Biomedical Multimedia Encryption by Fractional-Order Meixner Polynomials Map and Quaternion Fractional-Order Meixner Moments

Chaotic systems are widely used in signal and image encryption schemes. Therefore, the design of new chaotic systems is always useful for improving the performance of encryption schemes in terms of security. In this work, we first demonstrate the chaotic behavior of fractional order Meixner polynomials (FrMPs) for introducing a new two-dimensional (2D) chaotic system called FrMPs map. This system is very sensitive to any variation by 10−15 of its control parameters ( $\mu \textrm {and}\beta$ ).Next, we use FrMPs to introduce a new type of orthogonal transforms called quaternion fractional order Meixner moments (QFrMMs). The latter generalize the existing fractional order Meixner moments. To demonstrate the relevance of the proposed FrMPs map and QFrMMs in the field of signal and image processing, they are applied in the development of a new encryption scheme. The main advantage of this scheme is its applicability to the encryption of different types of biomedical data such as multi-biomedical signals, multiple grayscale medical images, color medical image, and grayscale medical image. Several simulation analysis (visual, histogram, runtime, correlation, robustness, etc.) are conducted to verify the efficiency of the proposed scheme. Simulation and comparison results confirm that our encryption method is effective in terms of high security level, high quality of the decrypted information, strong resistance to different types of attacks, etc. These findings support the suitability of the proposed scheme for the secure exchange of biomedical multimedia via a public communication channel.

Therefore, a high level of security can be predicted by using 70 FrMPs map in an encryption scheme. Moreover, the matrix 71 form of FrMPs map is exploited in the introduction of a 72 new type of discrete orthogonal moments called quaternion 73 fractional order Meixner moments (QFrMMs). Then, FrMPs 74 and QFrMMs are involved in the design of our unified 75 encryption scheme. Indeed, QFrMMs are used in the diffu-76 sion phase of four inputs. Then, FrMPs map is used in the 77 confusion phase. In both phases, QFrMMs and FrMPs map 78 parameters are given as security keys. Finally, the encrypted 79 data can be securely transmitted between different medical 80 analysis centers. In the decryption phase, a reverse process 81 of the encryption process is followed to recover the original 82 data with negligible reconstruction/decryption errors. In both 83 encryption and decryption phases, the same security keys 84 must be used to correctly recover the original input data. The 85 simulation results (see sections IV and VII.C) show that any 86 variation by the order 10 −15 of the security key parameters 87 leads to the failure in recovering the original input data, which 88 reflect the strong security level of our scheme. 89 The main contributions of the work presented in this paper 90 are summarized as follows: 91 New 2D chaotic system called FrMPs map is proposed, 92 which is very sensitive to any variation by the order 93 10 −15 of its control parameters (µ and β). 94 New quaternion fractional order Meixner moments 95 (QFrMMs) are proposed for the encryption of multiple 96 inputs in a holistic and compact way. 97 Introduce a novel unified encryption scheme based on 98 QFrMMs and FrMPs map for encrypting both bio-99 signal, grayscale medical image, color medical image, 100 multi-biomedical signals and multi-grayscale medical 101 images. 102 Provide experimental analysis and comparisons to prove 103 the validity, efficiency and superiority of the proposed 104 scheme. 105 The rest of the work is presented as follows: the second 106 section covers the related work. The third section briefly 107 presents the theoretical background of FrMPs. In the fourth 108 section, we present the proposed FrMPs map. The proposed 109 QFrMMs are presented in the fifth section. The details of 110 the suggested unified encryption scheme are delivered in the 111 sixth section. In the seventh section, the results of simulations 112 and comparisons are offered to valid the efficiency of the 113 designed scheme. Finally, conclusion and future work are 114 outlined in the last section.

260
From the analysis of the results shown in Figures (4)- (6), 261 we can see that the variation of the local parameters (µ and 262 β) by the order 10 −15 leads to a large variation of FrMPs 263 values, while a variation by the order 10 −15 of the fractional 264 order parameter (α) does not lead to a significant variation 265 on FrMPs. From these results, we can conclude that FrMPs 266 displays a chaotic behavior. Therefore, FrMPs is considered 267    The quaternion number (q) is firstly introduced by 284 Hamilton as follows: [42]: where a,b,c and d are real values with i,j and k are three 287 imaginary numbers satisfying the following rules: If a = 0 in Eq. (11), q is called a pure quaternion.

291
The q number can be used to compactly represent four 292 signals (S 1 , S 2 , S 3 and S 4 ) as follows [43]: Since the quaternion number is not commutative, we define 295 the right-side QFrMMs as follows: where µ is a pure unit quaternion selected in this paper as 299 µ = −(i+j+k)/ √ 3 . M α 1 n and M α 2 m represent FrMPs matrices 300 of fractional orders α 1 and α 2 , respectively.

316
The inverse transformation of the right-side of QFrMMs 317 can be computed by the following relation: whereŜ 1 ,Ŝ 2 ,Ŝ 3 andŜ 4 represent the reconstructed versions 325 of the original signals S 1 , S 2 , S 3 and S 4 , respectively.

326
To measure the reconstruction error between an original 327 signal (S) and its reconstructed form (Ŝ), we can use the 328 following mean-square error (MSE) criterion: The peak signal-to-noise ratio (PSNR) criterion is also uti-331 lized to compute the reconstruction error. This criterion is 332 given by the next relation: If the MSE value tends to zero (high PSNR), it means that the 335 original signal and its reconstructed form are very similar.

336
To quantity the difference between the reconstructed 337 1D signalf (i) and the original one f (i), we can use the fol-338 lowing Percentage Root Difference (PRD (%)) criterion [44]: 339 The next section presents the proposed unified encryption 341 scheme based on FrMPs and QFrMMs.

344
The diagram of the novel suggested encryption scheme is 345 presented in Figure 6, which show that our scheme involves which allows to obtain a quaternion matrix noted S of size 370 N ×M . The latter is then divided into non-overlapping blocks 371 each of size 8 × 8 to optimize the computation time of FrMPs 372 matrix in the next step.

373
Step 2: This step represents the input data diffusion process 374 of our scheme. For this purpose, the S matrix produced in 375 the previous step is divided into blocks each of size 8 × 8. 376 Next, Eq. (15) is used to compute QFrMMs of each block. 377 Then, QFrMMs corresponding to each block are concate-378 nated to produce a quaternion matrix named QM of size 379 N × M (e.g. 512 × 512, 1024 × 768, 1800 × 1200, 1024 × 380 1024 etc.). The resulting QM matrix represents the dif-381 fused input data. It is worth mentioning that the parameters 382 {α 1 , α 2 , µ 1 , µ 2 , β 1 , β 2 } of QFrMMs (Eq. (15)) are provided 383 as a security key noted KEY1. The optimal choice of these 384 parameters is conducted according to the systematic method 385 presented in [2], which is based on the Sine Cosine Algorithm 386 (SCA) [46]. The use of this method guarantees the good 387 quality of the reconstructed image when using our scheme. 388 To illustrate the relevance of this method, three medical color 389 images of various sizes are reconstructed by QFrMMs.Then, 390 we display in Figure 7 the reconstructed images with 391 the selected parameters by the method given in [2], and 392 the reconstruction error (PSNR) that corresponds to each 393 image.

394
From the results shown in Figure 7, we can notice that 395 the PSNR values are high, which indicates that the medical 396 images are reconstructed with high quality. These results 397 FIGURE 6. Diagram of the proposed scheme for multiple biomedical data encryption.  confirm that the method presented in [2] is successful in 398 selecting the optimal parameters of QFrMMs.

399
It is important to mention that FrMPs are computed only 400 for n,x = 0,1,. . . ,7. Therefore, the runtime of QFrMMs basis 401 polynomials is fast.

402
Step 3: This step is known as the scrambling process   The present phase is performed at the storage device 434 that receives the encrypted biomedical data. In this phase, 435 we perform the inverse process of the steps given in the 436 encryption phase to retrieve the original biomedical data. 437 Indeed, the following steps are involved:

438
Step 1: In this step, the inverse process to that described 439 in Step 2 of subsection VI.A is followed. Indeed, 2D chaotic 440 sequences C1 and C2 each of size L are generated via the 441 proposed FrMPs map using KEY2 as initial conditions of 442 this map. Then, C2 and C1 sequences are used to apply the 443 inverse confusion of QM * * columns and rows, respectively, 444 for recovering matrix.

445
Step 2: This step consists first of subdividing QM matrix 446 into 8 × 8 blocks. Then, the inverse of QFrMMs (IQFrMMs) 447 is computed for each block according to Eq. (16). Finally, the 448 computed IQFrMMs of the blocs are concatenated to retrieve 449 the quaternion matrix S of size N × M . In the current step, 450 KEY 1 is used for computing IQFrMMs.

451
Step 3: This step begins with separating the quaternion 452 matrix S into four components (S 1 , S 2 , S 3 and S 4 ). The 453 latter are then reshaped intoÎ 1 ,Î 2 ,Î 3 andÎ 4 matrices (or 1D 454 signals) that represent the decrypted medical data of sizes 455 It is worth mentioning that the four biomedical input data 457 are decrypted with very low reconstruction errors, which can 458 be measured using the reconstruction error criteria (MSE, 459 PRD, PSNR, etc.). It is also important to note that an efficient 460 transform-based encryption scheme requires the reconstruc-461 tion error to be close to zero (MSE, PRD 0).

464
In this section, we outline the strengths and capabilities of the 465 suggested method for encrypting multi-biomedical signals 466 and images. It should be noted that all the experiments of 467 the actual work are realized using Matlab 9.6 installed on a 468 2.4 GHz processor PC with 4 GB of RAM. To perform the following test, we use four biomedical signals 472 of different types selected from the PhysioBank database [49] 473 that contains more than 90,000 digitized physiological sig-474 nal records. The types of the selected bio-signals are ECG, 475 impedance pneumography respiratory (IPR), EEG and EMG, 476 and the size of each signal is N = 4096 samples (Figure 8)

511
Noting that the present test is performed by using the 512 following security key in both encryption and decryption 513 phases: 514 KEY = {α 1 , α 2 , µ 1 , µ 2 , β 1 , β 2 , α 3 , µ 3 , β 3 } To show the execution speed of our scheme, one measures 518 the execution time of the encryption and decryption phases 519 the four test signals (ECG, PPG, EEG, and EMG). Indeed, 520 each phase is executed 100 runs, and then the average time 521 of both phases is obtained and shown in Figure 9. From the 522 achieved results, we can observe that the average time to 523 encrypt and then decrypt the four input signals each of size 524 N = 4096 is 0.5842 sec. This encouraging result makes our 525 method promising for use in the encryption of bio-signals.  Figure 10. From this 539 figure, it is obvious that the quality and the visual representa-540 tion of the decrypted signal are significantly degraded when 541 a slight variation by the order = 10 −15 is performed on 542 one parameter value of the security key (KEY). This result 543 clearly designates that the proposed map is quite sensitive to 544 the slight deviation of the security key.

545
By considering the precision order of about 10 −15 for real 546 type value of double precision, the KEY size of our scheme 547 comes approximately equal to (10 15 ) 6 = 10 90 2 294 . This 548 key space is sufficiently higher than the minimum recom-549 mended key size that is 2 100 [50], which delivers adequate 550 security against exhaustive brute-force assaults.

552
Histogram analysis is frequently used to illustrate the tough-553 ness of an encryption scheme against statistical attacks. The 554 histograms of the original, encrypted and decrypted signals 555 are displayed in Figure 11. From this figure, we can clearly 556 note that the original and the decrypted signals histograms 557 are quite same, which specifies that the anticipated scheme 558 does not change the statistical characteristics of input signals. 559 On the other hand, we notice that the histograms of the real 560 and imaginary parts of encrypted signal are very different 561 from the histograms of the original/decrypted signals, which 562 means that our scheme can efficiently resist statistical attacks, 563 To assess the statistical dependence between input, encrypted 567 and decrypted signals, the correlation coefficient r XY is 568 widely used. The following relation defines this coefficient 569 for two input signals X and Y of the same size:
bio-signal encryption methods provided in [14], [51], [52], 604 and [53] are suitable for encrypting a single input bio -605 signal. For this purpose, the average time of 100 executions 606 is calculated for each method and then multiplied by four 607 to compare the obtained time with the mean running time 608 of the projected method, which is used for the simultaneous 609 encryption of four input bio-signals. From the comparison 610 results achieved in Table 3 it appears that, the suggested 611 scheme achieves improved performance than the competi-612 tive approaches in terms of statistical dependence between 613 the original, encrypted and decrypted signals. This can be 614 explained by the fact that FrMPs exhibit chaotic characters on 615 the one hand, and QFrMMs generate a decrypted signal with 616 negligible reconstruction errors on the other hand. Moreover, 617 we notice that the execution time of the suggested encryption 618 method is lower than the compared methods. The reason for 619 this can be explicated by the circumstance that our scheme is 620 block-based method, which reduces the computational time 621 in comparison to the compared methods.

622
After confirming the usefulness of our scheme for the 623 encryption of multi-biomedical signals, we show in the next 624 section the usefulness of our scheme in the encryption of 625 multiple grayscale medical images.

627
This section presents the tests that justify the competence of 628 the suggested scheme in the encryption of multiple medical 629 images. For this purpose, we arbitrary select four Magnetic 630 resonance imaging (MRI) from the [54] dataset and four 631 Computed Tomography (CT) images from the [55] dataset, 632 which contains over 32,000 labelled lesions detected on CT 633 images.

634
In the present test, the selected images of size 512×512 are 635 encrypted via the suggested method. Then, the four input 636 test images, encrypted and decrypted ones are displayed 637 in Figure 12. The results presented in this figure specify 638 that the quality of the decrypted CT images is very high 639 (PSNR>290). Therefore, the diagnosis of a specific pathol-640 ogy cannot be influenced by this very low degradation of the 641 derypted images. On the other hand, we can see that the real 642 and imaginary parts of the encrypted image fully hide the 643 visual information of the plaintext CT images. Therefore, the 644 VOLUME 10, 2022   ( Figure 13). Therefore, the statistical analysis is not success-649 ful when it is applied to the proposed scheme.

650
The present analysis is not sufficient to corroborate the effi- In attack analysis, the assailant employs two alike images 655 with a minor change in one pixel of these images. Then, the 656 attacker attempts to identify the resemblances between the 657 encrypted images trying to identify the used security key in 658 VOLUME 10, 2022 the encryption scheme [56]. The number of pixels change rate (NPCR) and the unified average changed intensity (UACI) 660 criteria are used to appraise the robustness of an encryption 661 scheme against differential attacks. NPCR and UACI criteria 662 can be defined as [57], [58] : where C and C are the original image and the changed one

684
That is, each input image is encrypted/decrypted with its 685 own security key. The use of this method guarantees the 686 resistance of suggested system against differential attacks.

687
That is, for the same input, the proposed algorithm generates 688 two different outputs in two successive iterations. In this 689 way, it is expected that the suggested encryption process 690 can avoid differential attacks. To test the efficiently of this 691 method against differential attacks, we use the test the images  Tables 4 and 5 and [62]. To perform the current test, we use 8-bit grayscale 705 medical images of various size that are shown in Figure 15.   and UACI are calculated and reported in Tables 6 and 7, 709 respectively.

710
The results of the current test show on the one hand that 711 all the compared methods meet the NPCR and UACI criteria 712 according to work presented in [59] since NPCR > 99.50% 713 and UACI > 33.33%. On the other hand, we can notice that 714 our method provides superior performance with respect to the 715 compared schemes. This superiority can be explained by the 716 fact that our scheme is based on FrMPs map and QFrMMs 717 that demonstrated a good chaotic behavior. In contrast, the 718 compared schemes are not very sensitive to the variation of 719 their control and fractional order parameters.    figure, it appears that the quality of the decrypted images 736 decreases proportionally to the increasing of k value. How-737 ever, the visual content of the decrypted images seems iden-738 tifiable. The archived results designate that the suggested 739 scheme can counterattack noise contamination.

741
Congestion or failure of a communication channel can occur 742 during the transmission of medical data (images, signals, 743 videos, etc.), which can lead to partial loss (cropping) of the 744 transmitted data. Therefore, it is requirement to assess the 745 robustness of our scheme against cropping. For this purpose, 746 we use an MRI image ( Figure 17) of size 512 × 512, which 747 is taken from the database [63]. This image is encrypted by 748 the proposed method. Then, the real and imaginary parts of 749 the encrypted image are cropped in the same area by various 750 occlusion values. Finally, the cropped images are decrypted 751 via our scheme. The results of the actual test are offered in 752 Figure 17, which show that the quality of the decrypted image 753 reduces (decrease PSNR) when the occlusion ratio increases. 754 However, we can see that the visual content of the decrypted 755 images is still presented, which specifies that the suggested 756 system can withstand cropping attacks.

758
In the following test, we use DTI images shown in Figure 18 [61], and [62]. The test results are given in 762 Table 5. From this table, we can observe that the coeffi-763 cients are tending towards 1 for the original images, which 764 shows the strong dependence between the adjacent pixels 765 VOLUME 10, 2022  University. Since 2011, he has been working as 1087 an Assistant Professor with the Department of Computer Engineering, 1088 Jamia Millia Islamia. He has published over 90 research papers in inter-1089 nationally reputed refereed journals and conference proceedings of the 1090 IEEE/Springer/Elsevier. He has more than 2200 citations of his research 1091 works with an H-index of 29, i-10 index of 60, and cumulative impact factor 1092 of more than 200. Recently, he is listed among World's Top 2% Scientists in 1093 a study conducted by Elsevier and Stanford University and report published 1094 by Elsevier. His research interests include multimedia security, chaos-1095 based cryptography, cryptanalysis, machine learning for security, image 1096 processing, and optimization techniques. He has served as a reviewer and 1097 a technical program committee member of many international conferences. 1098 He has also served as a Referee of some renowned journals, such as