The Sin map is represented by a mathematical expression which is as follows [47]: (1)X(r+1)=βsin⁡(πX(r)).where, X(1) is the initial values of Sin map, β is the parameter of Sin map.

The Logistic Map is represented by a mathematical expression which is as follows: (2)X(r+1)=αX(r)(1−X(r)).where, X(1) is is the initial values of Logistic Map, α is the parameter of Logistic Map.

One-dimensional chaotic maps have fewer parameters and smaller ranges, chaos will be reduced or even eliminated once the parameters exceed a certain range. Therefore, this paper proposes the 1D-MWSLC. The 1D-MWSLC is represented by a mathematical expression which is as follows: (3)X(r+1)=βsin⁡(e5παX(r)(1−X(r))).where, X(1) is is the initial values of 1D-MWSLC, X∈[−β,β], α (α∈(0,+∞)) and β (β∈(0,+∞)) are the parameters of 1D-MWSLC.

Chaotic attractors describe the changes in the output of a system. Systems with complex, chaotic behaviors have complex attractors occupying a large space. Fig. 1 describes the 2D chaotic attractor of 1D-MWSLC. Fig. 2 describes the 3D chaotic attractor of 1D-MWSLC. Note that x is randomly selected and can be any value. To show the chaotic attractor, this paper randomly select the initial value of 1D-MWSLC is x(1)=0.94905165561134.

Fig. 3 describes the 2D and 3D chaotic attractors of the Sin map and Logistic map. Set the same parameters and initial values as 1D-MWSLC. The comparison results show that under the same parameters and initial values, 1D-MWSLC has more complex attractors and occupies more space than the Sin and Logistic maps. 1D-MWSLC has good chaotic behavior and generates random sequences.

The Lyapunov exponent (LE) is applied to evaluate chaotic behavior of systems. Fig. 4 describes Lyapunov exponent of the 1D-MWSLC with different parameters β. Fig. 5 describes Lyapunov exponent of the 1D-MWSLC with different parameters α.

LE analysis shows that 1D-MWSLC is globally chaotic, and the value of LE tends to increase as β and α increase, indicating that the larger the values of α and β, the better the chaotic performance of 1D-MWSLC.

Fig. 6 describes the Lyapunov exponents of the Sin map and Logistic map. Because the Sin map and Logistic map have fewer parameters and smaller ranges, chaos will be reduced or even eliminated once the parameters exceed a certain range. Compared to Sin and Logistic maps, 1D-MWSLC has a larger chaotic range.

The ApEn comparison results for 1D-MWSLC, Sin map, and Logistic map are shown in Table 1. The ApEn analysis results show that under the same parameter conditions, the ApEn of 1D-MWSLC has a more stable value and a larger value than the ApEn of the Sin map and Logistic map. This indicates that 1D-MWSLC has a stable performance of generating many random numbers.

Unlike digital images, the data type of a 3D model contains arbitrary floating point numbers. Therefore, in the designed encryption algorithm, normalization processing is first performed to align the data type of the 3D model with the data type of the digital image.

TDM represents 3D model, TDM∈(−∞,+∞), and the size of TDM is L×3. The normalization process is as follows, (4)NTDM=255×TDM−minTDMmaxTDM−minTDM.

Now NTDM∈[0,255].

The secret key of the cryptosystem is generated by the new 3D model (NTDM), which are (5){KM1=∑d=1L⁡NTDM(d,1)×10−10f1(c+1)=3.99997×f1(c)×(1−f1(c))c=1,2,3,….,100,f1(1)=KM1,{KM2=∑d=1L⁡NTDM(d,2)×10−10f2(c+1)=3.99998×f2(c)×(1−f2(c))c=1,2,3,….,100,f2(1)=KM2. (6){KM3=∑d=1L⁡NTDM(d,3)×10−10f3(c+1)=3.99999×f3(c)×(1−f3(c))c=1,2,3,….,100,f3(1)=KM3,{F1=f1(100)F2=f2×100+1F3=f3×100+1.

F1,F2,F3 are the secret keys of the cryptographic system. During the stream cipher generation phase, F1,F2,F3 are the initial values and parameters of 1D-MWSLC.

Generate keystreams based on the secret key (F1,F2,F3) by 1D-MWSLC, which are (7){KS1:Xc+1=βsin⁡(e5παXc(1−Xc))β=F1,α=F2,X1=F3,{KS2:Xc+1=βsin⁡(e5παXc(1−Xc))β=F1,α=F3,X1=F2. (8){KS3:Xc+1=βsin⁡(e5παXc(1−Xc))β=F1,α=F2+F3,X1=F3,{KS4:Xc+1=βsin⁡(e5παXc(1−Xc))β=F1,α=F2,X1=F3+F2.

When 1D-MWSLC is sufficiently chaotic, the key stream is extracted, (9){KS1=KS1(floor(F1×50)+200:L×3−1)KS2=KS2(floor(F2×10)+200:L−1),KKS2=floor(KS2×1010)mod256KS3=KS3(floor(F3×10)+200:L−1),KKS3=floor(KS3×1010)mod256KS4=KS4(floor(F1×40)+200:L−1),KKS4=floor(KS4×1010)mod256.

In the scrambling phase, both the integer and fractional parts of the 3D model undergo simultaneous scrambling. In the diffusion phase, this paper suggest a bidirectional diffusion strategy that operates in parallel. All three coordinates of the 3D model are diffused simultaneously, starting from the center position of each coordinate axis. The diffusion operations are exclusively performed on the integer portions of the 3D model.

Input: NTMD, KS1,KS2,KS3,KS4

Output: CTDM

Step1: Convert NTMD (L×3) to NTMD (1×3L) by (10)NTDM=reshape(NTDM,1,L×3).

Step2: Generate Index Matrix LP, (11)LP=sort(KS1).where, KS1 is sorted from small to large and the index of its original matrix is found, recorded as LP.

Step3: The scrambling steps is, (12)SNTDM(c)=NTDM(LP(c)),c=1,2,3,….,L×3.

Step4: Convert SNTMD (1×3L) to SNTMD (L×3) by (13)SNTDMA=reshape(SNTDM,L,3).

Step5: During the diffusion process, only integer operations are performed on 3D models, and integer extraction of 3D models is performed through, (14){SNTDMA1=floor(SNTDMA)SNTDMA2=SNTDMA−SNTDMA1.

Step 6: In the parallel bidirectional diffusion stage, the starting point of bidirectional diffusion is PAW, where (15)PAW=floor(L/2).

Step 7: The specific steps for parallel bidirectional diffusion are as follows. Note that these three steps are performed simultaneously, (16){CAP(PAW)=SNTDMA1(PAW,1)+KKS2(PAW)mod256CAP(r)=SNTDMA1(r,1)+KKS2(r)+CAP(r−1)mod256, r=PAW+1:PACAP(r)=SNTDMA1(r,1)+KKS2(r)+CAP(r+1)+CAP(r+2)mod256,r=PAW−1:−1:1. (17){CBP(PAW)=SNTDMA1(PAW,2)+KKS3(PAW)mod256CBP(g)=SNTDMA1(g,2)+KKS3(g)+CBP(g−1)mod256),g=PAW+1:PACBP(g)=SNTDMA1(g,2)+KKS3(g)+CBP(g+1)+CBP(g+2)mod256,g=PAW−1:−1:1. (18){CBP(PAW)=SNTDMA1(PAW,3)+KKS3(PAW)mod256CBP(b)=SNTDMA1(b,3)+KKS4(b)+CCP(b−1)mod256),b=PAW+1:PACBP(b)=SNTDMA1(b,3)+KKS4(b)+CCP(b+1)+CCP(b+2)mod256, b=PAW−1:−1:1.

Step 8: The ciphertext CTMD of the 3D model is (19){CTDM=[CAP;CBP;CCP]CTDM=CTDM+SNTDMA2.

The proposed 3D model encryption algorithm is a symmetric encryption algorithm. Therefore, the decryption process is the inverse of the encryption process. More specifically,

Input: CTDM, KS1,KKS2,KKS3,KKS4.

Output: NTMD

Step1: Decompose CTDM into CAP, CBP, CCP and SNTDMA2.

Step 2: The inverse process of diffusion is (20){SNTDMA1(PAW,1)=CAP(PAW)−KKS2(PAW)mod256SNTDMA1(r,1)=CAP(r)−KKS2(r)−CAP(r−1)mod256, r=PAW+1:PASNTDMA1(r,1)=CAP(r)−KKS2(r)−CAP(r+1)−CAP(r+2)mod256, r=PAW−1:−1:1. (21){SNTDMA1(PAW,2)=CBP(PAW)−KKS3(PAW)mod256SNTDMA1(g,2)=CBP(g)−KKS3(g)+CBP(g−1)mod256),g=PAW+1:PASNTDMA1(g,2)=CBP(g)−KKS3(g)+CBP(g+1)+CBP(g+2)mod256, g=PAW−1:−1:1. (22){SNTDMA1(PAW,3)=CBP(PAW)−KKS3(PAW)mod256SNTDMA1(b,3)=CBP(b)−KKS4(b)−CCP(b−1)mod256),b=PAW+1:PASNTDMA1(b,3)=CBP(b)−KKS4(b)−CCP(b+1)−CCP(b+2)mod256, b=PAW−1:−1:1

Step 3: The reverse process of scrambling is NTDM(LP(c))=SNTDM(c),c=1,2,3,….,L×3.where SNTDM=SNTDMA1+SNTDMA2, and LP is generated by KS1.

Due to its two control parameters, the 1D-MWSLC offers a larger parameter space for the cryptographic system. The algorithm generates a key stream for both confusion and diffusion. The 3D model is regularized (TDM→NTMD) and mapped to [0,255]. In the confusing stage, random confusion is applied. The key stream generates an index matrix that confuses both the integer and decimal parts of the 3D model, as illustrated in Fig. 8. In the diffusion stage, parallel bidirectional diffusion is employed simultaneously to diffuse the integer parts of the three coordinates of the 3D model, as depicted in Fig. 9.

In order to assess the security and viability of the suggested algorithm, an algorithmic test was performed on a 3D model obtained from the Stanford 3D scanning repository database (https://graphics.stanford.edu/data/3Dscanrep/). The evaluation findings, presented in Fig. 10, demonstrate that the plaintext information in the encrypted image cannot be identified. This validates that the proposed encryption technique is highly effective in preventing the leakage of plaintext information content.

The secret key of the algorithm includes F1(F1∈(0,1)), F2(F2∈(0,100)), and F3(F3∈(0,100)). If the calculation accuracy of the computer is 10−15. The key space of the algorithm is KeySpace=1015×1017×1017=1047≈2156.

When the secret key space of the algorithm is greater than 2100, it is considered that the algorithm can resist violent attacks. Therefore, the proposed algorithm has good resistance to violent attacks.

For a secure cryptographic system, the more sensitive the encryption algorithm is to the secret key, the more difficult it is to decipher it with various opportunities for plaintext analysis. In dragonStandRight_0, the initial key is KY1={F1=0.7629504319360213F2=0.3561951297073947F3=0.5143438570224240.

Use the new secret key to decrypt the encryption algorithm, which are KY2={F1=0.7629504319360213+10−15F2=0.3561951297073947F3=0.5143438570224240,KY3={F1=0.7629504319360213F2=0.3561951297073947+10−15F3=0.5143438570224240,KY4={F1=0.7629504319360213F2=0.3561951297073947F3=0.5143438570224240+10−15.

The decryption results are shown in Fig. 11. The secret key analysis shows that the proposed encryption algorithm is sensitive to the secret key.

Information entropy can indicate the level of pseudorandomness in an image. A higher information entropy value indicates good pseudorandomness. Table 2 presents the analysis results of information entropy. Furthermore, Table 3 compares the results of our proposed algorithm with algorithms in Ref. [48] and Ref. [49].

The comparison shows that our proposed algorithm achieves similar information entropy as the theoretical value, and the information entropy is closer to the theoretical value than algorithms in Ref. [48] and in Ref. [49]. The coordinate distribution in the generated ciphertext image demonstrates good randomness.

In the original 3D model, adjacent coordinates have a strong correlation, meaning accessing a small amount of plaintext information can reveal the entire message. Fig. 12 shows a correlation analysis of plaintext and ciphertext images (dragonStandRight_0). Obtaining a correlation image of the ciphertext shows a scattered pattern, indicating that the correlation between adjacent pixel values is minimal.

Reducing complexity and enhancing security: Table 4 presents the correlation analysis results obtained using the proposed algorithm. The analysis indicates that the pixel correlation in the ciphertext is weak, with a correlation coefficient close to zero. Therefore, it is unlikely for an attacker to extract meaningful information from the ciphertext pixel values.

In addition, Table 5 provides a comparison between the correlation results obtained using the proposed algorithm and those from other algorithms, such as those reported in the cited references [48,49]. The comparison reveals that the proposed algorithm produces lower correlation values, indicating that it offers good security.

The NIST Statistical Test Suite includes various tests, such as the Frequency (Monobit) Test, Block Frequency Test, Runs Test, Longest Run of Ones in a Block Test, and Random Excursions Test. These tests can be applied to both plaintexts and ciphertexts to evaluate the strength and security of a cryptographic algorithm. Table 6 presents the NIST results for plaintexts and ciphertexts. All of the tests resulted in a value of 0 for the plaintexts, indicating that they failed to meet the statistical randomness requirements of the NIST tests. However, the ciphertexts passed all of the tests, indicating that the proposed algorithm is highly resistant to statistical attacks on the ciphertext, which is a desirable property for a secure encryption algorithm.

Histogram analysis provides a deeper understanding of the central tendency, variability, and skewness of data. The histogram analysis is shown in Fig. 13. The uniform distribution of the ciphertext’s histogram indicates that the encryption algorithm’s ciphertext has a high degree of randomness and unpredictability. This can enhance the security of the encryption algorithm because an attacker cannot gain any information about the plaintext or key by analyzing the histogram of the ciphertext.

The chi squared test of histograms is a statistical method used to evaluate the difference between observed and expected values. The chi squared test is shown in Table 7.

The table shows the chi-square test results performed on plaintext and ciphertext data for different 3D models. The values in the table represent the chi-square statistic for each model. For all models, the chi-square statistic for the plaintext is much higher than that of the ciphertext, indicating that the distribution of the plaintext data is not uniform. Overall, the results suggest that the encryption algorithm used in the study produces ciphertext with a more uniform distribution than plaintext, which is desirable for ensuring the security of the data.

Robustness analysis can help determine the applicability of encryption algorithms in practical applications, as encrypted images may be subject to various unexpected and intentional attacks in practical scenarios. By subjecting the encrypted image to various attacks, such as noise attacks, the robustness of the encryption algorithm against these attacks can be evaluated. This paper evaluates the robustness of the algorithm using dragonStandRight_0, as shown in Fig. 14. If the encryption algorithm can still protect the confidentiality of the image in the face of various attacks, it can be considered to have good robustness. The results of the robustness analysis indicate that the tested encryption algorithm can protect the confidentiality of the image in the face of attacks and has strong security.

Differential attacks are one of the most common methods used to break encryption algorithms. In image encryption, a differential attack involves identifying the changes in the pixel values between two similar images. The differential attack then uses this information to derive the encryption key or recover the plaintext. As suggested by Wu in Ref. [50], if the NPCR value is greater than 99.5893% and the UACI value falls between 33.3730% and 33.5541%, then the algorithm is considered resistant to differential attacks. The differential attack analysis is shown in Table 8.

The analysis of differential attacks shows that the NPCR values of all models are above 99.59%, while the UACI values are between 33.41% and 33.54%, indicating that these algorithms can resist differential attacks. Therefore, all models have passed the test, demonstrating the algorithm’s excellent ability to resist differential attacks.