another possible direction for future work could regard
investigating additional runtime efficiency refinements. Finally, proving the NP-
hardness of EPRP by deriving a suitable poly-time reduction would result in a
substantial breakthrough in computer science.
REFERENCES
[1] A. Shamir, “How to share a secret.,” Communications of the ACM, vol. 22, no.
11, pp. 612–613, 1979.](https://image.slidesharecdn.com/space-efficientverifiablesecretsharing-150511112348-lva1-app6892/85/Space-efficient-Verifiable-Secret-Sharing-Using-Polynomial-Interpolation-10-320.jpg)
![[2] G. Blakley, “Safeguarding cryptographic keys,” in Proceedings of the 1979
AFIPS National Computer Conference, (Monval, NJ, USA), pp. 313–317, AFIPS
Press, 1979.
[3] M. Mignotte, “How to share a secret,” in Proceedings of the 1982 Conference
on Cryptography, (Berlin, Heidelberg), pp. 371–375, Springer-Verlag, 1983.
[4] C. Asmuth and J. Bloom, “A modular approach to key safeguarding,” IEEE
Trans. Inf. Theor., vol. 29, pp. 208– 10, Sept. 2006.
[5] P. Tysowski and M. Hasan, “Hybrid attribute- and reencryption- based key
management for secure and scalable mobile applications in clouds,” Cloud
Computing, IEEE Transactions on, vol. 1, pp. 172–186, July 2013.
[6] B. Chor, S. Goldwasser, S. Micali, and B. Awerbuch, “Verifiable secret sharing
and achieving simultaneity in the presence of faults,” in Foundations of Computer
Science, 1985., 26th Annual Symposium on, pp. 383–395, 1985. [7] M. Stadler,
“Publicly verifiable secret sharing,” in Advances in Cryptology EUROCRYPT 96
(U. Maurer, ed.), vol. 1070 of Lecture Notes in Computer Science, pp. 190–199,
Springer Berlin Heidelberg, 1996.
[8] E. Fujisaki and T. Okamoto, “A practical and provably secure scheme for
publicly verifiable secret sharing and its applications,” in Advances in Cryptology
EUROCRYPT’98 (K. Nyberg, ed.), vol. 1403 of Lecture Notes in Computer
Science, pp. 32–46, Springer Berlin Heidelberg, 1998.](https://image.slidesharecdn.com/space-efficientverifiablesecretsharing-150511112348-lva1-app6892/85/Space-efficient-Verifiable-Secret-Sharing-Using-Polynomial-Interpolation-11-320.jpg)
![[9] B. Schoenmakers, “A simple publicly verifiable secret sharing scheme and its
application to electronic,” in Proceedings of the 19th Annual International
Cryptology Conference on Advances in Cryptology, CRYPTO ’99, pp. 148–164,
Springer-Verlag, 1999.](https://image.slidesharecdn.com/space-efficientverifiablesecretsharing-150511112348-lva1-app6892/85/Space-efficient-Verifiable-Secret-Sharing-Using-Polynomial-Interpolation-12-320.jpg)
Space-efficient Verifiable Secret Sharing Using Polynomial Interpolation
- 1. SPACE-EFFICIENT VERIFIABLE SECRET SHARING USING POLYNOMIAL INTERPOLATION Abstract—Preserving data confidentiality in clouds is a key issue. Secret Sharing, a cryptographic primitive for the distribution of a secret among a group of n participants designed so that only subsets of shareholders of cardinality 0 < t _ n are allowed to reconstruct the secret by pooling their shares, can help mitigating and minimizing the problem. A desirable feature of Secret Sharing schemes is cheater detection, i.e. the ability to detect one or more malicious shareholders trying to reconstruct the secret by obtaining legal shares from the other shareholders while providing them with fake shares. Verifiable Secret Sharing schemes solve this problem by allowing shareholders verifying the others’ shares. We present new verification algorithms providing arbitrary secret sharing schemes with cheater detection capabilities, and prove their space efficiency with regard to other schemes appeared in the literature. We also introduce, in one of our schemes, the Exponentiating Polynomial Root Problem (EPRP), which is believed to be NP- Intermediate and therefore difficult. EXISTING SYSTEM:
- 2. In this Section, we discuss related work. We begin by reviewing commitments, and then proceed analyzing hashing, the schemes based on homomorphic commitments proposed by Feldman Pedersen and Benaloh and the set coherence verification method, introduced by Harn and Lin. A commitment is a statement that proves knowledge of some information, without revealing the information itself. A formal definition follows: Definition 2.1 (Commitment). Given a value x, a commitment c(x) is a value such that the following conditions are satisfied: • Hiding: By knowledge of c(x), it is impossible (or very difficult) to obtain x — c(x) hides x; • Binding: It is infeasible or impossible to find another value y for which c(y) = c(x) — c(x) binds to x. The two properties just defined may refer to the computational or to the unconditional security setting: if an attacker with infinite computing power can break the former or the latter, the scheme is said to be, respectively, computationally hiding or computationally binding. Otherwise, a commitment scheme is said to be unconditionally hiding or unconditionally binding. More precisely, it can be proved that a commitment scheme cannot be simultaneously
- 3. unconditionally hiding and unconditionally binding. Commitments can be implemented via one-way functions, as a basis for verification schemes. PROPOSED SYSTEM: One important issue in the design of a secret sharing protocol is its robustness against cheaters: common solutions proposed in literature rely on checking consistency of the secret information after reconstruction from more than one group of shareholders, or on adding helpful data to the shares in order to detect and/or identify mistrustful behavior. Verifiable Secret Sharing (VSS) is therefore secret sharing augmented with features that allow only detection or also identification of any cheater in a coalition, unconditionally or with respect to the scheme parameters(threshold value, total number of dishonest shareholders, etc.). Several VSS schemes have been proposed, including, for instance, Publicly Verifiable Secret Sharing (PVSS) or schemes focusing on Asynchronous Verifiable Secret Sharing (AVSS) such as. In this work, we present new verification algorithms based on commitments providing arbitrary secret sharing schemes with cheater detection capabilities, and prove their data efficiency with regard to other schemes appeared in the literature. Our approach belongs to the Honest-Dealer VSS scheme category, since it requires a one-time honest dealer.
- 4. Our contribution is three-fold: (i) we present space efficient verification protocols that does not even require storing public data for verification; (ii) our schemes can be used in conjunction with arbitrary secret sharing schemes, and provide cheater detection capabilities; (iii) we also introduce, in one of our schemes, a new computational problem, namely the Exponentiating Polynomial Root Problem (EPRP), which generalizes the Discrete Logarithm Problem (DLP). Module 1 Privacy and Security in cloud This section addresses the core theme of this chapter, i.e., the security and privacy- related challenges in cloud computing. There are numerous security issues for cloud computing as it encompasses many technologies including networks, databases, operating systems, virtualization, resource scheduling, transaction management, load balancing, concurrency control and memory management. Therefore, security issues for many of these systems and technologies are applicable to cloud computing. For example, the network that interconnects the systems in a cloud has to be secure. Furthermore, virtualization paradigm in cloud computing leads to several security concerns. For example, mapping the virtual machines to the physical machines has to be carried out securely. Data security
- 5. involves encrypting the data as well as ensuring that appropriate policies are enforced for data sharing. In addition, resource allocation and memory management algorithms have to be secure. Finally, data mining techniques may be applicable for malware detection in the clouds – an approach which is usually adopted in intrusion detection systems (IDSs) (Sen & Sengupta, 2005; Sen et al., 2006b; Sen et al., 2008; Sen, 2010a; Sen, 2010b; Sen 2010c). There are six specific areas of the cloud computing environment where equipment and software require substantial security attention (Trusted Computing Group’s White Paper, 2010). These six areas are: (1) security of data at rest, (2) security of data in transit, (3) authentication of users/applications/ processes, (4) robust separation between data belonging to different customers, (5) cloud legal and regulatory issues, and (6) incident response. For securing data at rest, cryptographic encryption mechanisms are certainly the best options. The hard drive manufacturers are now shipping self- encrypting drives that implement trusted storage standards of the trusted computing group (Trusted Computing Group’s White Paper, 2010). These self- encrypting drives build encryption hardware into the drive, providing automated encryption with minimal cost or performance impact. Although software encryption can also be used for protecting data, it makes the process slower and
- 6. less secure since it may be possible for an adversary to steal the encryption key from the machine without being detected. Module 2 Space-efficientverifiability In this Section, we introduce our construction of a new verification method for threshold secret sharing. It is not designed for a particular scheme, nor does it require any assumption on the shares. The designed verification algorithm is non- interactive (verification does not require receiving additional data from other shareholders, besides the shares), requires a one-time honest dealer, and belongs to the family of commitment-based methods, since it relies on one way functions. It will be shown that, under certain hypotheses, it is more space-efficient than the already illustrated homomorphic VSS extensions. Definitions Notations related to mathematical and string operators are listed below. The following convention will be used: any operator defined for a bitstring is valid for an unsigned integer type, and vice-versa. Module 3
- 7. Designing a space-efficientVSS extension The verification scheme that is going to be designed will be the result of incremental refinements of partially secure techniques. The main goal to achieve during the design will be the reduction of verification data. Labels of the form VSS- X will be used to better identify and distinguish the variants obtained. Moreover, since the final result is a commitment scheme, the security analysis will develop around the two security properties of hiding and binding. Module 4 Security assumptions • There is a single, one-time, honest dealer, that distributes data to all of the n shareholders involved in the scheme instance; • There is no trusted shareholder in the underlying network, and no storage of shared or public data. That is, once provided with their shares and verification data, shareholders do not need any other information for secret reconstruction and cheater identification;
- 8. • Secure bidirectional channels can be established between pairs of entities - any external attacker can only be passive, so man-in-the-middle attacks are not considered in this model; security against these kinds of attack is assumed to be addressed by the protocols that establish communication between the parties over a network (e.g., TLS or SSL); • Client machines are fully trusted. All of the entities (the dealer and the shareholders) run their respective protocol steps on their client machines where keys and certificates required for encryption/ decryption and authentication are stored. If a CSP (Cloud Service Provider) has to be used for share storage, shareholders encrypt their shares using a symmetric cipher before uploading them. Similarly, shareholders download shares from CSPs to their clients and decrypt them before engaging in secret reconstruction and cheater identification; • CSPs are semi-trusted and modeled as Honest- But-Curious adversaries. Therefore, they act according to their prescribed actions in all of the protocols they are involved in (they do not, as malicious users do, try to alter stored data and communications), but it is assumed that CSPs are interested in learning the contents of shares stored by shareholders, and can fully access everything stored on their cloud storage infrastructure. Module 5
- 9. Designfeatures The main features design attempts will insist on, are summarized below: • Commitments on shares: Verification routines ensure that shares are legal independently from the secret they are generated from– unlike homomorphic commitment schemes, that guarantees that a share correspondsto some secret; • Non-interactivity: Verification algorithms can be carried out in one interaction, that is, no further communication with other parties is required after receiving the shares; • Private verification: each shareholder is able to verify the others’ shares, but not its one: this is not necessary since this interaction model assumes a one-time honest dealer; moreover, verification is performed differently by each shareholder – by taking as additional input a secret parameter. CONCLUSIONS We have presented new verification schemes enhancing arbitrary secret sharing schemes by adding cheater detection capabilities. Our main effort was devoted to reducing the amount of verification data for a secret sharing scheme without
- 10. worsening the security properties; a new computational problem, EPRP, supposed to be harder than the DLP, has been introduced, but the derived verification schemes, missing the homomorphic property, are not extensible to additional shareholders, and the dealer must be a trusted entity, since any malicious behavior of this party cannot be detected. Further research should be carried out on the possibility of modifying the proposed problem in order to augment it with the homomorphic property, so that a resulting VSS scheme would present shareholder extensibility, and to investigate if this kind of problem can be also exploited in interactive proofs for authenticating the dealer’s integrity and in publickey based cryptosystems. Another possible direction for future work could regard investigating additional runtime efficiency refinements. Finally, proving the NP- hardness of EPRP by deriving a suitable poly-time reduction would result in a substantial breakthrough in computer science. REFERENCES [1] A. Shamir, “How to share a secret.,” Communications of the ACM, vol. 22, no. 11, pp. 612–613, 1979.
- 11. [2] G. Blakley, “Safeguarding cryptographic keys,” in Proceedings of the 1979 AFIPS National Computer Conference, (Monval, NJ, USA), pp. 313–317, AFIPS Press, 1979. [3] M. Mignotte, “How to share a secret,” in Proceedings of the 1982 Conference on Cryptography, (Berlin, Heidelberg), pp. 371–375, Springer-Verlag, 1983. [4] C. Asmuth and J. Bloom, “A modular approach to key safeguarding,” IEEE Trans. Inf. Theor., vol. 29, pp. 208– 10, Sept. 2006. [5] P. Tysowski and M. Hasan, “Hybrid attribute- and reencryption- based key management for secure and scalable mobile applications in clouds,” Cloud Computing, IEEE Transactions on, vol. 1, pp. 172–186, July 2013. [6] B. Chor, S. Goldwasser, S. Micali, and B. Awerbuch, “Verifiable secret sharing and achieving simultaneity in the presence of faults,” in Foundations of Computer Science, 1985., 26th Annual Symposium on, pp. 383–395, 1985. [7] M. Stadler, “Publicly verifiable secret sharing,” in Advances in Cryptology EUROCRYPT 96 (U. Maurer, ed.), vol. 1070 of Lecture Notes in Computer Science, pp. 190–199, Springer Berlin Heidelberg, 1996. [8] E. Fujisaki and T. Okamoto, “A practical and provably secure scheme for publicly verifiable secret sharing and its applications,” in Advances in Cryptology EUROCRYPT’98 (K. Nyberg, ed.), vol. 1403 of Lecture Notes in Computer Science, pp. 32–46, Springer Berlin Heidelberg, 1998.
- 12. [9] B. Schoenmakers, “A simple publicly verifiable secret sharing scheme and its application to electronic,” in Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology, CRYPTO ’99, pp. 148–164, Springer-Verlag, 1999.