AN EFFICIENT THRESHOLD CRYPTOGRAPHY SCHEME FOR CLOUD ERP DATA

International Journal on Cryptography and Information Security (IJCIS), Vol. 10, No.1, March 2020
DOI:10.5121/ijcis.2020.10101 1
AN EFFICIENT THRESHOLD CRYPTOGRAPHY
SCHEME FOR CLOUD ERP DATA
Arnold Mashud Abukari1
, Edem Kwedzo Bankas2
and
Mohammed Iddrisu Muniru3
1
Computer Science Department, Tamale Technical University, Tamale, Ghana
2
Computer Science Department, University for Development Studies, Navrongo, Ghana
3
Mathematics Department, University for Development Studies, Navrongo, Ghana
ABSTRACT
Cloud ERP is a new concept that has caught the attention of researchers and industry players. One of the
major challenges towards migrating ERP data to cloud is security and confidentiality. A number of secret
sharing schemes (SSS) have been proposed Mignote, Asmuth-Bloom, Shamir and Blakley but not without
challenges. In this research we proposed an efficient threshold cryptography scheme to secure cloud ERP
data. We modified the Asmuth-Bloom SSS and applied it to cloud ERP data.
KEYWORDS
Threshold Cryptography, Secret Sharing Scheme (SSS), CRT, Cloud ERP.
1. INTRODUCTION
Threshold Cryptography is a concept for sharing very sensitive data among a group of users. In
our case, the group of users will be referred to as a group of n Cloud Enterprise Resource
Planning (ERP) providers. Researchers over the years have succeeded in proposing Secret
Sharing Schemes (SSS) including Asmuth-Bloom SSS [1] which is based on the Chinese
Remainder Theorem (CRT). In [1] special sequence of integers and sequence of pairwise integers
were used to share a randomly chosen secrets to different users. The proposed scheme in [1]
failed to handle data redundancy and not applicable to cloud. Shamir [2] also developed a SSS
based on polynomial interpolation. In 1983, Mignotte [3] proposed a weighted SSS based on the
CRT.
The scheme in [3] allowed the distribution of load among server (clouds) based on existing
resources and the choice of a set of pairwise coprimes. In threshold cryptography, the most
important requirement is the availability of a computable function without compelling parties to
disclose their secret shares. This concept is known as function sharing problem and requires each
part of the computation is carried out by different users without disclosing their secrets to each
other. Several proposed schemes can be found in literature [[4],[5],[6],[7]] but most of them are
built from Shamir [2] published in 1979. The concept of cloud computing has spread very fast
and caught the attention of researchers and organisations due to the security implications on big
data [8]. One of the new emerging solutions for cloud providers is Cloud ERP [9]. Cloud ERP,
just like the on-premise ERP generates and store highly sensitive data and thus requires maximum
data protection from hackers and cloud providers. ERP data stored in cloud uses computational
resources provided by the cloud provider and this raises questions on confidentiality, reliability
and safety of the Cloud ERP data. In this research, we propose Threshold Cryptography (TC)
scheme to help reduce redundancy and enhance the security of Cloud ERP data. To ensure
confidentiality and integrity of cloud ERP data, our TC Scheme is applicable.

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2. THRESHOLD CRYPTOGRAPHY
A cryptosystem is referred to as a threshold cryptosystem, if it requires several parties usually
more than some threshold number in order to decrypt an encrypted message or data[6]. The
security operations such as encryption, decryption, signature generation and verification can be
performed by a group of processes without reconstructing the shared secrets.
Considering a general problem of t-out-of-n secret sharing in threshold cryptography, and we
have n-parties to share x = (x1; : : : ; xn), two main important properties needs to be guaranteed.
These properties are: Recoverability: Given any t-shares, we can recover x Secrecy: Given any ¡t,
x can never be recovered
3. SECRET SHARING SCHEME
Secret Sharing Scheme is applied to build a storage system using n-number of clouds
independently and separately so that any of the clouds can recover their respective data without
knowing the secret shares of the others. The shared data can only make meaning when a certain
number of the cloud providers conspire. In this research, we examined the Asmuth-Bloom
scheme [1], Shamir [2] and Mignotte’s scheme [3] to see how it can be applied to Cloud ERP
Data. We propose Threshold Cryptography (TC) scheme to help reduce redundancy and enhance
the security of Cloud ERP data.
3.1. Shamir Secret Sharing Scheme
The first scheme for sharing a secret was proposed by Shamir (Shamir, 1979) based on
polynomial interpolation.
To obtain a (t, n) secret sharing, a random polynomial
is generated over where p is a prime
number and a0 = d is the secret.
The share of the ith party is . If t or more parties come together, they can
construct the polynomial by Lagrange interpolation and obtain the secret, but any smaller
coalitions cannot.
3.2. Blakley Secret Sharing Scheme
Blakley gave a general construction of a threshold scheme using projective geometry.
Construction.
• i. In PG (t, q), randomly select a hyperplane X to be the secret.
• ii. Choose n points in the hyperplane that are in general position in X. Hence, any t points
form a basis for X.
• iii. To reconstruct the secret, we simple take the span U of the participating points and see
if U = X

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3.3. Asmuth-Bloom Scheme
The Asmuth-Bloom scheme uses special sequence of integers and pairwise relatively prime
integers p0[1]. In this scheme,p1< p2 < : : : <pn are selected such that:
Where k is the number of distinct shares. A secret S is chosen as a random element of the set Zp0
Let the shares Ii be chosen as Ii=(S+.p0) mod pi, 8 1 _ i _ n where is an arbitrary integer or a
positive integer generated randomly such that S+.p0 2 Zpi;: : : pk . The Asmuth-Bloom scheme
assumes four conditions as follows:
The Asmuth-Bloom scheme requires n-times the computational resources for data storage for data
like Cloud ERP data[1]. Given k distinct shares Ii1, Ii2,. . . ,Iik, the secret S can be obtained since
S=x0 modp0, where x0 can be calculated using standard variant of the Chinese Remainder
Theorem with a unique modulo pi1; : : : ; pik of the system.
To make the scheme become asymptotically ideal SSS and also to secure the confidentiality of
cloud ERP data incase there is cloud conspiracy, we introduce constraints and compactness to the
Asmuth-Bloom Scheme such that p0 < p1 < p2; ::; pn< 2p0.
4. PROBLEM STATEMENT
The concept of Threshold Cryptography in the application to cloud computing has not received
adequate research after Shamir, Blakley, Asmuth-Bloom and Mignotte made strides in applying it
to secret sharing. The desire of organisations for centralised, accurate and timely information that
will help in analysing data in order to make strong strategic decisions and gain a favourable
competitive advantage has given birth to Cloud Enterprise Resource Planning (ERP). The size of
encrypted ERP Data in cloud, the execution time and memory usage during encryption and
decryption deserves much attention and improvement and has been the focus of our research.

4
5. PROPOSED SCHEME
In this research we propose a new Scheme based on CRT to help reduce the execution time for
encryption and decryption as well as memory usage. Our scheme also reduces the size of
encrypted data. We propose a Scheme to provide security for cloud ERP data and also to
significantly reduce the redundancy of Cloud ERP Data.
Considering our modification on the four conditions that needs to be satisfied to fulfill the
Asmuth-Bloom secret sharing scheme as stated in (Asmuth and Bloom, 1983), we propose a new
Scheme to provide security for cloud ERP data and also to significantly reduce the redundancy of
Cloud ERP Data when the Mignotte scheme (Mignotte, 1983) is applied. The proposed scheme
also applied the Asmuth-Bloom scheme to prove its threshold cryptographic properties. We
considered the following notations in building the our Scheme: D – Cloud ERP Data, p1; p2; p3; :
: : ; pn being pairwise coprime numbers, p0 is an arbitrary integer also called the PROPOSED
Key that corresponds with the above coprime
We considered the following notations in building the Scheme:
• D – Cloud ERP Data
• p1,p2,p3,…,pn being pairwise coprime numbers, p0 is an arbitrary integer also called the
PROPOSED Key that corresponds with the above coprime numbers.
• We propose that the Cloud ERP data be stored among n number of cloud service
providers.
We propose that
Where is the PROPOSED key, t is the threshold for the cloud providers in order to recover the
secret or ERP data.
To calculate the ERP data shares for each cloud service provider we use:
,
C is the cloud service providers shares, D is the data or secret, is the PROPOSED key and is
an arbitrary integer randomly generated.
5.1. Conditions For Our Scheme
The conditions that needs to be satisfied by our scheme are:
•
•
• p0<p1<p2<…<pn<2p0 (Constraint compactness)
•
The condition four(4) is necessary to confirm the threshold property of our scheme needed for
secret sharing schemes.

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5.2. Proposed Algorithms

6
5.3. Cloud Erp Data Redundancy With Our Scheme
Considering po being the PROPOSED key, pi being the ERP data shares based on coprime
randomly selected numbers and D is the ERP data then the redundancy of Cloud ERP data based
on our proposed scheme is defined as follows per (Mignotte, 1983):
Where n is the number of cloud service providers.
6. RESULTS AND DISCUSSION
We used a pdf file of size 14,752,894bytes (14mb) and the number of cloud service providers n to
be 5 whiles the k-value is 3. These common parameters were used for all three Secret Sharing
Schemes namely Shamir SSS, Blakley SSS and our Scheme.
Figure 1: Parameters for testing
6.1. Encryption Execution Time Analysis
Despite no differences in memory usage compared with Blakley and Shamir , our scheme uses
lesser time during the encryption process and thus per the simulation results is faster during
encryption compared to the Shamir, Blakley and Asmuth-Bloom Schemes. Faster execution time
means less computational resources are used during the encryption process. This lesser time for
encryption will compensate for the high memory usage since the memory will be released within
the shortest time frame compared to others.

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Figure 2: Execution Time Analysis
6.2. Encryption Memory Usage Analysis
Our scheme performed equally in the memory used in the encryption process as Blakley and
Shamir but performed better than Asmuth-Bloom scheme. The simulation results averagely
suggests our Scheme is performing well in memory space just as the other schemes.
Figure 3: Encryption Memory Usage Analysis
6.3. Size On Disk After Encryption Analysis
Encrypted Data size after the encryption process was also investigated by comparing all the three
schemes. The simulation results averagely suggests that our scheme performs better in terms of
the size of encrypted data. Comparing the sizes of the encrypted data from the schemes reveals
the size of the our encrypted data is relatively smaller compared to others. This suggests that our
scheme improves the efficient use of space on the servers.

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Figure 4: Size on Disk Analysis
6.4. Decryption Execution Time
In the decryption process, Our scheme performs better in memory usage and time of execution.
This suggests that our scheme saves time and computational resources during the decryption
processes.
Figure 5: Decryption Execution Time Analysis

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7. CONCLUSION
In this research work, we propose a new secret sharing scheme which is based on the Chinese
Remainder Theorem (CRT). The main objective behind our scheme is to help avoid cloud
collusion using threshold cryptography and to reduce the execution time and memory usage as
well reduce the size of encrypted data in cloud. The PROPOSED scheme is an improvement to
the Shamir, Blakley and Mignotte schemes since it reduces the exection time and computational
resources. The test runs results show an improved execution time and less computational
resources when our scheme is applied
REFERENCES
[1] C. Asmuth and J. Bloom, “A modular approach to key safeguarding,” IEEE Transactions on
Information Theory, vol. 29, no. 2, pp. 208–210, March 1983.
[2] A. Shamir, “How to share a secret,” Commun. ACM, vol. 22, no. 11, pp. 612–613, Nov. 1979.
[Online]. Available: http://doi.acm.org/10.1145/359168.359176
[3] M. Mignotte, “How to share a secret,” in Cryptography, T. Beth, Ed. Berlin, Heidelberg: Springer
Berlin Heidelberg, 1983, pp. 371–375.
[4] A. De Santis, Y. Desmedt, Y. Frankel, and M. Yung, “How to share a function securely,” in
Proceedings of the Twenty-sixth Annual ACM Symposium on Theory of Computing, ser. STOC ’94.
New York, NY, USA: ACM, 1994, pp. 522–533. [Online]. Available:
http://doi.acm.org/10.1145/195058.195405
[5] H.-F. Huang and C.-C.Chang, “A novel efficient (t, n) threshold proxy signature scheme,”
Information sciences, vol. 176, no. 10, pp. 1338–1349, 2006.
[6] Y. Desmedt, “Some recent research aspects of threshold cryptography,” in International Workshop on
Information Security. Springer, 1997, pp. 158–173.
[7] V. Shoup, “Practical threshold signatures,” in International Conference on the Theory and
Applications of Cryptographic Techniques. Springer, 2000, pp. 207–220.
[8] C. P. Chen and C.-Y. Zhang, “Data-intensive applications, challenges, techniques and technologies: A
survey on big data,” Information Sciences, vol. 275, pp. 314–347, 2014.
[9] C. Songsheng and Y. Peipei, “Economic benefits of enterprise resources planning (erp)-based on
empirical evidence from chinese listed companies,” in Logistics Systems and Intelligent Management,
2010 International Conference on, vol. 3. IEEE, 2010, pp. 1305–1308.

AN EFFICIENT THRESHOLD CRYPTOGRAPHY SCHEME FOR CLOUD ERP DATA

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