SVD‐based image compression, encryption, and identity authentication algorithm on cloud

IET Image ProcessingVolume 13, Issue 12 p. 2224-2232 Research ArticleFree Access SVD-based image compression, encryption, and identity authentication algorithm on cloud Changzhi Yu, Changzhi Yu University of Jinan, Jinan City, Shandong Province, People's Republic of ChinaSearch for more papers by this authorHengjian Li, Corresponding Author Hengjian Li ise_lihj@ujn.edu.cn University of Jinan, Jinan City, Shandong Province, People's Republic of China Department of Computer Science and Technology Shandong Provincial Key Laboratory of Network based Intelligent Computing University of Jinan, Jinan, People's Republic of ChinaSearch for more papers by this authorXiyu Wang, Xiyu Wang University of Jinan, Jinan City, Shandong Province, People's Republic of ChinaSearch for more papers by this author Changzhi Yu, Changzhi Yu University of Jinan, Jinan City, Shandong Province, People's Republic of ChinaSearch for more papers by this authorHengjian Li, Corresponding Author Hengjian Li ise_lihj@ujn.edu.cn University of Jinan, Jinan City, Shandong Province, People's Republic of China Department of Computer Science and Technology Shandong Provincial Key Laboratory of Network based Intelligent Computing University of Jinan, Jinan, People's Republic of ChinaSearch for more papers by this authorXiyu Wang, Xiyu Wang University of Jinan, Jinan City, Shandong Province, People's Republic of ChinaSearch for more papers by this author First published: 17 September 2019 https://doi.org/10.1049/iet-ipr.2018.5912Citations: 12AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Based on singular value decomposition (SVD), an image compression, encryption, and identity authentication scheme is proposed here. This scheme can not only encrypt image data which would store in the cloud but also implement identity authentication. The authors use the SVD to decompose the image data into three parts: the left singular value matrix, the right singular value matrix, and the singular value matrix. The left singular value matrix and right singular value matrix are not as important as the singular value matrix. They propose a logistic-tent-sine chaotic system to encrypt them. In this scheme, they proposed a novel authentication value calculation algorithm, which can calculate the authentication value according to related data. According to the authentication value calculated from the ciphertext, the algorithm has the perfect authentication performance, so as in the scenarios if the image is cropped or added noisy. Theoretical analysis and empirical evaluations show that the proposed system can achieve better compression performance, satisfactory security performance, and low computational complexity. 1 Introduction As a kind of important media for information exchange among people, digital images have wide applications in many fields. Especially for the aspect of privacy image, the confidentiality and security of information have become more and more important in the cloud [1]. Due to the increasing popularity of mobile devices, the number of digital images are increasing now, and simple small-capacity storage devices are not able to meet the storage needs of users [2]. Summarising the above situation, how to store more digital images while ensuring data security has become a challenge. Cloud outsourcing services are not only cheap but also have a lot of storage space. Users can store their data and download it from the cloud when they need it. Although cloud storage services have so many advantages [3-5], its security still needs to be strengthened. In 2014, Apple iCloud service appears accidently that the user's privacy image was leaked. In fact, for the users, the security of cloud storage is guaranteed by ID and login password. Once the account and password is compromised, the personal privacy data stored by the user in the cloud will have no security at all. In addition, there are many storage security vulnerabilities in the cloud [6, 7]. Cloud computing security has attracted more and more scholars' attention during recent years [8-10]. Parakh and Kak [11] proposed a secure way to assign a secret implicit data based on K-thresholding, which does not require encryption. In 2010, Wang et al. [12] proposed a scheme for using trusted third party to audit cloud-computing data. This kind of auditing scheme only needs to verify the data users, and this method does not require the data provider to provide raw data, thus which can prevent cloud computing data from being tampered by cloud computing service providers and protect data confidentiality and data integrity. J Tharaud et al. [13] adopted digital watermarking technology and encryption protocol to protect the privacy of data by enhancing the control of data usage information. Itani W et al. [14] studied the problem of privacy protection in cloud computing environment and concluded that privacy protection should also be provided to users as cloud services. In 2017, Xiao Di et al. [15] proposed privacy cloud protection and identity authentication based on compressive sensing, which can achieve personal identity authentication on the cloud. However, the authentication cannot distinguish a specific user, and its authentication results will be only two cases where there exists watermark in the image or not. Image content protection techniques with authenticity and credibility are mainly divided into two categories: digital image watermarking and image hashing [16]. The digital watermarking techniques are utilised to detect whether an image has been tampered with or not. It can also detect and localise the tampered areas from an image. An image authentication scheme based on singular value decomposition (SVD), embedding the watermark by computing the singular values, is proposed [17, 18]. However, these fragile watermarking schemes are mainly used for image tamper detection. Also, the Gaussian–Hermite moments are employed for image hash with robustness and uniqueness [19]. Based on tensor decomposition, Tang proposed image hashing where image hash generation is viewed a compact representation from a tensor [20]. These image hashing methods are mainly for image classification between robustness and discrimination. However, these methods are mainly employed for image authentication in plaintext domain and cannot be directly employed for cloud authentication. The researches basically consider the security of data which is stored in the cloud, but if personal cloud account and password were stolen, the data stored in the cloud cannot continue to ensure its security. In order to solve this problem, we proposed the image security storage scheme based on authentication value algorithm. The main idea of our algorithm is to compress and encrypt the private image then we use authentication value calculation algorithm (AVCA) to calculate authentication. At last, this authentication returns to the user. This authentication method can ensure that the privacy images stored in the cloud are not maliciously downloaded and deleted when the individual user's cloud account and password are stolen. The rest of this paper is organised as follows: Fundamental knowledge is presented in Section 2. Section 3 describes our proposed image compression, encryption, and identity authentication algorithm on cloud. The final experimental results and security analyses are given in Section 4. In Section 5, we make a conclusion about this paper. 2 Fundamental knowledge For the SVD, the eigenvalue can represent intrinsic structure of images [21-23]. It has been used for image classification, such as face recognition. The chaos is usually employed for image encryption for its good characters. However, the security of a simple and basic chaos system is limited [24-27]. In order to maintain high security, some basic chaotic systems are employed to form a complex chaotic system [28]. Generally speaking, if the Lyapunov exponents of chaotic system is high, it has high security level. Therefore, inspired by the Logistic-Sine system (LSS) [28], we proposed a novel chaotic system, named as logistic-tent-sine chaotic system (LTSS) for its lager Lyapunov exponents. A brief overview of the concepts is employed in our algorithm in this paper: the SVD algorithm and the proposed the LTSS. 2.1 Singular value decomposition In many cases, a small portion of the image data contains most information. There are many matrix decomposition techniques in linear algebra that can represent matrices in new and easy-to-handle forms [29]. One of the most common methods is SVD for any matrix M, which size is , this matrix can be decomposed into three parts: (1)SVD divides M into three matrices: U, S, and V. Matrix satisfies . Matrix satisfies , where is an unit matrix. The resulting S is a diagonal matrix, where diagonal elements are singular values and tell us the important features of the data. We compress the image data by choosing some diagonal elements. When all the elements are selected, SVD can reconstruct the original image without error. If k elements are chosen, the compression efficiency is: (2) 2.2 Logistic-tent-sine chaotic system One-dimensional chaotic maps have lots of applications because of their simple structures. As we all know, the most classic and widely studied 1D chaotic system is logistic chaotic system [24] whose definition equation as (3). Fig. 1 is the logistic chaotic and Lyapunov exponent of logistic map is shown in Fig. 2. (3)where n is the iteration step, are the parameters of logistic chaos mapping. When , the entire logistic sequence tends to chaos. Fig. 1Open in figure viewerPowerPoint Bifurcation diagrams of logistic map Fig. 2Open in figure viewerPowerPoint Lyapunov Exponent of LTSS The sine chaotic [26] system also has the same chaotic performance as the logistic chaotic system, which is defines as (4). Fig. 3 is the sine chaotic (4)where parameter . Fig. 3Open in figure viewerPowerPoint Bifurcation diagrams of sine map The tent chaotic [26] system is a kind of 1D chaoticsystem, which is defines as (5).Fig. 4 is the tent chaotic. (5) These traditional chaos systems, such as logistic chaos system,sine chaotic system, and so on, have some disadvantages including: (i) a limitedor/and discontinuous ranges of chaotic behaviours [26, 27]; (ii) thevulnerability of low-computation-cost analysis using iteration and correlationfunctions. In order to solve the shortcomings of 1D chaotic system, inspired bythe LTS, we have proposed LTSS chaotic system. The definition of LTSS chaoticsystem refers to (6) (6) As we can see from Fig. 5,LTSS is composed of LTSS. The effect of mod () is to ensure that the sequence generated by LTSS iswithin (0, 1). Fig. 4Open in figure viewerPowerPoint Bifurcation diagrams of tent map Fig. 5Open in figure viewerPowerPoint Proposed LTSS chaotic system As shown in Fig. 6, the LTSS has more complex chaotic properties. When one of its seed maps is out of chaotic range, LTSS can still have excellent chaotic behaviours. A sample chaotic system has limited data ranges within [0, 1] while the output sequences of our proposed LTSS system spread out in the entire data range between 0 and 1. This property ensures the proposed system is well suitable for image security. Second, the LTSS has a wider chaotic range. Even if one seed map is out of the chaotic range, the proposed system still keeps chaotic behaviours. Fig. 6Open in figure viewerPowerPoint Bifurcation diagrams of LTSS More important, Lyapunov exponents of different chaotic systems are calculated and illustrated in Fig. 2. Usually, the bigger Lyapunov exponent values mean the more obvious the chaotic character, the higher the chaotic degree, which provides more security in encryption domain. As shown in Fig. 2, the Lyapunov exponents of LTSS system is larger than logistic system, tent system, and sine system. Furthermore, the Lyapunov exponents of the LTSS is close to 1.3, while the Lyapunov exponents of logistic system are only about 0.5. The Lyapunov exponent of LSS [28] is about 0.7. This indicates that our proposed chaotic system has excellent chaotic performance and has more complex chaotic properties, therefore, providing high security level. 3 ISSAV model For the general image storage in the cloud, if the cloud accounts and passwords stolen which occurs in the last periods, the image can be downloaded for its plaintext. In order to solve the problem that, we proposed the following model illustrated in Fig. 7, our whole model can be divided into three parts which are upload part, cloud part, and download part. For each of the three parts, we describe the following section, respectively, in detail. Fig. 7Open in figure viewerPowerPoint Proposed image compression, encryption and identity authentication model An original image is decomposed by using SVD, and then the left and right singular value matrices are scrambled and encrypted. Different from the image digital watermarking techniques and image hashing, we encrypt the singular value sequence twice and extract the authentication value from ciphertext data. When download part want to download the ciphertext data, the download part needs to upload the authentication value and the key to the cloud at first. Only if download part pass the authentication, then the ciphertext data can be downloaded from the cloud then decrypt and decompress this ciphertext. For the cloud part, what needs to be done is to authenticate the identity of the download part. If we download the data fully and have the correct key, we can also recover and reconstruct the original image. 3.1 Upload part Assume that the size of original image I is . Then, if an original image is decomposed by SVD, then we can obtain the left singular value matrix , right singular value matrix , and the singular value matrix . Therefore, we have the following formula: (7)As illustrated in Fig. 8, we compress the data by choosing a number of singular values. If we select m singular values from the singular matrix, then the original data is compressed into , , . We use the LTSS chaotic system to encrypt and . (8) (9)where means taking B to generate LTSS chaotic sequence, and using LTSS chaotic sequence to encrypted A. Fig. 8Open in figure viewerPowerPoint Upload part of the proposed model As we all know the singular value matrix is a diagonal matrix, only diagonal elements are considered. The diagonal element sequence Sdiag are extracted from the compressed singular value matrix . Although the size of Sdiag is small, it plays an important role in image reconstruction. Therefore, we use bit scrambling method to scramble and encrypt Sdiag sequence. Our proposed bit encryption method is shown in Fig. 9 and Table 1. Table 1. Proposed bit encryption method The proposed bit encryption method step1: converting theSdiag decimal data to binary dataD. step2: using key to generate a chaotic sequence L. In order to generate a randomised sequence R, we define a threshold operation: where t is the threshold, R is the randomised sequence. step3: using Exclusive OR (XOR) operation to encrypt the binary data D: . step4: getting by using chaotic sequence to scrambling. step5: converting the binary data into decimal data. Fig. 9Open in figure viewerPowerPoint Bit encryption operation Since bit scrambling encryption algorithm requires the input data to be an integer, we decompose the sequence of singular values into integer parts and fractional parts, as in the formula: (10)We turn the singular value sequence Sdiag into binary then encrypt these data, the mathematical definition can be expressed in the following equation: (11)In order to achieve authentication, we proposed our own AVCA method, then we use this algorithm to extract the authentication value from the ciphertext data. The calculated authentication returns to the user as a token when the receive part download the data. (12)where and is the reverse order of . In the above formula, the effect of the Mc item is to amplify the position information features of the original sequence. The effect of the Vc item is to measure the change of the data relative to the entire sequence at each position of the sequence. Finally, the using of correlation coefficient is to further improve the ability of this model to reflect the size of the data at different positions in the sequence. In order to further improve the security performance, we encrypt the sequence of encrypted singular values twice. The final encrypted sequence of singular values is . (13)Finally, the encrypted data , , and are uploaded to the cloud. It should be noted that data which we upload is the ciphertext data; therefore, the privacy of personal data security can be ensured. 3.2 Cloud part As shown in Fig. 10, cloud part can verify the identity of the download part by using the key and authentication value which are upload by the download part. We assume the authentication value and the key are and . The cloud part will use these data to partially decrypt the ciphertext data: (14)We obtain the decrypted data , and we calculate a new authentication value by using our proposed authentication value calculation model. If , then the download part has access to the ciphertext data and then cloud part transmitter the ciphertext data to the download part. Once download part authentication fails, the download part cannot download any data from the cloud. This method can also protect the data further. If the download part cannot download any data, especially for ciphertext data, attacker cannot crack ciphertext data violently. Fig. 10Open in figure viewerPowerPoint Cloud part of the proposed model 3.3 Download part The download part can decrypt the data and reconstruct the SVD data, which is illustrated in Fig. 11. First, download part reconstruct by using Key1, Key2 and . (15)The ciphertext data and are decrypted by using Key1. Finally, the download part can obtain the correct decrypted and decompressed image by using SVD reconstruction. (16)Based on the cloud computing background, our proposed algorithm can achieve the compression then encryption of the image, as well as the authentication. The image is decomposed by using the SVD, and the data is compressed by selecting the number of singular values. For the data encryption problem, we use the chaotic system to scramble and encrypt the left and right singular value matrix. We use bit encryption algorithm to process the singular value matrix. The advantage of bit encryption operation is that it can scramble the data and change the size of the data, and the encryption strength is high. The proposed AVCA model can extract authentication values from data. The extracted authentication values can be used as authentication evidence. Fig. 11Open in figure viewerPowerPoint Download part of the proposed model 4 Experimental results and analysis We experimentally evaluate the performance of our proposed algorithm in this part, including compression performance and computation efficiency, key sensitivity and security analysis and performance of authentication value algorithm on big image databases. In the following experiment, both client and cloud server computing are performed on the same workstation to ignore the communication delays between the client and the cloud part. The setting of our workstation is shown as following: Intel(R) Core i5 CPU running at 2.2 GHz and 4 GB RAM. The normal standard test images, such as Lena, Barbara, Camera, and Pepper, are selected as our test images, where the size is 256 × 256. In order to evaluate the performance of ACVA further, a challenging set of 256 object categories [30] and VOCtrainval_06-Nov-2007 [31] databases is employed. 4.1 Compression performance and computation efficiency We select different numbers of singular values to compress the image data, the decrypt images are shown in Figs. 12-15. Fig. 12Open in figure viewerPowerPoint Reconstructed Lena (a) 10 singular values, (b) 50 singular values, (c) 100 singular values, (d) 150 singular values, (e) 200 singular values Fig. 13Open in figure viewerPowerPoint Reconstructed Barbara (a) 10 singular values, (b) 50 singular values, (c) 100 singular values, (d) 150 singular values, (e) 200 singular values Fig. 14Open in figure viewerPowerPoint Reconstructed Camera (a) 10 singular values, (b) 50 singular values, (c) 100 singular values, (d) 150 singular values, (e) 200 singular values Fig. 15Open in figure viewerPowerPoint Reconstructed Pepper (a) 10 singular values, (b) 50 singular values, (c) 100 singular values, (d) 150 singular values, (e) 200 singular values The peak signal-to-noise ratio (PSNR), ERROR, normalised correlation (NC), compression ratio (CS), structural similarity (SSIM), feature similarity index for image quality assessment (FSIM) [32], and the perceptual SIMilarity (PSIM) measure [33] are used to evaluate the quality of reconstructed images. Also, data encryption and authentication value extraction time and identity authentication time are used to measure the time complexity of the proposed algorithm. These evaluation indexes are listed in Tables 2–5 for different images. Table 2. Reconstructed quality of Lena Num PSNE, dB ERROR CS NC SSIM FSIM PSIM , s , s T 10 22.6476 0.1433 8.22% 0.9179 0.62 0.81 1.00 0.027135 0.025310 94.1229696379658 50 31.0476 0.0545 39.53% 0.9886 0.85 0.94 1.00 0.022492 0.034938 342.341955410369 100 37.9491 0.0246 78.67% 0.9977 0.95 0.99 1.00 0.027720 0.039740 −1733.3792338450 150 44.6742 0.0113 117.81% 0.9995 0.99 1.00 1.00 0.032130 0.050316 −756.72785608433 200 53.6823 0.004 156.95% 0.9999 1.00 1.00 1.00 0.043431 0.072912 1617.02564626375 Table 3. Reconstructed quality of Barbara Num PSNE, dB ERROR CS NC SSIM FSIM PSIM , s , s T 10 21.7542 0.1646 8.22% 0.9065 0.55 0.79 1.00 0.031766 0.023800 22.3294577518329 50 28.5432 0.0753 39.53% 0.9812 0.82 0.91 1.00 0.027099 0.032449 353.099158907498 100 34.4907 0.0380 78.67% 0.9952 0.93 0.96 1.00 0.028923 0.042390 392.325511903861 150 41.9759 0.0160 117.81% 0.9992 0.98 0.99 1.00 0.033775 0.055211 -942.38490835976 200 52.8114 0.0046 156.95% 0.9999 1.00 1.00 1.00 0.037779 0.068762 -2445.4284876871 Table 4. Reconstructed quality of Camera Num PSNE, dB ERROR CS NC SSIM FSIM PSIM , s , s T 10 21.2988 0.1638 8.22% 0.9359 0.58 0.72 1.00 0.029117 0.028092 −28.92352132026 50 30.1058 0.0594 39.53% 0.9918 0.84 0.90 1.00 0.032586 0.034626 −172.76178858155 100 37.7338 0.0247 78.67% 0.9986 0.94 0.98 1.00 0.026963 0.044874 −1261.5479189429 150 46.8272 0.0087 117.81% 0.9998 0.99 1.00 1.00 0.029425 0.058472 −569.03218938324 200 58.1514 0.0024 156.95% 1.0000 1.00 1.00 1.00 0.037674 0.073723 3152.33411209607 Table 5. Reconstructed quality of Pepper Num PSNE, dB ERROR CS NC SSIM FSIM PSIM , s , s T 10 22.1868 0.1478 8.22% 0.9278 0.63 0.79 1.00 0.017720 0.027301 −17.81343291936 50 32.0231 0.0476 39.53% 0.9928 0.88 0.94 1.00 0.026167 0.051462 232.59099519530 100 38.6526 0.0222 78.67% 0.9984 0.96 0.99 1.00 0.026152 0.061457 −516.03121182734 150 45.1521 0.0105 117.81% 0.9996 0.99 1.00 1.00 0.028885 0.073545 −1209.03258930099 200 54.3598 0.0036 156.95% 1.0000 1.00 1.00 1.00 0.038878 0.106323 −2927.15730180338 From Tables 2–5, with the increase of the number of singular values, PSNR, NC, SSIM, and FSIM of the correctly decrypted images are improved continuously. This indicates that the quality of the reconstructed image is getting better. Specifically, when the number of singular values is 150, we can obtain the image reconstruction with a very good quality. However, this may produce a large amount of data redundancy. Taking the Lena image as an example, when the number of singular values that we select is 10, the PSNR and SSIM of reconstructed image are 22.6476 dB and 0.62; the reconstructed image is blurred. However, this compression data is only the 8.22% of the original data. When the number of singular values is 100, PSNR, SSIM, FSIM, and PSIM of reconstructed images can reach 37.9491, 0.95, 0.99, and 1.00 dB. In this case, the quality of the reconstructed image is high, while it achieves good compression efficiency. Therefore, in real applications, the quality of the reconstructed image and compression efficiency should be balanced. The computational burden of SVD and the authentication value (12) is not complex and time-consuming. As for the time-consumption of the model, we tested data encryption and authentication value extraction time and identity authentication time . The time-consumption only takes 0.03 s on average, which is very short for the real-time applications. 4.2 Key sensitivity and security analysis We use Key1 to encrypt left and right singular value matrices and singular value sequences and then we use Key2 to encrypt the encrypted singular value sequences twice. The key space size of Key1 is: . The key space size of Key2 is . So, we can calculate the key space of our proposed algorithm as: The key space is large enough against any brute-force attack, which ensure the security of the proposed algorithm. In order to test the sensitivity of the authentication value, the Lena image is used as an example. The wrong (Key1 + ) is used to calculate the authentication value. The decrypted image is shown in Fig. 16 and the authentication value is listed in Table 6. Table 6. Authentication values Number T T′ 10 94.1229696379658 15.611833255405 50 342.341955410369 −108.795565417743 100 −1733.37923384503 2841.46524684822 150 −756.727856084332 −1216.16555950187 200 1617.02564626375 −3159.99025883821 Fig. 16Open in figure viewerPowerPoint Decrypted image with a wrong keys (a) 10 singular values, (b) 50 singular values, (c) 100 singular values, (d) 150 singular values, (e) 200 singular values From Table 6, if a wrong Key2 is used to decrypt the ciphertext image, even when the error is , the authentication value calculated by our algorithm varies greatly. This variety shows that our proposed algorithm has enough sensitivity. As we know, the singular values are arranged in ascending order for the image matrix SVD; therefore, the singular values vary greatly. In order to verify the bit encryption algorithm efficiently and security strength, only the singular value sequence is encrypted. The corresponding encryption results are shown in Figs. 17 and 18. From Fig. 17, with the increase of the number of singular values, the singular value changes drastically from and gradually becomes 0.01. If we use bit encryption to encrypt this set of data, as shown in Fig. 18, the trend of data changes significantly and becomes random, which inspired for encryption and higher security. Fig. 17Open in figure viewerPowerPoint Singular value Fig. 18Open in figure viewerPowerPoint Encrypted singular value 4.3 Performance under cropped and noisy conditions If an image is cropped or added noise and the image is very different, then the authentication values should be different. In order to test the performance under cropped and noisy conditions of the proposed, a wrong Key2 with the error is used. The reconstruction results of the original image after cropped and Gauss noise pollution are demonstrated. The cropped ratios are 25, 50, and 75%, respectively. The corresponding experimental results are illustrated in Figs. 19-21 and Tables 7–9. The Gauss noise levels are 0.01, 0.05, and 0.1, respectively. The corresponding experimental results are illustrated in Figs. 22-24 and Tables 10–12. Table 7. Authentication values with image cropped 25% Number T T′ 10 −18.3309186211663 −777.657668844548 50 −236.212402193319 −202.789496075353 100 −309.732326875458 364.02900866787 150 −1621.66631211814 1890.38235702511 200 12,407.5634405695 −1615.58003707756 Table 8. Authentication values with image cropped 50% Number T T′ 10 −36.6869438457192 −15.9225584934262 50 −392.338602017247 −2424.74046619486 100 −516.372580566944 364.02900866787 150 −793.323023223889 2252.57222413361 200 −7404.24825100671 −3126.24520036072 Table 9. Authentication values with image cropped 75% Number T T′ 10 −15.6410785919942 −486.911532729906 50 184.899993037763 7590.05717306007 100 −862.862833956905 2603.87535192358 150 −718.725643379496 −529.656202123734 Table 10. Authentication values with by Gaussian noise (Mean = 0, Var = 0.01) Number T T′ 10 −45.61509071552 21.1141558193425 50 −138.981343529603 217.008917028534 100 −548.500386017135 879.176170697387 150 −11,244.8591014598 2794.69458614614 Table 11. Authentication values with Gaussian noise (Mean = 0, Var = 0.05) Number T T′ 10 73.0932701663422 −17.8496055541322 50 −6058.27724585819 4578.34904325208 100 4133.70971239969 671.070984265625 150 2572.6532683029 5906.521