Research on computer 3D image encryption processing based on the nonlinear algorithm

3.1 Piecewise linear chaotic mapping

The simplest chaotic map is the piecewise linear chaotic mapping (PLCM), and the fixed-point technique may be used to materialize it with limited precision. It offers a certain advantage in terms of implementation and speed, and therefore it piques the attention of researchers. Because the number of segments in the PLCM in use is so small, chaotic maps must be iterated for many hours for cryptosystems to be secure and take advantage of the chaotic feature. As a result, encryption speed falls as a result. We also do not have the option of choosing between information obtained in connection and encryption speed. PLCM, the encryption system using this map, has the characteristics of a statistical principle. The definition of its four-segment linear chaotic map is as follows (1):

In the above formula, y ₀ is the initial value of the state, where y _n ∈ [0, 1]; P is the control parameter of the encryption system, p ∈ (0, 0.5]. Its basic characteristics are as follows:

1. The system is chaotic, and the information output by the system has certain ergodicity, mixing, and other characteristics in the defined interval;

2. The distribution characteristics are uniform and will not change;

3. Its output sequence autocorrelation function is δ-like [18].

3.2 Algorithm description of cryptographic scheme

The core idea of the algorithm is to generate the corresponding chaotic sequence from the external input key, change the chaotic sequence into a key sequence that can be used for image encryption, and then encrypt the image with multiple rounds of pixel value substitution [19]. The specific flow of the algorithm is as follows:

1. Transform the image to be encrypted into a 1D sequence p, p = {(si) = 1.2 M × N}, where m × n is the size of the image.

2. Generate the initial value and parameter p required by the system. There are m external keys input, i.e., keys K = k ₁, k ₂, k ₃…k _m , where k _j is the ASCII code value corresponding to the visible characters in the key string. That is, the following two formulas are used to process k _j to obtain the initial state value and initial parameter value of the corresponding k _j :

   y j ( 0 ) = 1 256 k j + ∑ j = 1 m k j mod 256 ,   j = 1 ,   2 , … , m ,

   (2) y j ( 0 ) = 1 128 k j + ∑ j = 1 m k j mod 256 ,  mod  1   j = 1 ,   2 , … , m .

3. Replacing the key sequence Key(i) with the image pixel value.

   Based on the calculation of the total number of pixels in the corresponding image, the total number of keys needed by all PLCM is calculated as follows:

   (3) h = S m .

   When the formula is not divisible, the generation of the remainder key is completed by using the mth mapping in PLCM using y _j (0) and y _j (0) calculated in the previous step, a chaotic sequence with a length of H is generated by completing H iterations of the formula. All chaotic sequences obtained are as follows:

   (4) { F ( i ) | = 1 , 2 , … , S } .

   Then, the chaotic real number in the sequence is transformed by the following method to obtain a key sequence with the integer value {Key(i) \ = 1, 2, S}:

   (5) Key ( i ) = F i ⁎ 10 n mod 256 .

   In the above formula, the value of n is a positive integer, and the specific value and the actual calculation accuracy shall prevail. In this article, according to the requirement of calculation accuracy, n is taken as 6, that is, n = 6.

4. The pixel value of the encrypted image is subjected to multiple rounds of alternative encryption: in the first round, the first point of the plain text pixel of the image is subjected to separate encryption processing, and it is converted to any point of the image in the 1D sequence.

Encryption of p(i) is completed by using plaintext value p(i) and current Key(i), cipher text value C(i − 1), and previous Key(i − 1) of the previous pixel of the image. This encryption method is a form of encryption using the feedback of cipher text. In the k round, the cipher text encrypted in the previous round is regarded as the plain text encrypted in this round, and then the encryption is completed in the same way as in the first round. Here, the processing mode of the first point of the image pixel sequence is slightly different from that of the first round.

At this time, the encryption calculation is completed by using the value of the first point, Key(1), and the value C(S) after the last pixel encryption in the previous round. The 1D pixel sequence in the original image is P(i), and the sequence corresponding to the image is represented by C(i) after encryption. This alternative encryption method can use the following expression of flow according to code:

The first round:

where K = (k > 1), and

where C(S) represents the value of the last pixel in the last round of image encryption during encryption. When using this kind of multi-round encryption, it usually has a good effect after three rounds. Generally speaking, the more rounds, the better the encryption effect will be.

1. The obtained matrix is transformed into a 2D matrix, and the corresponding cipher text image can be obtained. The decryption process is similar to the encryption process, which is the reverse operation of encryption. After the key stream sequence is generated by PLCM; the decryption work can be completed by the decryption formula. When decrypting, first decrypt the encrypted sequence in the last step of the encryption process, then decrypt the penultimate step, and so on, until the decryption is completed.

Therefore, it can be known that the pixels in each round of decryption start from the pixel at the end of the image and then are decrypted to the first pixel in the image. The range of the plain text value p(i), the key Key(i), and the cipher text value C(i) of the image pixel point is [0,255], and the decryption method of the (k − 1)th round obtained from the encryption sequence of the (k)th round can be deduced from the above formula, specifically as follows:

Then, the plain text sequence before encryption of the first round is obtained from the cipher text sequence of the second round. The decryption of this part is as follows:

After that decryption is finished, the obtained sequence P is converted into an m × n matrix, and the image version corresponding to the original image can be obtained at this time. The initial external key used in decryption is consistent with that used in encryption, and the decrypted image is consistent with the original image.

4.1 Experimental simulation and result analysis

In the experiment, the image used is an image with a size of 256 × 256 and a picture gray value of 8 bits. In the experiment, the system parameters are set as follows: initial iteration value x0 = 0.27, control parameters p = 0.24, L = 1,000, x2 = 178, α = 112, and β = 10.

The experimental environment of this article is MATLAB7.0.

4.2 Analysis of anti-attack of selected plain text and selected cipher text

When the image is encrypted, its control parameters and initial values will change according to the scrambled sequence t and the gray value of the image. Therefore, the parameter values mentioned above are determined by the scrambling sequence and the gray value of the pixel. For different image encryptions, the key sequence streams used are also different, so when performing partial decryption, the corresponding key stream is needed, otherwise it is difficult to decrypt correctly. Moreover, to decipher, it is necessary to analyze the scrambling sequence T, system parameters α and β, and the key stream used for encryption. Because it is necessary to decipher the scrambled sequence when deciphering, the algorithm in this article has a good effect on resisting plaintext attacks and is proved as follows:

Let the cipher text to be deciphered be C = {C1, C2, C3,…, CM × N}.

1. In the first round of decryption, according to the above formula, the values of parameters α and β are needed to decrypt the gray value of pixels in the image. However, the key stream d can only be obtained by using the initial value. The initial value of the system is determined by the parameters α and β, so the pixel gray value cannot be decoded by direct operation, and the exhaustive attack can only be used.

2. In the second round of decryption, as in the first round, the decipherer cannot decipher the pixel gray value by direct operation but can only decipher it by an exhaustive method.

3. When scrambling, the decipherer can recover the cipher text only on the premise of possessing the scrambling sequence T, and the scrambling sequence T needs a series of parameters, but these parameters are all keys, so the deciphering can only be done by exhaustive methods.

To sum up, when deciphering cipher text, it can only be done in an exhaustive way, so the encryption algorithm in this article plays a very good role in resisting attacks.

4.3 Correlation of adjacent pixels

In an image, if the correlation between adjacent pixels is very large, it can be explained that the redundancy of the image is greater, and the possibility of obtaining the original image from the cipher text is lower. Therefore, for the encryption algorithm, the correlation of the original data should be considered, even if this correlation is no longer shown in the cipher text, so that the correlation between image contents is concealed. The correlation between image elements is an important parameter, so it is necessary to deal with the correlation between cipher text and plain text so that the head system between them no longer shows some regularity.

Randomly select 1,000 pairs of horizontally, vertically, and diagonally adjacent point pairs. See Table 1 and Figure 1 for related statistics before encryption and Table 2 for related statistics after encryption. It can be seen that the correlation between adjacent pixels of the image before encryption is very strong, while the correlation between pixels of the cipher text image is very weak, which can be regarded as independent of each other, indicating that the encryption algorithm of Scheme II has good statistical characteristics.

Figure 1

Correlation between adjacent pixels of the image before encryption.

Table 3 states the time spent on encryption and decryption. Three images of various pixel sizes have been taken and their encryption and decryption times were considered. In this article, Akhshani’s method is used to analyze the correlation of image elements. Table 4 is the comparison of the correlation between image elements analyzed from horizontal, vertical, and diagonal angles under different algorithms. From the data in the table, it can be seen that the cipher text obtained by this algorithm has a weak correlation in the adjacent pixels of the image, and the adjacent pixels are basically irrelevant. Therefore, we can know that the algorithm in this article spreads the statistical features in the image to the random cipher text. Therefore, the algorithm in this article has higher security.

According to the data in the table, the cipher text generated by this technique has little correlation with the neighboring pixels of the picture, and the adjacent pixels are essentially unimportant. As a result, we may conclude that the technique in this study distributes statistical information in the picture to random cipher text. As a result, the method in this work is more secure.