An asymmetric and optimized encryption method to protect the confidentiality of 3D mesh model

An asymmetric and optimized encryption method to protect the confidentiality of 3D mesh model

Advanced Engineering Informatics 42 (2019) 100963 Contents lists available at ScienceDirect Advanced Engineering Informatics journal homepage: www.e...

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Advanced Engineering Informatics 42 (2019) 100963

Contents lists available at ScienceDirect

Advanced Engineering Informatics journal homepage: www.elsevier.com/locate/aei

Full length article

An asymmetric and optimized encryption method to protect the confidentiality of 3D mesh model

T

Yaqian Lianga, Fazhi Hea,b, , Haoran Lia ⁎

a b

School of Computer Science, Wuhan University, Wuhan 430072, China State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China

ARTICLE INFO

ABSTRACT

Keywords: Asymmetric encryption 3D mesh model Multi-objective optimization Integrity checking

3D models are widely used in computer graphics, design and manufacture engineering, art animation and entertainment. With the universal of acquisition equipment and sensors, a huge number of 3D models are generated, which are becoming the major source of engineering data. How to preserve the privacy of the 3D models is a challenge issue. In this paper, an asymmetric and optimized encryption method is presented to protect the 3D mesh models. Firstly, we propose an asymmetric encryption method for 3D mesh models to overcome the drawbacks of traditional symmetric encryption. The primary benefit is that our approach can enhance the security of the key. Secondly, we extend the typically asymmetric encryption algorithm from integer domain to float domain. In our method, we present a normalization function to map the float DC (Discrete Cosine) coefficients to integer domain. Thirdly, considering that the shape error and encryption/decryption computation cost are contradictory in the normalization mapping, we formulate the contradiction as a multi-objective optimization problem. And then, we propose a multi-objective solution to find an optimized mapping range for encryption/decryption efficiently. Furthermore, benefiting from the proposed asymmetric encryption framework, we continue to put forward a method to check the integrity of the encrypted 3D mesh model, in which the digest is encrypted twice to generate digital signature more safely. The proposed method has been tested on 3D mesh models from Stanford university and other sources to demonstrate the effect of the proposed encryption method and optimization mechanism.

1. Introduction Since the population of acquisition equipments and sensors, such as camera, radar, and laser, 3D models have been created easier than before [1–7] and become one major source of engineering data in industry [8,9]. There are many kinds of 3D models, including 3D mesh model [10], 3D point cloud model [11–13], 3D CAD model [14–17], 3D NURBS model and so on. In this article, we mainly discuss the 3D mesh model. The 3D mesh model, composed of facets, has many advantages in application areas of smart city [18], 3D animation [19], 3D simulation, 3D virtual reality [20], and manufacture. Usually, the large scenes often contain a lot of 3D mesh models and each of them also has a lot of triangular facets. Along with the widely use of 3D mesh models, the security of them is becoming a challenging issue. In areas of medical care, industry [21], manufacture [22,23], entertainment and 3D printing [24–26], the shape contents of 3D models are becoming the property and



competitiveness of modern enterprises, which should be protected against illegal copying and stealing. More seriously, in some special applications, the content of 3D models must be securely protected, otherwise the leakage of models will lead to serious consequences. For example, the confidential models like weapons must be encrypted before being stored and transmitted to ensure the security of data and prevent illegal replication. At the same time, with the using of large-scale 3D models in engineering and industry, more and more collaborative designs for 3D models have already emerged [27–31], in which the “confidentiality preservation” is one of the major concerns [32]. In order to protect the security of 3D models, many approaches for encrypting 3D models have been proposed in the history of this area [33–38], which will be surveyed in the section of related work in detail. Based on the survey, although there are many security protection approaches in the historical development of this area, there are still serious challenging issues, which inspire the research motivation behind this manuscript:

Corresponding author at: School of Computer Science, Wuhan University, Wuhan 430072, China. E-mail address: [email protected] (F. He).

https://doi.org/10.1016/j.aei.2019.100963 Received 27 January 2019; Received in revised form 13 May 2019; Accepted 4 July 2019 1474-0346/ © 2019 Elsevier Ltd. All rights reserved.

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• Firstly, in the previous encryption methods for 3D mesh models, the





2. Related work

symmetric encryption algorithm is generally applied, in which the single secret key is used to encrypt and decrypt the models. Most of exist methods only consider the security of the method itself, which ignore the problem of key leakage in public network transmission. So, another security threat comes from the secure storage and transmission of the key in the open network. However, as best as we know, there is no public literature seriously discusses the problem. In this work, we present a novel method that could encrypt 3D mesh model as well as protect the security of the key, which is based on the asymmetric framework, such as RSA algorithm. Secondly, unfortunately the existing asymmetric method cannot be directly used to encrypt the vertex information of 3D mesh models. The mechanism of asymmetric encryption method (such as RSA) is based on the factorization of large integers, so it is impossible to directly encrypt the float vertex information (DC coefficients) using the asymmetric encryption algorithm. Thus, we innovatively present an idea to map the float DC coefficients to the format which can be handled by the asymmetric encryption algorithm. Thirdly, after proposing the idea, another challenging issue comes into being. It is difficult to find a suitable mapping range for there are many kinds of 3D models and the application users often have different demands. And if we set a fixed mapping range, there will occur a contradiction between shape loss and computation cost: on the one hand, if a large mapping range is set, the shape loss will be small but at cost of the expensive computation; on the other hand, if a small mapping range is set, the computation cost will be small but at cost of a big shape loss.

2.1. The security approach for product data With the growth of the knowledge-based economy and the increasing of collaboration in global relationship, ubiquitous digital communication techniques as well as tough competition has led to an increasing importance of intellectual property protection [39]. The security requirements of product data include the following four aspects: confidentiality, integrity, discriminability and non-repudiation. In order to meet the above requirements, there are many kinds of methods for intellectual property protection in the different stages of data interaction as follows: (1) Before distributing the data In the era of information sharing, the data leakage is inevitable in the process of distributing the data. In order to minimize the loss of information leak as much as possible, there should be fewer key technologies and key information in the transmitted data. Therefore, the data should be filtered before distributing the data. (2) In the data transmission In the process of transmission, the data is transmitted through the open network. In order to protect the security of data, using Data Leakage Prevention is a good method to stop unauthorized data leakage over known interfaces. (3) After the end of data transmission After the data transmission, the data comes to the client terminal. In this stage, there should be the terminal server solutions ensuring that the data is only presented to the legitimate recipient. (4) The phase of copyright verification At present, digital watermarking is a widely used method of copyright verification, especially for multimedia data. For texts, images and 3D models, there are corresponding digital watermarking method to allow the users to identify the copyright of the data. (5) The control over the whole process In 2011, a successive project group was founded, the Enterprise Rights Management (ERM) Open group aiming to increase interoperability and simplify the integration of ERM solutions. Using ERM to control over the whole process of data distribution is also an effective method to protect intellectual property.

In this paper, in order to encrypt the 3D mesh models more secure, we propose an asymmetric and optimized encryption method for 3D mesh models. The contribution of the proposed approach is as follows. (1) An asymmetric encryption framework is proposed to enhance the security of key for 3D mesh models encryption. (2) A mapping method is presented to map the float DC coefficients to integer domain, in which the transformed DC coefficients could be handled by the asymmetric encryption algorithms. (3) An optimized asymmetric encryption method for 3D mesh models is proposed. Firstly, we define MSE (Mean Squared Error)-based measurement to quantify the shape loss between the original 3D mesh model and recovered 3D mesh model, and LoEC (length of encryption content) to measure the computation cost. Secondly, we propose a multi-objective optimization method to find PS (Pareto Solution) set to balance the shape loss and the computation cost. Thirdly, we construct a preference function to facilitate the users to choose a best solution from Perato Solution set from engineering and industry. (4) In additional, benefiting from the proposed asymmetric encryption framework, a method is proposed to check the integrity of 3D models with double encryption of digital signature, in which the potential collision and crack problem of MD5 digest can be avoided.

2.2. The encryption approach for 2D images and 3D models Compared with traditional data, multimedia data has new encryption methods. In this subsection, we mainly discuss about the security protection for 2D images and 3D models. For the 2D images and 2D engineering graphics [40–43], image encryption is used to ensure the confidentiality of images, which is the core of image security technology. The image encryption approaches mainly include the following two kinds:

• Image encryption methods based on modern cryptography. In this

This paper is organized as follows. In Section 2, we will introduce related work of encryption methods for graphs and 3D models. In Section 3, we present an asymmetric encryption approach for 3D models. In Section 4, we propose an optimization method to balance the contradictory problem of the shape loss and computation time arisen in Section 3. In Section 5, we present an integrity verification method for the 3D models. Experimental results and the evaluation of the proposed method will be shown in Section 6. Finally, Section 7 shows the conclusion and future work.



2

kind of methods, the images are regarded as plaintext, and are encrypted by various classical encryption algorithms under the control of the key. However, it does not combine the features of image data, so it is difficult to meet the requirement of practical application. Image encryption methods based on the features of image. These kind of encryption methods encrypt the images based on the features of the images [44]. By changing the features of the image through some transformation rules, the meaningful original image will become “disorderly” and lose its original appearance in this process. They are specially for image encryption, including encrypt the image using matrix transformation [45,46], encrypt the image

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• Elsheh et al. [33] proposed two secret sharing approaches for 3D

based on chaos [47–50], encrypt the image based on frequency domain [51], and encrypt the image based on Secret sharing [52]. All above encryption methods make a good use of the features of image and they also establish the foundation for the encryption algorithms of 3D models.



Because there are similarities between 3D models and the 2D images, many security measures for 3D models are extensions of image security technology. The security methods mainly include the following points:

• Encrypt the 3D models using common methods in images encryp•



tion. This kind of methods encrypt the features of 3D models such as vertex coordinates, vertex sequences, etc. to protect the security of 3D models. We will discuss about the encryption methods for 3D models in detail in next subsection. Similar to images, there are methods using Secret Sharing to protect the security of 3D models in the process of distribution and transmission. Elsheh et al. [33] proposed two secret sharing approaches for 3D models based on Blakely and Thien and Lin schemes. These methods are the cryptography-based approaches. In (k, n) schemes, a secret is shared among n parties such that any k of them can collaborate to obtain the secret, but fewer than k cannot learn anything about the secret.

mesh models based on Blakely and Thien and Lin schemes. In the proposed (k, n) schemes, a secret is shared among n parties, and any k parts of them can obtain the secret collaboratively. But the parts with fewer than k cannot learn anything about the secret model. This is an early research for the content secrecy of 3D models. Marc et al. [34] proposed a method to encrypt 3D models by scrambling the order of vertices. The key idea of this method was inspired by the method of scrambling pixels in image encryption. This method was the pioneer to apply encryption to processing the vertex information of the 3D models. Continuing the idea of processing the vertex information, Ngoc-Giao Pham et al. [35–37] proposed a family of methods to encrypt 3D models based on various vertex transformation methods. In [35–37], the basic idea is to transform and encrypt the vertex coordinates of 3D models. [35] proposed an encryption method based on encrypting the clustered vertex coordinates of the 3D models. [36] proposed an encryption method based on encrypting the vertices coordinates of each face in the 3D models. And [37] proposed an encryption method based on encrypting the features obtained by the inverse interpolation and geometric distortion to the vertex coordinates.

In all above methods, researchers employed the symmetric encryption framework (using the single secret key). However, it is difficult to ensure the security of the key transmitted on the open network. Once the key is leaked, all encrypted contents may be completely exposed. Consequently, there may be some security problems with previous methods in network transmission and cloud computing.

Moreover, digital watermarking technology has achieved outstanding results in ensuring the copyright of 2D images [53] and 3D models. At present, adding watermarking to an image or a 3D model is a common way to authenticate the copyright. In the next subsection, we will discuss about encryption methods for security protection of 3D models in detail.

3. The proposed asymmetric encryption method The RSA algorithm is the fundamental theory for asymmetric encryption in many applications and engineerings. The proposed asymmetric encryption method for 3D mesh models also adopts RSA algorithm. However, there are two notes that should be pointed out. Firstly, the asymmetric encryption for 3D mesh models is more difficult than it seems at first sight because the formats of 3D mesh models are different from the strings or the 2D images, which have more dimensions and more complicated contents (infinite non-circulation floats resulting from coordinate transformation by discrete cosine transform, DCT). Clearly, this problem cannot be solved by the simple application of the RSA algorithm; on the contrary, it requires a more sophisticated approach. In order to introduce the RSA algorithm into 3D mesh model encryption, there should be some approaches to map the float DC coefficients into integer domain first, the format which can be handled by the asymmetric algorithms. Secondly, the proposed asymmetric encryption framework is not limited to RSA algorithm. In our framework, the other fundamental asymmetric encryption algorithms can be also used.

2.3. Encryption approaches for 3D model There are many different kinds of technologies in the domain of secure protection the contents of different kinds of 3D models. For the 3D point cloud models, Xin Jin et al. [38] proposed two methods to encrypt the 3D point cloud models by chaotic mapping. In the first method, each random vector is sorted to randomly shuffler each coordinate of the 3D point clouds. In the second method, each 3D point is projected to another random place using a random invertible rotation matrix and a translate vector which are generated by the logistic mapping. For the 3D CAD feature models, Cai et al. [54,55] proposed an encryption approach for CAD models, which is based on geometric transformation encryption mechanisms for features of CAD models. The key content of this approach is an enhanced encryption transformation matrix. But he only used around 50% faces to be the encryption feature, so this method may only change a part of the shape of the model. This paper is mainly about encryption methods for security protection of 3D mesh models. Along with the widely application of 3D mesh models, the security of them have become significant. A lot of related works for 3D mesh models have been proposed [33–38]. Both our work and related work are step-by-step historical advances in this area. In the early stage of research, researchers used digital watermarking algorithms to verify the copyright of the 3D mesh models. This kind of methods could resist the attack of model rotation, translation, affine transformation and other deformation. However, the digital watermarking algorithms are not suitable for preserving the confidentiality of 3D models. After embedding digital watermark, the overall geometric shapes of the 3D models have not been changed. So, the shape contents of 3D models may still be leaked. Therefore, the content secrecy of 3D models becomes another important issue. Many encryption algorithms for shape content of 3D models have been proposed.

3.1. Overview of encryption and decryption method RSA encryption algorithm is usually used to encrypt the texts or the images [56]. For the texts, RSA algorithm is usually used to encrypt the strings. And for the images, RSA algorithm is usually used to scrambles the pixels [57]. But all of these are not suitable for 3D mesh models. Different from the previous methods, the vertices coordinates of 3D mesh models are transformed by DCT, and are encrypted by RSA algorithm in the proposed method. According to the proposed method, the receiver adopts the RSA mechanism to generate a pair of keys firstly: the public key (e1, n1) and the private key (d1, n1). Secondly, the sender uses the receiver’s public key to encrypt the 3D mesh models. Because the private key is uniquely held by the receiver, only the receiver can decrypt the encrypted 3D model with the private key. 3

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Fig. 1. The encryption process.

Traditional RSA algorithm applications, such as in text encryption and image encryption, are designed to encrypt the integer numbers. However, in context of 3D models, DC coefficients of transformed coordinates are floats, which cannot be encrypted/decrypted by the RSA algorithm directly. In order to solve this problem, we propose a normalization function to map the DC coefficients to integer numbers. The proposed method is described in Fig. 1 and Fig. 2.

M = {Fi |i [1, m]} , where m is the number of triangular mesh patches, Fi = {vi1, vi2, vi3 |i [1, m]} refers to the ith triangle patch containing vertices vi1, vi2, vi3 . The detailed encryption process is as follows: (1) Extract facets from 3D mesh model. Use the three vertices of facet Fi (i [1, m]) to construct the matrix Ai (i [1, m]), as shown in Eq. (3.1). Then transform the matrix Ai to frequency domain by DCT, and get the matrix DAi as shown in the Eq. (3.2).

3.2. The encryption process

Ai =

The 3D mesh models contain many facets, described as:

x i1 x i 2 x i3 yi1 yi2 yi3 z i1 z i2 z i3

Fig. 2. The decryption process. 4

(3.1)

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DAi = DCT (Ai ) =

EAi = IDCT (DAi ) =

dx i1 dxi2 dx i3 dyi1 dyi2 dyi3 dz i1 dz i2 dz i3

(3.2)

(among the formula i

(2) Map the floats dx ij, dyij , dz ij to integer domain using the normalization function, as shown in Eq. (3.3), Eq. (3.4), Eq. (3.5). Then organize the mapped DC coefficients to construct matrix DAZi , as shown Eq. (3.6). The mapping range will be described in detail later. (3.3)

E Fi = {exij, eyij, ez ij |i

max = maxinum {dxij, dyij, dz ij}

(3.4)

EM = {E Fi |i

dxij

min

max min

range + min

(among the formula d

dx ij, i

dxz i1 dxz i2 dxz i3 DAZi = dyz i1 dyz i2 dyz i3 dzz i1 dzz i2 dzz i3

(among the formula i

(3) In the DCT domain, the mapped DC coefficient dxz ij is encrypted by the encryption formula EC () , as shown in Eq. (3.7), where the encryption key is the public key generated by the receiver. The coefficient dyz ij, dzz ij are encrypted using the similar equation as Eq. (3.7). Then organize the dxz ij, dyz ij, dzz ij to construct the matrix DAZi , as shown in Eq. (3.8).

(among the formula i

(among the formula i

DAi =

dxz ij

min

range

(max

In the process of encryption and decryption as described in the above subsection, the first step is to transform the vertex coordinate matrices into discrete cosine domain. Because the DC coefficients are floats, which cannot be encrypted by RSA algorithm directly, this work presents a mapping method in which the float DC coefficients are mapped into the integer domain. However, the mapped DC coefficients must be rounded off the float part to obtain the integer. In this way, this mapping will result in the shape loss between the original 3D mesh model and recovered 3D mesh model after encryption/decryption process. On the one hand, if a large mapping range is set, the mapped DC coefficient can be rounded into the integer by taking off a small float part, and the shape loss will be small but at cost of the expensive computation. On the other hand, if a small mapping range is set, the mapped DC coefficient can be rounded into the integer by taking off a large float part, and the computation cost will be small but at cost of a big shape loss.

[1, 3])

min) + min

(3.9)

dxi1 dx i2 dx i3 dyi1 dyi2 dyi3 dzi1 dz i2 dzi3

[1, 3])

3.4. A contradict problem

(4) Apply the inverse normalization to the encrypted matrixes, and transform them back to the original float state, as shown in Eq. (3.9), Eq. (3.10). Then all DC coefficients are performed inverse DCT in order to generate the encrypted vertices coordinates matrix EAi , as shown in Eq. (3.11).

dx ij =

[1, m], j

(3.14)

(3) Apply the inverse normalization to the decrypted matrix DAZi to construct the floats matrix DAi . Then perform inverse DCT to matrix DAi in order to get the decrypted vertex matrix Ai . (4) The decrypted vertex matrices are organized to construct the decrypted 3D model.

(3.7)

(3.8)

[1, m], j

(3.12) (3.13)

[1, m]}

dxz ij = DEC (d1, n1, dxz ij ) = (dxz ij )d1 modn1

dxz i1 dxz i2 dxz i3 DAZi = dyz i1 dyz i2 dyz i3 dzz i1 dzz i2 dzz i3

[1, 3]}

(1) Extract facets from 3D mesh model EM and use three vertices of face E Fi (i [1, m]) to construct the matrix EAi (i [1, m]). Convert the matrix EAi to discrete cosine domain by DCT, and get the matrix DAi (i [1, m]). (2) First the DAi will be mapped into integer domain by normalization function to construct the matrix DAZi . Then the mapped DC coefficients are decrypted by the decryption function DEC () as shown in Eq. (3.14). All the decrypted DC coefficients dxz ij are organized to form the matrix DAZi .

[1, 3])

dxz ij = EC (e1, n1, dxz ij ) = (dxz ij )e1 modn1

[1, m], j

The decryption process is the inverse process of the encryption. The specific decryption process is as follows:

[1, 3])

(3.6)

[1, m], j

[1, 3])

3.3. The decryption process

(3.5)

[1, m], j

[1, m], j

(3.11)

(5) Organize the encrypted vertex matrices to construct the encrypted face E Fi , as shown in Eq. (3.12). The encrypted faces are combined to construct the whole 3D model, as shown in Eq. (3.13). Finally, the encrypted 3D model EM is got.

min = mininum {dxij, dyij, dz ij}

dxz ij =

ex i1 ex i2 ex i3 eyi1 eyi2 eyi3 ez i1 ez i2 ez i3

(3.10)

5

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optimization methods [61,62]. In this paper, we present a multi-objective method to balance the contradiction between the shape loss and the computation cost, which are contradictory with each other. A typical multi-objective optimization problem can be described as in Eq. (4.3).

Minimize : y = {y1 = f1 (x ), y2 = f2 (x ), …, yq = fq (x )}

(4.3)

where x Rn is a vector with n solution variables, constituting a decision space, y Rn is a vector with q objective functions, constituting a goal space. In the process of optimization, it is not possible to simultaneously optimize all objects. And the optimization algorithm will give a large number of alternative solutions lying near the Pareto-optimal front, such solutions are called non-dominant solutions or Pareto solutions (PS) [63]. Therefore, we present a multi-objective optimization for 3D model encryption/decryption in Eq. (4.4). Fig. 3. Pareto solutions of the Bunny model.

Minimize : y = {y1 = MSE (x ), y2 = LoEC (x )}

where the mapping range is the solution variable, expressed as x, constituting a decision space; MSE(x) and LoEC(x) are two contradictory objective functions depending on the mapping range. There are different kinds of solving methods for multi-objective optimization problems. In this paper, we adapt the classical NSGA-II method. The first reason to choose NSGA-II method is that it has many theoretical advantages, such as high efficiency and simple calculation. Secondly, we also test the feasibility of NSGA-II method in context of this research by the experiments as follows. The experiments are implemented on a typical 3D model, the famous Bunny model from Stanford University. We utilize the Eq. (4.4) to find the PS solutions (“possible best mapping range”) in the encryption and decryption process. The distribution of the PS solutions is shown in Fig. 3, any one of which can be the best solution for the given user’s requirements. However, for the ordinary users who don’t understand the principle of multi-objective optimization encryption/decryption, it is a tedious task for them to choose one solution(“best mapping range”) from a lot of PS solutions.

At first sight, it is difficult to simultaneously minimize both the shape loss and the computation cost [58,59]. Therefore, in this paper, the above contradictory problem is modeled as a multi-object optimization problem, which will be discussed in following section. 4. An optimized method for asymmetric encryption/decryption of 3D mesh model 4.1. Optimization fitness functions of multi-objective for 3D encryption/ decryption For the encryption/decryption of 3D models, both the computation cost and the shape loss are important. However these two aspects are theoretically contradictory. In order to solve this problem, we propose an optimized encryption approach in this section. The proposed multi-object optimization model has two optimization objectives, which are evaluated with two fitness functions as follows. The first optimization object is about the shape loss. In the research area of 3D CAD, CAE, CAM, graphics and animation, MSE (Mean Square Error) is usually used to evaluate the deformation degree of the 3D models. Here we use MSE to evaluate the shape loss between the recovered 3D model and the original 3D model. The calculation equation of MSE is shown as Eq. (4.1).

MSE =

n

4.3. Posteriori decision based on user’s preference for 3D encryption/ decryption In this subsection, we construct a preference function to facilitate the users to choose a best solution from Perato solution set (“possible best”). If the users have the clear accuracy requirements and time requirements for the 3D models encryption/decryption, they can select the most suitable solution from Perato solution set (“possible best”) as the point determined by the definite MSE and LoEC on the Perato frontier. However, in most cases, the number of vertices and facets are different for the given 3D models, and the users often have inconsistent demands for model accuracy and computation cost. In order to reflect the user’s requirements, a user preference function is designed as shown in Eq. (4.5).

n

1 3

(vix

vix )2 + (viy

viy )2 + (viz

viz )2

i=1

(4.1)

where n refers to the number of model vertices. The second optimization object is about the computation cost. The encryption/decryption computation cost is closely related to the content length being encrypted [60]. If the content length being encrypted is increased, the computation cost required in the encryption/decryption process will be increased. Therefore, in this paper, the length of encryption content LoEC is adopted to measure the computation cost as follows. The L (a) indicates how many digits are included in the DC coefficient a, and a can be dxz ij, dyz ij, dzz ij . LoEC is the sum of all the length of encryption content, as shown in Eq. (4.2). m

3

i=1

j=1

LoEC =

L (dxz ij ) + L (dyz ij ) + L (dzz ij )

(4.4)

S( ) =

MSE + (1

)

(4.5)

LoEC

where the MSE refers to the shape loss of 3D models, LoEC refers to the computation cost, and is the user’s preferences, which can be easily set by the ordinary users. For a multi-objective optimization model, it can be transformed into a single-objective optimization model using aggregation function as shown in Eq. (4.6)

(4.2)

where m is the number of triangular mesh patches. 4.2. An multi-objective optimization method for 3D encryption/decryption

y = w1

Many science and engineering problems can benefit from

y1 + w2

y1 = w1

f1 (x ) + w2

f2 (x )

(4.6)

where the w1 + w2 = 1. Two coefficients w1 and w2 are assigned 6

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Fig. 4. Three typical coefficient selection schemes.

• When the value of MSE of one solution is too large to out of the • Fig. 5. The original Bunny model.

Finally, we give an example to demonstrate the proposed approach. The original Bunny model has many vertices, faces and also many details as shown in Fig. 5. In order to test the method of posteriori decision, we set a serious of experiments by setting different and different feasible range. As shown in the Table 1, the best solution can be get from PS set under the specific requirements. We select several typical optimal solutions as the mapping range to encrypt and decrypt the Bunny model, the results are shown in Fig. 6.

according to users’ preference. There are three typical selection schemes for coefficient setting, including w1 < w2, w1 = w2 and w1 > w2 . Fig. 4 shows the final solutions when choosing different schemes. We will analyze these situations as follows. In situation (a), the coefficients are assigned as w1 < w2 , the absolute value of the gradient of the asymptote |K L1 | is more than 1. Therefore, in the intersection of line L1 and Pareto frontier N1, f1 (x ) will be larger than f2 (x ) . In situation (b), the coefficients are assigned as w1 = w2 , the absolute value of the gradient of the asymptote |K L2 | is equal to 1. In situation (c), the coefficients are assigned as w1 > w2 , the absolute value of the gradient of the asymptote |K L3 | is less than 1. Therefore, in the intersection of line L3 and Pareto frontier N3, f1 (x ) will be smaller than f2 (x ) . In this way, in an optimization problem for minimization, if the user more emphasizes some target, he will want the value of this target smaller, so the coefficient corresponding to the goal should be assigned larger than 0.5. Therefore, there are two typical user’s preferences as follows:

5. Integrity checking for 3D mesh model based on double encryption digital signature In the process of network transmission, contents may be lost due to the quality of the network, or be forged by the third party. In the previous works, in order to eliminate the key escrow problem of identity-based cryptography, a concrete hierarchical certificateless signature scheme is proposed. MD5 digest algorithm is a common method in generating the digital signature. It is a typical proof of the authenticity of the information, and can be used to verify the integrity of the file. However, MD5 may be attacked by hash collision [64]. In this situation, a file can be forged with equal digest value but with inconsistent file content. In order to overcome the problem of MD5 be cracked in the point to point network transmission, we propose to generate the digital signature by encrypting MD5 digest twice.

• The first situation is that the users put more emphasis on the shape •

loss of 3D models, they should set > 0.5, and gradually enlarge according to the importance. The second situation is that the users put more emphasis on the computation cost of encryption/decryption, they should set < 0.5, and gradually decrease according to the importance.

In both above situations, two extreme cases maybe occur as follows:

Table 1 The experimental results of the Bunny model by setting different parameters.

• Along with the decrease of the shape loss of 3D models, the com•

feasible range, the value of MSE will be adjusted very large so that the value of preference function S (a) will be larger than all the other solutions. In this way, this solution will be made particularly poor in the whole PS set, then it will be abandoned. When the value of LoEC of one solution is too large out of the feasible range, the value of LoEC will be adjusted very large so that the value of preference function S (a) will be larger than all the other solutions. In this way, the solution will be made particularly poor in the whole PS set, then it will be abandoned.

putation cost will be enlarged. If the computation cost is too high to exceed the user’s acceptance range, this solution should be abandoned. Along with the decrease of the computation time of encryption/ decryption, the shape loss will be enlarged. If the shape loss is too high to exceed the user’s acceptance range, this solution should also be abandoned.

(1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4

If these two extreme cases occurs, we suggest an effective approach to restrict all objectives and keep them within the feasible range as follows. 7

0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6 0.6

)

Feasible range

Optimal solution (Mapping range)

MSE MSE MSE MSE MSE MSE MSE MSE MSE MSE MSE

780 1184 188 6189 1290 122 2484 122 1184 1184 64

0.05 0.1 0.15 0.1 0.05 0.05 0.1 0.15 0.15 0.1 0.05

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Fig. 6. The experimental results of the Bunny by setting different parameters.

5.1. The proposed integrity checking method for 3D mesh model

The implementation of digital signature is described in Fig. 7. And the verification process is shown in Fig. 8.

In the proposed method, the digest is encrypted for the first time with the sender’s private key for the authentication and non-repudiation of the sender’s identity. Then, the digest is encrypted with the receiver’s public key for the second time to ensure that only the receiver can get the real digest value. In this way, the third party cannot know and collide with the digest, thus the forging of the original content can be prevented.

5.2. The process of double encryption digital signature of 3D mesh model The detailed process of generating digital signature is as follows. (1) Generate the digest C from the encrypted 3D model EM , as shown in Eq. (5.1).

Fig. 7. Generate the digital signature. 8

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Fig. 8. Integrity checking.

Table 2 The experimental models and the experimental results Name

Cow Dragon Bunny Kitten Welsh-Dragon Hair Ball Man Buddha

Facets

5804 209227 69662 49912 2210673 2880000 412669 1087474

Vertices

2903 104855 34833 24956 1105352 2940000 207285 543524

(3) Receiver decrypts the encrypted digest value C once more using the public key (e2 , n2 ) generated by the sender, as shown in Eq. (5.6).

Enropy(dB) Proposed Method

Pham’s method

Marc’s method

818682 39251343 12073963 8434789 482399384 638346416 81056839 227284534

72566 3698026 1120728 778981 46592263 61797995 7698185 21806626

33394 1748775 525566 364535 22191129 63172911 3660913 10355210

C = MD5(EM )

C=C

6.1. Experimental results (1) The typical 3D mesh data set The test data set of this paper includes Bunny, Buddha and Dragon models from the Computer Graphics Laboratory of Stanford University; the Welsh Dragon model from the Bangor University in Eurographics 2011; the Hairball model from Samuli; and some common and publicly available 3D models on the Internet. The detail information of models is shown in Table 2. (2) The typical methods being compared We compare our method with the following two state-of-the-art methods, which are both latest approaches for encrypting the features of 3D models. Marc et al. proposed the method to encrypt 3D models based on scrambling the order of the vertices in the Proceeding of Conference: Compression at Repre sentation Signal Audio in 2018. Ngoc-Giao Pham et al. proposed methods to encrypt the vertices coordinates of the 3D models based on various transformation methods in Journal of Korea Multimedia Society in 2018. (3) The experimental results compared with other typical methods In order to let the readers understand the effect of the proposed approach more clearly, we conduct experiments on the 3D mesh models mentioned above. There is one table in one page. The experimental results of Bunny, Dragon, Buddha and WelshDragon are shown in Table 3, and the experimental results of other models are shown in Table 4. In each table, four models are placed in each row. They are the original model, the encrypted model using proposed method, the encrypted model using Pham’s method and the encrypted model using Marc’s method respectively. According to the experimental effect, the 3D mesh models are distorted severely after being encrypted by the proposed approach. Compared with the other algorithms, the encrypted 3D model using

(5.1)

(5.2)

(5.3)

(4) Send the encrypted digest D to the recipient as the digital signature. 5.3. The process of verifying the signature of 3D mesh model The detailed process of verifying the signature is as follows. (1) The receiver generates the digest value C0 of the received 3D model EM using the MD5 digest algorithm, as shown in Eq. (5.4).

C0 = MD5(EM )

(5.4)

(2) The received digital signature D is decrypted by the private key (d1, n1) generated by receiver to get the encrypted digest C , as shown in Eq. (5.5).

C = D d1 modn1

(5.6)

6. Experimental results and security analysis

(3) The encrypted digest C is encrypted once again by the public key (d1, n1) generated by the receiver to obtain the encrypted digest D, as shown in Eq. (5.3).

D = C e1 modn1

modn2

(4) Compare whether the digest value C0 and C are equal. If they are equal, the received 3D model can be verified correct and original.

(2) The digest C is encrypted by the private key (d2 , n2 ) generated by sender to obtain the encrypted digest C , as shown in Eq. (5.2).

C = C d2 modn2

e2

(5.5) 9

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Table 3 The experimental results.

our approach is more difficult to be recovered without the key, which will be further analyzed quantificationally in next subsection.

the RSA system will be quite safe. So, the generating mechanism of the secret key used in encryption/decryption is quite safe. At the same time, benefiting from the proposed asymmetric encryption framework, the secret key is no longer need to be transmitted through the network. All of these guarantee the security of the proposed encryption approach. (2) The security of the encrypted 3D model Because it is hard to judge which method provides better encryption based on the visualization, we use the quantitative analysis to measure the security of encryption methods. The entropy can be used to describe the uncertainty of information sources and measure the degree of confusion. In this way, the entropy could measure the difficulty of recovering the model violently without the key. In this paper, we utilize the entropy as the evaluation function of the security of the encrypted 3D model. If the

6.2. Security analysis (1) The security of the encryption approach The security of our approach mainly depends on the proposed asymmetric encryption framework, such as RSA algorithm. As we know, the security of RSA algorithm depends on the factorization of large integers. The difficulty of attacking RSA system is equivalent to the difficulty of large integer factorization, which is generally considered to be an NPC problem. The increase of the key length will increase the time required for factorization of large integers. If the length of n reaches a certain requirement, and the parameters p, q and e are selected properly, 10

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Table 4 The experimental results.

entropy is high, the security of the encrypted 3D model will be high. As can be seen from Section 3, the entropy of the encrypted 3D model relys on both the secret key K and the number of model patches |M|. For the convenience, we set the secret key K to be consistent for all the 3D models. In this way, the entropy of the 3D model is positively correlated with the number of patches |M| of the 3D model. In this paper, we encrypt all the coordinates of the vertices of each patch, and our entropy calculation formula of encrypted 3D model is shown in Eq. (6.1).

HM = H (k ) + HM = k

log2 |K | + (9

M)

log2 |9

M|

transmission, even if the infringers illegally obtain the 3D models encrypted by the proposed method, the difficulty of reconstructing the original 3D models by only using the encrypted models will be much higher. In this way, from the aspect of numerical analysis, the encrypted 3D model using our approach is more difficult to be recovered violently compared with the other algorithms. (3) The security of the digital signature MD5 digest algorithm has the characteristic of anti-modification, that is, any change to the original data will make the digest value change greatly. In Section 5, we propose to generate digital signature by encrypting the digest twice to avoid the cracking of MD5 digest algorithm by collision. In this way, MD5 digest is still a trustworthy evidence to verify the integrity of the 3D mesh model. In the process of generating digital signature, we use the encrypted 3D model as the original content to generate the digest directly. And in the process of checking the integrity of the 3D model, the

(6.1)

From the Table 2, the entropy of the encrypted 3D model is increased greatly after encrypted by the proposed method compared with other encryption methods. So, in the process of network 11

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encrypted 3D model is also used as the original content to generate the digest. In this way, if the decrypted MD5 digest and the regenerated MD5 digest are consistent, the integrity of the encrypted 3D model could be confirmed. In order to avoid the cracking of MD5 digest algorithm, in our approach, the MD5 digest has been encrypted twice by the private key of the sender and the public key of the receiver separately. Firstly, because the digest is encrypted by the receiver’s public key, there is no one could decrypt the digital signature to get the digest of the original encrypted 3D model except the receiver. The third party cannot collide the MD5 digest, and the safety of the digest could also be ensured. Secondly, the MD5 digest of the encrypted 3D model is encrypted by the sender’s private key. Therefore, if there is a dispute between the sender and the receiver, the sender cannot deny that the 3D model has been sent by himself,and the arbitration agency can also solve it easily. In this way, the security of digital signature is ensured.

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7. Conclusion In this paper, we propose an asymmetric and optimized encryption method for 3D mesh models. The proposed approach enhances the security of the encrypted models by constructing an asymmetric encryption framework in which the secret key is no longer transmitted on the open network. More important, the proposed method balances the shape loss and computation time to achieve an optimal encryption result. In addition, the receiver could use the proposed approach to check the integrity of the encrypted 3D mesh models, which will be more safely in point-to-point transmission. In the future, we will continue the research in at least four directions but not limited. Firstly, the shape loss between the recovered 3D model and the original 3D model partly comes from the process of DCT. For the large-scale 3D models, the shape loss caused by discrete cosine transform will be bigger because of the longer distances between the coordinates of the vertices. Thus, we try to extend the proposed asymmetric and optimized encryption method to process the segmented model. In this way, the shape error caused by the discrete cosine transform will be reduced, and the parallel encryption/decryption process can be accelerated adapting multiple core CPU and many-core GPU [65–67]. Secondly, thanks to our general and optimized encryption framework, we will extend it to other asymmetric encryption algorithms, such as Elgamal and ECC. And then we will apply the proposed encryption method to other engineering data, such as text, image, video and material data [68–72]. Thirdly, we will extend the proposed encryption method for information security in collaboration design and social computing [73–76]. Fourthly, the typical engineering processes involve the integration of different kinds of physical and digital informatics, therefore we will extend the proposed encryption method to the whole process of design, transmission and application [22,77]. Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgment This research has been funded by the National Science Foundation of China (Grant No. 61472289) and the National Key Research and Development Project of China (Grant No. 2016YFC0106305). References [1] H. Wang, P. Wang, X. Wang, T. Peng, B. Zhang, A 3d reconstruction system for large scene based on rgb-d image, International Conference on Intelligent Science and Big Data Engineering, Springer, 2018, pp. 518–527.

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