A Hybrid Improved Zhou and Wornell’s inspired Fully Homomorphic Encryption Scheme for Securing Big Data Computation in Cloud Environment

The process of performing smart computations in the big data and cloud computing environment is considered to be highly essential in spite of its complexity and cost. The method of Fully Homomorphic encryption is considered to be the effective approach that provides the option of working with the encrypted form of sensitive data in order to preserve high confidentiality that concentrates on deriving benefits from cloud computing capabilities. In this paper, a Hybrid Improved Zhou and Wornell’s inspired Fully Homomorphic Encryption (HIZWFHE) Scheme is proposed for securing big data computation, when they are outsourced to cloud service. This HIZWFHE scheme is potent in encrypting integer vectors that permit the computation of big data represented in the contextual polynomial form in the encrypted form with a bounded degree of limits. This HIZWFHE scheme is determined to be highly applicable and suitable and applicable in cloud big data computation in which the learning process of low dimensional representations is of high concern.


Introduction
The cloud computing is considered as a successful computing model with a considerable number of merits facilitated to the providers and clients [1]. This cloud computing possesses a dramatic advantage of delegating the complex bog data computations by deriving the benefits of optimal technologies that provides maximum computation power under lowest expensive [2]. This cost merits facilitated by the cloud computing is determined as the major statements that provides the justification behind the utilization of it in a diversified number of industries [3]. However, the emergence of security issues under the process of managing the computation of data is considered as the major challenge [4]. In the recent decade, a number of research works were contributed to secure the data stored in the cloud [6][7]. However, the fully homomorphic encryption schemes are considered to be vital in ensuring maximum security without the knowledge of data and computation function [8]. Most of the fully homomorphic encryption schemes are considered are affected by a vulnerability in the secret key [9].Further, the noise free-based fully homomorphic encryption schemes are considered to be highly suitable and applicable to superior cloud data computation with reduced time [10].
In this paper, a Hybrid Improved Zhou and Wornell's inspired Fully Homomorphic Encryption (HIZWFHE) Scheme is proposed for facilitating superior security under big data computation. This proposed HIZWFHE scheme is also potent in encrypting integer vectors that permit the computation of big data that are represented in the bit vector form with the threshold degree of limits.The secrecy degree of the proposed HIZWFHE scheme is investigated for identifying its potential in securing big data computations in the cloud environment.The experimental investigations of the proposed HIZWFHE scheme was also conducted using time incurred in single operations(seconds), the time incurred per operations for a single bit of encrypted data (seconds), percentage increase in data retrieval, percentage increase in response time, percentage increase in Unmask performance time and percentage decrease in the memory consumption rate under a varying number of dimensions.

Related Work
In this section, the most recent approaches of the fully homomorphic encryption schemes are presented with the merits and limitations Initially, Kim et al. [11] has presented hybrid data mining based approach for identifying DDoS attacks and to mitigate it by various means. This method comprises of two different modules automated selection module and classification module. The hybridization of data mining capability is due to crucial data flow in the DDoS attack detection. The gathering and utilization of the data flow is performed based on the Netflow. Scherrer et al. [12] in 2014 recommended a method in order to extract features of the DDoS attack and based on the features how the detection and mitigation of DDoS. This Polly Cracker-based FHE is integrated with the merits of Polly Cracker for attaining superior performance in computation with a reduction in time. This Polly Crackerbased FHE is determined to secret in nature. But, the memory consumptions of this Polly Crackerbased FHE is comparatively low compared to the Gentrys' Scheme proposed for Homomorphic encryption. Further, Lee et al. [13] in 2008 implemented a method which acts as aproactive measure for DDoS attacks. This proactive protocol has been proposed based on the distributed architecture and also based on selecting various variable related to the attack occurring features. Cluster based mechanism has been presented for mitigating DDoS attacks. Nguyen and Choi [14] in 2016 have presented a method which aids in identification of DDoS attacks based on the network conditional parameters. With reference to the key features the key variable are chosen and based on thenearest neighbor method the network conditions and performance are classified. The unmask performance time of this Non-Deterministic FHE scheme was determined to be maximum to a degree of 93% with reduced overhead in computations.
Tsai and Lin [15] introduced a method for mitigating DDoS which is referred as Triangle Area Based Nearest Approach. The accuracy of this approach is evaluated based on the false negative and false positive rate. Bhange et al. [16] illustrated a novel methodology to mitigate DDoS attack and its influence in the network performance. The distributed network traffic is analyzed based on the parameters of the network. This analysis is performed in order to identify the difference in the anomaly traffic and normal traffic. Tan et al. in 2014 [17] recommended more comfortable and efficient approach for detecting and isolation of DDoS attack in the networking environment. This novel detection scheme is based on the MCA in order to protect from DDoS attacks. This modified Gentrys' scheme utilized 0.47 seconds of computation time for the user and 0.1 seconds for the server. But, the memory consumption rate of this modified Gentrys' scheme is only half, leaving the remaining memory cycles idle during the process of data computations in the clouds.
Luo et al. [18] has developed a mathematical model based estimation for the mitigation of DDoS attacks for the networking environment. This method results in an efficient and effective way of detecting and mitigating DDoS attacks. This Ghostshell-based FHE is effective in the process of SIMD operations exploitation with a single time chosen MAC identifier. This Ghostshell-based FHE incorporated the multiplication depth of 2 with the benefits of Hamming distance that supports 2400 bit of data. The unmask performance time, memory consumptions of this RN-FHE was also estimated to be 23% lower than the DORACS-FHE scheme. In addition, a FHE that focuses on the provision of security against Adaptive Chosen Ciphertext Attack (CCA) was proposed for handling the issues that are the most common in cloud storage, cloud computation, electronic voting and multi-party computation [19]. This FHE scheme included the property of residual classes by imposing the composite degree of modular arithmetic with determined superior security. This proposed DORACS-FHE improves the unmask performance time, memory consumptions and time consumptions with increased scalability of data used for computations. However, the data retrieval rate is determined is comparatively 19% lower than the existing approaches of the literature.

Encryption (HIZWFHE) Scheme
The Hybrid Improved Zhou and Wornell's inspired Fully Homomorphic Encryption (HIZWFHE) Scheme for facilitating efficient security over the big data computations is proposed for permitting the possible computations over the encrypted data. This HIZWFHE Scheme aids in sustaining the function secrecy in big data computations of cloud environment by encrypting the data and the computation function, which enables the cloud to compute the function without the knowledge of data and computation function. This method of computing the function without the knowledge of data and computation function is contradictory to the existing Gentry-based necessitate the computation function to be made available in the unencrypted form over the clouds, since the computing requirements need to be transformed into a circuit.
Initially, this proposed HIZWFHE Scheme computes a ciphertext vector n V Z C  from the plaintext vector and a secret key ( Where e and w represents the error term and a large integer that possess elements that are smaller than 2 w . In this proposed HIZWFHE Scheme, the size of the secret key is considered to be very small compared to the utilized large integer ). This consideration is determined to be vital in maintaining the error terms as small as possible in order to apply possible operations in the encrypted field. The decryption of the ciphertext vector is determined in a straightforward manner with the knowledge of the secret key based on Equation (2) ) ( Further, the vital concept that is used for performing the most significant operations in the encrypted field is the key switching mechanism. The process of key switching process aid in choosing a new secret key In this context, wJ is the secret key and plain text with the zero error term. Then, the process of key switching is employed on wJ to determine the new secret key and its associated ciphertext

Key switching mechanisms of the proposed HIZWFHE Scheme
In this key switching process, the original secret key and ciphertext pairs are transformed into a modified secret key and ciphertext pairs with the initially considered secret key still used for the process of encrypting the original plaintext [19]. This process of key switching is efficient for simplifying the implementation and investigation of the operations that are feasible in the cloud This process of key switching is partitioned into two steps which converts the original cipher text and secret key into an intermediate form in the first step. Further, the switching process of bit representation is facilitated for determined the required secret key in the second key. Furthermore, Step 1: Select a value r such that Further, a matrix representing derived from the original secret key, which is represented in a vector highlighted in Equation (6) ] . Then, the value of Step 2: In this step, the process of converting the bit representation of the original is transformed into a new secret key-cipher text pair. Then, a new key switch matrix pair is constructed by satisfying the condition specified in Equation (7

Weighted inner products of the proposed HIZWFHE Scheme
In this step, the process of weighted inner product is determined by considering this operation as the product of matrix-vector. If This process of weighted inner product aids in improving the dimensions of the ciphertext from order n to order 2 n . However, the dimensions are brought down to a manageable level by the utilization of the key switching process used before this step.

Polynomial generations of the proposed HIZWFHE Scheme
Finally, the arbitrary polynomials can be produced after the incorporation of three primitive operations that corresponds to addition, linear transformation and weighted inner product. The weighted inner product operation [20] is considered as the key component during the process of synthesizing polynomials in order to employ a slight enhancement for inheriting its inhomogeneous degree. In this context, the plaintext vector Under the secret key Then, the method of the inner product is applied for calculating the polynomial of degree 2 based on Equation (10) ) ( ) ( Where G is the appropriately chosen for working with the maximum lowest enhancement in the existing data. For instance, the second order polynomial can be represented as the weighted inner product of

Secrecy Investigation of the proposed HIZWFHE Scheme.
In this section, secrecy inherent to the estimation of computation function of the cloud data (the function which is sent to the clouds by the collaborating clients) is investigated, since it is considered as the indispensable characteristics of the homomorphic encryption approach. In this context, the enforcement of encryption over the computation function ' g ' is represented as a key switch matrix SM K . This key switch matrix SM K is considered as the data, which is sent to the cloud by the interacting clients. At this juncture, the matrix SM K must be capable enough for hiding any significant information associated weighed inner product coefficient or the linear transformation coefficient. Hence, the secrecy of the proposed HIZWFHE Scheme depends on the potential in handling the issue of Learning with errors. This proposed HIZWFHE Scheme is considered to more secret and hard, if it is possible to estimate the equation If suppose, the elements in the matrices are considered from the set of value Z , then the elements of p Z for some prime number can be defined. Thus the consideration of special case under the condition can be translated into Equation (11) e s k s Hence, it is accessible to 'n' samples of ( Thus, the issue of Learning with errors is made hard since solving Equations (11) and (12) are equivalently hard. It is highly difficult to recover the new secret key  ) ( x V s based on the estimated key switch matrix SM K . Hence, the malicious adversaries that intercept the communication with the cloud and the and the cloud itself cannot recover the secret key, plaintext, G and H corresponding to the weighted inner products and linear transformation processes. Therefore, the proposed HIZWFHE Scheme is determined to be highly secret.

Experimental Results and Discussion
In this section, the proposed HIZWFHE Scheme is investigated with the existing GENTRY-FHE [20], SMART-VEC [19] and BATCH-DGHV [18] fully homomorphic encryption schemes    Funding This research work has not received any funding from any organization.

Data Availability Statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of interest
The authors declare that ther is no conflict of interest.