In this paper, we provide a new approach to data encryption using generalized inverses. Encryption is based on the implementation of weighted Moore–Penrose inverse

The level of security of data stored on the cloud is primarily based on the identification of sensitive and confidential databases, and it is necessary to apply additional protection, encryption, and monitoring. It is important to consider whether it is possible to encrypt data at all levels, where they are designed, and how encryption algorithms are tested. Data encryption became of great importance in many fields including healthcare [

The major contribution of this paper regarding the issues of security and efficiency may refer to multiple different encryptions based on the random key (in form of matrix), while data encryption is based on different inversions (in this paper we have presented one of them). We present a novel method of data encryption based on matrix calculations and Weighted Moore–Penrose inverses (MP Encryption). The Moore Penrose inverses have found many applications in various areas of research. This proposed MP encryption method can be applied effectively in the encryption and decryption of images in multi-party communications.

The structure of the present paper is as follows. In the second section are exposed the similar research from the field of problems application matrices in cryptography. Also, we have presented some similar research in the field of data encryption in a cloud environment. The third section consists of the basic properties of weighted Moore–Penrose inverse (MP inverse) and presented ways to compute source matrix which can presented text or image. In the fourth section, are listed the examples for the encryption method based on weighted Moore Penrose inverse and Hermitian positive definite matrix as a cryptographic key in image encryption cases. Also, in this section, we have provided the National Institute of Standards and Technology (NIST) quality assurance tests for random generated Hermitian matrix (a total of 10 different tests and additional analysis with approximate entropy and random digression). The fifth section contains the comparative analysis of encryption methods between machine learning methods. Machine learning algorithms could be compared by achieved results of classification concentrating on classes. Sensitivity and specificity are mostly used performance measuring of complex data during classification. In this research sensitivity and specificity define achieved results of classifying Advanced Encryption Standard (AES) and MP Encryption, respectively. The sixth section lists the conclusions and suggestions for further works.

Cloud computing, a recently emerged paradigm faces major challenges in achieving the privacy of migrated data, network security, etc. Too many cryptographic technologies are raised to solve these issues based on identity, attributes, and prediction algorithms yet. These techniques are highly prone to attackers. This would raise a need for an effective encryption technique, which would ensure secure data migration [

The importance of the application of matrix computations in the encryption procedure is stated in [

For any matrix

and

Let the matrices

Then

where

Different variants of calculation of the weighted MP inverse in combination with relational databases are given in the paper [

For calculation of

The partition method of Wang for calculation of the weighting MP inverse has been extended to a set of rational and polynomial matrices with one variable [

In this case,

This is the process by which we will restore the encoded image value to the actual one.

Let

From

So,

and

This can be rewritten as,

In order to simplify the expression and calculation process, without affecting the protection degree in this way we can assume that the matrix M is given as an identity matrix from order

Because

Cloud security can be automated by a combination of a number of services available, with the goal of creating an integrated platform for monitoring, reporting, and responding to events that could compromise the security of cloud data.

The method for data encryption has four phases (see the general model framework on

Loading data (text or image)

Generation of Hermitian positive definite matrix (cryptographic key)

Converting the input data into a binary string.

Application of weighted Moore Penrose inverse in data encryption.

After converting the Base64 string into a binary record, is applied the weighted MP inverse where we use the Hermitian positive definite matrix (key), where obtained a new binary string which is converted to the Base64 string, which in this case represents the cipher of the image.

M_8 × 8 =

{{339, −87, −110, 119, 9, −41, −20, 10},

{−87, 514, −119, 10, 48, −55, −360, 45},

{−110, −119, 395, −225, −30, 81, −16, −129},

{119, 10, −225, 392, 43, −8, 180, 109},

{9, 48, −30, 43, 473, 93, −188, 90},

{−41, −55, 81, −8, 93, 552, −3, −44},

{−20, −360, −16, 180, −188, −3, 691, 0},

{10, 45, −129, 109, 90, −44, 0, 611}}.

The first phase is loading image and application of the Base64 image encoder converting of the received Base64 string into a binary string. Then, the next phase is the generation of the Hermitian positive definite matrix (in this case order 8).

The third phase is the application of the weighted Moore Penrose inverse in image encryption where we use the Hermitian positive definite matrix as key, where we get a binary string that represents the cipher of the image (see

In the reverse case it is needed base64 encode

In other cases, if we use all correct parameters, then image decoding is successful (see

In our paper [

In order for this encryption method to provide a high secrecy we used prescribed statistic NIST tests. NIST tests is applied only binary sequences. Therefore, in our testing, we first need to convert the cryptological key to binary (matrix from Example 1). The NIST quality assurance test results for randomly generated matrix (cryptographic key) are given in

NIST—Quality assurance test | 8 × 8 random matrix (2504 bits key) |
---|---|

Frequency test | P = 0.8752, success |

Block frequency test | P = 0.7851, success |

Runs test | P = 0.0128, success |

Longest runs of one’s test | P = 0.0121, success |

FFT—Fourier transform | P = 0.4877, success |

Non-periodic templates | P = 0.1893, success |

Linear complexity | P = 0.2896, success |

Serial test |
P1 = 0.0111, success; P2 = 0.0127, success |

Cumulative sums |
Forward = 0.7999, success; Reverse = 0.7985, success |

After the test, we can conclude that all tests met the condition

Input parameters | ^{2} |
||
---|---|---|---|

0.05281 | 77.002 | 0.9841 | |

0.11077 | 138.2502 | 0.9217 |

Input parameters | State (x) | Output P | Conclusion |
---|---|---|---|

ε = 1000000 bits binary extension n = 1000^{2} J = 8 × 8 = 64 × 8 = 512 bits |
−4 | 0.1789 | |

−3 | 0.1745 | ||

−2 | 0.1425 | ||

−1 | 0.0110 | ||

+1 | 0.2563 | ||

+2 | 0.1078 | ||

+3 | 0.3327 | ||

+4 | 0.1996 |

In the additional testing of the quality of the random matrix generated, we can conclude that the results of our advanced analysis (such as

Machine learning algorithms could be compared by achieved results of classification concentrating on classes. Finding classification performance is a challenging part if we use inadequate data. By comparing the means of misclassified instances, we can make a comparison between machine learning methods. Several machine learning methods are used in order to distinguish two types of ciphertexts: (1) encrypted by the AES algorithm, and (2) encrypted by the proposed encryption method based on weighted MP inverse. The basic questions of the analysis are:

Is it possible to identify the type of encryption method by machine learning models learned only from information in encrypted text?

Are there significant differences between the AES and the proposed MP encryption method?

Hence, the most commonly used measure which is not focused on different classes quantity of right labels is accuracy:

On the other hand, two measures that distinctly approximate a classifier’s presentation on diverse classes are

where are correctly classified:

Specificity and sensitivity are mostly used performance measuring of complex data during classification. In a comparative analysis, we give results of classifying of AES algorithm and encryption method based on Moore–Penrose inverse, respectively. In this study, we used two datasets which are obtained by extraction and decoding of a message in combination with different machine learning techniques. In order to produce an efficient machine learning algorithm, which will be able to satisfy all requirements, we tested both datasets on different types of machine learning methods.

In this experiment, we used both datasets in combination with different ensemble machine learning methods. The result of this experiment is given in the following tables. We apply different machine learning techniques on both datasets without any feature extraction. Results obtained in that way are presented in

ADTree | 62 | 54 | 58 | 54 | 62 | 58 |

AttributeSelectedClassifier | 18 | 96 | 57 | 96 | 18 | 57 |

Random tree | 52 | 60 | 56 | 50 | 40 | 45 |

Decision table | 14 | 96 | 55 | 84 | 14 | 49 |

MultiBoostAB (DecisionStump) | 48 | 62 | 55 | 62 | 48 | 55 |

ANN | 36 | 52 | 44 | 42 | 36 | 39 |

SVM | 36 | 42 | 39 | 42 | 38 | 40 |

As you can see from the table, the best result with an average accuracy of 58% is achieved using the

After we load data and check data distribution, we see that there is a lot of features which does not have any value for any row. So, we decided to apply some feature selection methods before we introduce classification methods. For the feature selection method, we applied

ADTree | 22 | 96 | 59 | 96 | 22 | 59 |

AttributeSelectedClassifier | 18 | 98 | 58 | 98 | 18 | 58 |

Random tree | 22 | 96 | 59 | 96 | 22 | 59 |

Decision table | 14 | 98 | 56 | 98 | 14 | 56 |

MultiBoostAB (DecisionStump) | 22 | 84 | 53 | 84 | 22 | 53 |

ANN | 28 | 84 | 56 | 84 | 28 | 56 |

SVM | 44 | 62 | 53 | 90 | 22 | 56 |

In

As you can see from the presented results, accuracy is increased for almost all machine learning methods which are applied. Some of the machine learning technologies achieved slightly better accuracy, for 1%, but for some of the methods, we achieved accuracy which is higher 14% than the previous one when we used all features. We have a similar situation in terms of accuracy distribution between classes, so again for MP Encrypt class, we have much greater accuracy in comparison with accuracy achieved for AES class.

When we use feature selected database, there are no classifiers which produce balanced accuracy for both classes. As it is obvious, all classifiers except

With the development of cloud and computer technologies, tools and software are being developed that violate the security of cloud computing resources. The layered cloud storage architecture is used primarily because different types and kinds of data may have different requirements in terms of storage. It is important to point out that there are often requirements related to encryption and data security. Mathematical systems found a wide application in encryption. The calculation of the weighted MP inverse represents one of those matrix system applications in cryptography. The basic precondition for developing of cryptologic systems with the public key is the efficient generation of a parameter which generates the key.

In this paper is presented a new idea in the form of applications matrix computations and generalized inverses in cryptography. We have provided a new way of encryption of text or images, where the whole process is based on the use of weighted MP inverse over the Hermitian positive definite matrix order 8 which presented key. The number of different combinations of the Hermitian positive definite matrices order 8 is huge so this solution represents a strong and secure key. Also, it was done performed the tests for the Hermitian positive definite matrices-keys generation through several aspects. In the experimental part of this paper, we give a comparison of encryption methods between machine learning methods. Machine learning algorithms could be compared by achieved results of classification concentrating on classes. In a comparative analysis, we give results of classifying of AES algorithm and encryption method based on Moore–Penrose inverse, respectively. Security problems are one of the most important issues related to cloud technologies. Data security and physical access to the location where the equipment was located needed to be constantly improved, as security threats to data and systems are becoming more serious day by day. Progress continues and more people are turning to these technologies because security will be improved without any doubt. Cloud computing is changing the business logic in the world. Due to the transition of the company to cloud computing, the client-server life on it will improve. Larger companies will need more time to move to cloud storage. Security issues are a big problem for them, as well as control over sensitive data.

The future of clouds will be slower in large companies as well as in large urban areas. The directions of our further development of the proposed method could refer to a connection with database management systems and matrix computations using PHP and MySQL technologies [

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