The weighted visual cryptographic scheme (WVCS) is a secret sharing technology, where weights are assigned to each shadow (participant) according to its importance. Among WVCS, the random grid-based WVCS (RGWVCS) is a frequently visited subject. It considers the premise of equality of all participants, without taking into account the existence of privileged people in reality. To address this problem of RGWVCS, this paper designs a new model, named as (
Visual cryptography scheme (VCS) sometimes also called visual secret sharing was formally proposed by Naor and Shamir at Eurocrypt’94 [
As can be seen from the (
Due to the high security and concealment of VCS, it is essential in the transmission of highly confidential information in a completely hostile channel. More precisely, the problem here is to transmit highly confidential information or authentication information through one or more insecure channels which are under full control of the adversary. Although the emergence of identity authentication technology has made a momentous contribution to keep the identity security, the filch or illegal decoding of smartcard and certificate will pose a high safety hazard. Therefore, the visual cryptography technology is combined with the identity authentication to store the shares generated when the identity information is encrypted in the authentication device and the database, respectively. The authentication of the user needs to be decrypted through the superposition of the stored shares, while the attacker cannot obtain any valuable information of the secret image from the individual share through existing analytical methods, and it is impossible to decrypt by forging user shares. In uncontrolled channels, this problem is difficult to solve, and visual authentication [
The loss of contrast and resolution of the secret image enables the VC to achieve security confidentiality, so in general, the quality of the secret recovered image is lower than that of the original secret image. With the gradual maturity of VC technology, efficiency and security have become the focus [
Traditional VCSs mostly devote to optimizing the quality of secret recovered images [
To address the issue of weight leakage, Yang et al. [
In this paper, we propose a novel RGWVC method for a (
Therefore, the proposed scheme can be better applied to practical models such as management pyramid, and the experimental results and the comparison with the existing schemes are shown that the scheme we designed has several highlights as follows: (1) no codebook design, (2) no pixel expansion, (3) no weight leakage, (4) a weighted VCS, (5) (
The rest of this paper is organized as follows. Section
In this section, we provide some preliminaries including background, definitions, notations, and conditions that will be used later. For more details about information theory and the definition of secret sharing, see, e.g., [
The weighted random grid-based visual cryptographic scheme (WRGVCS) is the basis of the proposed scheme. This method can assign weights to each participant of different importance so that each share has different ability to restore the secret image. Our visual cryptography scheme inherits the advantages of WRGVCS and considers the possible privileged role in the reality, which increases the practicality of the scheme and makes it more comprehensive.
Suppose a binary secret image S with a size of Randomly select Generate Set Arrange Repeat the Steps 1–4 to fill in the above
Monte Carlo method is a numerical simulation method. It takes probabilistic phenomena as its research objective. In equipment effectiveness evaluation, it is often used to determine the efficiency index with random factors, such as the probability of discovery, the probability of hit, and the average number of damaged targets.
The simulation processes are described as follows. (1) Construct a simple and feasible stochastic or probabilistic model to describe the problem. The solution of the proposed problem is bound to some features of the random variables in the model (such as probability, mean value, and variance); (2) Generate a sufficient amount of random numbers according to the different distribution of each random variable in the stochastic or probabilistic model; (3) Design a sampling method suitable for the probabilistic model and random variable distribution; and (4) According to the established model, the simulation test and calculation are carried out to obtain the random solution of the problem.
In this section, we will introduce some notations and definitions to prepare for the further work. Later, the symbols
(contrast). The contrast can describe the quality of the recovered secret image, denoted as
Figure
The heatmap of the contrast
In this part, we will introduce two necessary conditions to verify the feasibility and security of the proposed scheme, in terms of secure encryption and visual decryption.
In the visual cryptography scheme, the ultimate goal is to realize the visualization of secret under the condition of meeting the requirements of the scheme. Therefore, a qualified visual cryptography scheme must meet the following condition in visual recognition:
(visually recognizable). The recovery image S′ is visually recognizable (
In the visual cryptography scheme, the security of encrypted secret is of capital importance. Therefore, a secure visual cryptography scheme must meet the following condition in secret recovery.
(security). The secret recovery image S′ is visually unrecognizable (
Based on the basic WRGVCS described in the previous section, we designed a new (
The proposed (
For every pixel, assign the
As can be seen from Figure
Initialize For any pixel Assume the If If Select Generate Repeat steps 2–5
Shadow encrypting architecture of the designed RGWVCS. The left half of the encryption architecture describes the generation process of the subpixel and encrypted secret sharing matrices, while the right half is the decryption process of secret sharing matrices.
After the overall grasp of the shadow encryption architecture in the previous section, this section will analyse the details of the right half of Figure
For the sake of clarity, (2, 2)-threshold VCS is taken as an illustrative example where
One black-white pixel
Encryption structure for a single black-white pixel, and the sets
Pixel | Strategy 1 | Strategy 2 | Subpixel matrix | |
---|---|---|---|---|
Share 1 | ||||
Share 2 | ||||
Share 1 | ||||
Share 2 |
Pixel expansion refers to the number of subpixels in shares encrypted from a secret pixel. In the example above, each pixel in the secret image is encrypted as a subpixel in each share. Therefore, there is no pixel extension. The degree of pixel expansion is negatively correlated with the actual resolution of the share image. A VCS with the lowest pixel extension indicates that its encryption is optimal.
Through the above analysis of the shadow encryption system, we can have a sense of the system. In this section, we will demonstrate the detailed process of encryption and decryption using an instance (3, 2, 4)-RGWVCS. Thus, we will have a clearer and more specific understanding of the shadow encrypting architecture. In the (3, 2, 4)-RGWVCS, the secret image needs to be encrypted into 4 secret sharing matrices with the same size. During the decryption side, it requires at least 2 shares to complete the secret recovery when the privileged participate in, whereas at least 3 shares are required.
For each pixel of S, the same position of the 4 secret sharing matrices,
In the proposed RGWVCS scheme, step 1 (initialize
In the proposed scheme, we use the Monte Carlo method to achieve secret layering according to the weight, in which 1000 groups of random cast experiments are conducted on four targets with equal probability by the Monte Carlo method. The experimental results are listed in Figure
The unit square plane in Figure
Distribution and statistical histograms of 1000 random points in four regions of equal area indicate that the Monte Carlo method is reliable: (a) random point density graph; (b) statistical histogram of casting point probability.
After the detailed introduction and analysis of the relevant algorithms, we will carry out theoretical analysis on the security and visual recognition of the designed scheme from Conditions
In the proposed (
In the proposed (
In the proposed (
Firstly, when If If Secondly, when If If
In conclusion, if we superpose
The designed scheme is an effective (
Firstly, based on Lemma
The scheme is designed to take the possible existence of the privileged into account.
Through the analysis of the designed method, a larger weight leads to a larger probability of covering
There is a lot of class division in real life, and being high class also means having certain privileged rights. The traditional visual secret sharing scheme ignores the existence of real privileges and becomes unsuitable for all real situations. The weighted visual secret sharing scheme with privileges can solve this problem well. The following company is taken as an example.
In this case of the company above, there are two classes, including manager and clerk. Managers are superior to their clerks and thus have a greater capacity of secret recovery. It can be seen from Figure
Schematic of an instance of privilege in a company situation, consisting of two distinct classes of manager and clerk. The two classes have different secret image recovery ability: (a)
In this section, we will show the performance of RGWVCS through experimental results, in which the validity of the scheme is verified from the quality and characteristics of the encrypted and decrypted images, and the superiority of the scheme is illustrated by comparing with other schemes.
In the designed RGWVCS for (3, 2, 4) threshold, secret image S is divided into four encrypted images
It can be seen from Figure
The outcomes of the proposed RGWVCS for (3, 2, 4) threshold where
So, according to the above analysis, we can draw the following conclusions: The experimental results of The quality of secret recovery is proportional to the total weight of participants.
In this section, we will first use the contrast as the main indicator to accurately describe the quality of the recovered secret images from different combinations. Then, we will analyse the relation between contrast and weight. Finally, we will compare the contrast between our scheme and relevant schemes.
From Table
The contrast and weight of the decrypted image formed by superimposing shares.
Superimposed encrypted image | Contrast | Weight |
---|---|---|
— | 0.1 | |
— | 0.2 | |
— | 0.3 | |
— | 0.4 | |
— | 0.3 | |
— | 0.4 | |
— | 0.5 | |
0.5 | ||
0.6 | ||
0.7 | ||
0.6 | ||
0.7 | ||
0.8 | ||
0.9 | ||
1 |
In Figure
The relation between the contrast and total weight of the decrypted image with different combinations.
For fear of the influence of randomness of the secret image in above part, we carried out much more identical tests with the designed scheme and other related schemes, in which we selected randomly 100 binary images with different sizes and patterns for encryption and decryption. Finally, we calculated the average contrast for different combinations in 100 tests to prove the superiority of the designed scheme. In the experiment, we set that
Table
The contrast of the proposed scheme under the (3, 2, 4) threshold, compared with Tu et al. [
Collected shadows | Ours | Tu et al. [ | Fan et al. [ | Yang et al. [ |
---|---|---|---|---|
— | 0.1035 | 0.1105 | 0.1096 | |
— | 0.1226 | 0.1526 | 0.1531 | |
0.1505 | 0.1993 | 0.1985 | ||
— | 0.1681 | 0.1981 | 0.1986 | |
0.1872 | 0.2486 | 0.2491 | ||
0.2962 | 0.2176 | 0.3021 | ||
0.2249 | 0.2488 | 0.2485 | ||
0.2523 | 0.3038 | 0.3040 | ||
0.2841 | 0.3623 | 0.3642 | ||
0.3192 | 0.4263 | 0.4278 | ||
0.3873 | 0.4988 | 0.4996 |
In this section, we will compare the quality of the recovered image and a series of characteristics of the proposed scheme with some typical WVCS schemes with admirable features, such as Yang et al. [
In Table
A comparison of series of representative features between the designed scheme and relevant VCSs.
Proposer | Threshold | Recovery efficiency (complexity) | No pixel expansion | No codebook design | No weight leakage | Weighted |
---|---|---|---|---|---|---|
Shamir et al. [ | ✗ | ✗ | ✓ | ✓ | ||
Yang et al. [ | ✓ | ✗ | ✓ | ✓ | ||
Tan et al. [ | ✓ | ✓ | ✗ | ✓ | ||
Liu et al. [ | ✓ | ✓ | ✗ | ✓ | ||
Fan et al. [ | ✓ | ✓ | ✓ | ✓ | ||
Tu et al. [ | ✓ | ✓ | ✓ | ✓ | ||
Ours | ✓ | ✓ | ✓ | ✓ |
From Table
After comparing the characteristics of different schemes, we use the objective evaluation index recall to quantitatively compare our scheme with others. The formula of recall is as follows, where
A comparison of recall between the designed scheme and relevant VCSs.
The traditional (
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
The authors declare that they have no conflicts of interest regarding the publication of this paper.
This work was supported by the National Key R&D Program of China under grant no. 2018YFB1003205; by the National Natural Science Foundation of China under grant nos. U1836208, U1536206, U1836110, 61972207, 61602253, and 61672294; by the Engineering Research Center of Digital Forensics, Ministry of Education; by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) Fund; and by the Collaborative Innovation Center of Atmospheric Environment and Equipment Technology (CICAEET) Fund, China.