We propose a new image encryption algorithm based on DNA sequences combined with chaotic maps. This algorithm has two innovations: (1) it diffuses the pixels by transforming the nucleotides into corresponding base pairs a random number of times and (2) it confuses the pixels by a chaotic index based on a chaotic map. For any size of the original grayscale image, the rows and columns are fist exchanged by the arrays generated by a logistic chaotic map. Secondly, each pixel that has been confused is encoded into four nucleotides according to the DNA coding. Thirdly, each nucleotide is transformed into the corresponding base pair a random number of time(s) by a series of iterative computations based on Chebyshev’s chaotic map. Experimental results indicate that the key account of this algorithm is 1.536 × 10127, the correlation coefficient of a 256 × 256 Lena image between, before, and after the encryption processes was 0.0028, and the information entropy of the encrypted image was 7.9854. These simulation results and security analysis show that the proposed algorithm not only has good encryption effect, but also has the ability to repel exhaustive, statistical, differential, and noise attacks.
With the development of science, technology, and society, the computer industry, in which a small branch of digital images applications has become increasingly pervasive, has come to occupy a dominant position worldwide. Digital images have become one of the most popular media types and are now used extensively in various fields such as politics, economics, defense, and education [
Various common image encryption algorithms are available, including text encryption technology, SCAN language-based encryption technology, quadtree image encryption technology, vector quantization encryption technology (VQ), encryption technology based on pseudorandom sequences, encryption technology based on the “key image” chaotic encryption technology, and image encryption technology based on DNA computing [
In 1994, Adleman first introduced DNA computing into the encryption field, which created a new stage of information processing. DNA encryption is a new frontier and is presently at the forefront of international cryptography research [
In summary, our study successfully combines chaotic encryption technology and DNA coding techniques in a method that has been verified via a large number of experiments and security analyses to prove the security and rationality of the algorithm.
DNA sequencing is the process used to map the nucleotide sequence forming a strand of DNA. Four bases, adenine (A), thymine (T), guanine (G), and cytosine (C) form the building blocks of genetic code. “A” binds with “T” and “G” binds with “C” [
The rules of DNA encoding.
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|
00-A | 00-A | 00-C | 00-C | 00-G | 00-G | 00-T | 00-T |
01-C | 01-G | 01-A | 01-T | 01-A | 01-T | 01-C | 01-G |
10-G | 10-C | 10-T | 10-A | 10-T | 10-A | 10-G | 10-C |
11-T | 11-T | 11-G | 11-G | 11-C | 11-C | 11-A | 11-A |
We assume that the size of the original grayscale image
In accordance with the principle of the complementary base, we set the nucleotide string (AT)(TC)(CG)(GA), (AT)(TG)(GC)(CA), (AC)(CT)(TG)(GA), (AC)(CG)(GT)(TA), (AG)(GT)(TC)(CA), (AG)(GC)(CT)(TA).
During the pixel diffusion, the nucleotides will be substituted using the DNA complementary rule. We randomly select one rule from the six available to achieve a complementary substitution. Thus, we can achieve our objective of pixel diffusion.
In this paper, we use two types of chaotic maps: logistic chaotic map and Chebyshev’s chaotic map. The logistic chaotic map is a polynomial map of depth two [
Therefore, our algorithm confuses the pixels using the logistic map and suppose that the size of the original grayscale image
The primary objective of this step is to find the index of the largest number from the sequence with size
The expression of Chebyshev’s map is as follows:
We generated iterations of complementary replacements of encoding DNA using Chebyshev’s map, which is mainly used to diffuse the pixels of an image. The main steps are as follows.
Chebyshev’s map has two initial values,
The overall encryption process is depicted in Figure
Block diagram of the proposed model.
If we suppose that the size of the original grayscale image Input: image Output: the encrypted image.
switch case 0, do not change case 1, case 2, case 3,
The complementarily substituted DNA sequence is
The decryption and encryption algorithm processes are reversed and operate as follows: Input: encryption image Output: original grayscale image.
In our experiment, we first set the initial values and parameters of the logistic map:
The original and the encrypted images.
Lena
Permuted rows and columns
Encrypted image
Shrek
Permuted rows and columns
Encrypted image
3D color Lena
Encrypted Lena
Any chaotic system is sensitive to the initial values. To make the encryption algorithm highly secure, the key space should be large enough to make any brute-force attack ineffective. Here, all the keys are from the process of confusing and diffusing the pixels. Our encryption algorithm actually does have some of the following secret keys: the initial values of the logistic map, the initial values of Chebyshev’s map, DNA coding and complementary rules and random integers
The sensitivities to the initial values and parameters of the logistic map are both considered to be
In the case of Chebyshev’s map, when the change in the initial value is small,
We encrypted Lena’s image with two similar initial values (
Encryption of Lena’s image using similar initial values and their differences.
The difference between (a) and (b)
There are only eight kinds of coding DNA that meet the complementary rule, and there are altogether six kinds of legal complementary rules. We used coding DNA rules twice and DNA complementary rules once. Therefore, the key space of the random integers is
Following statistical analysis of the original and encrypted images, we constructed grayscale histogram analysis of Lena and Shrek and their encrypted images (Figure
The histograms of Lena’s and Shrek’s images.
Grayscale histogram of Lena’s image
Grayscale histogram of Lena’s encrypted image
Grayscale histogram of Shrek’s image
Grayscale histogram of Shrek’s encrypted image
The correlation coefficient is the evaluation criterion used to find the degree of linear correlation between two random variables. The range of our correlation coefficient
Comparison of the correlation coefficients of Lena’s images.
Correlation | Horizontal | Vertical | Diagonal |
---|---|---|---|
Original image | 0.9765 | 0.9139 | 0.9437 |
Encrypted image | 0.0002 | 0.0024 | −0.0032 |
Zhang et al. [ |
0.0036 | 0.0023 | 0.0039 |
Zhang et al. [ |
0.0046 | 0.0040 | 0.0017 |
Information entropy is the average information from which the redundant part has been excluded. Information entropy is the most important feature of randomness. Let
Information entropy values.
Image | Lena | Shrek |
---|---|---|
Original image | 7.4224 | 7.3104 |
Encrypted image | 7.9854 | 7.9479 |
A general requirement for all image encryption schemes is that the encrypted image be significantly different from its original version. This difference can be measured by means of two criteria: namely, the number of pixel change rate (NPCR) and the unified average changing intensity (UACI). NPCR is the change rate of the encrypted image pixels when the image changes one pixel in the process of encryption. The larger NPCR is, the stronger the resistance is of the algorithm to plaintext attack. UACI is the change rate of the average strength of the original image and the encrypted image. The larger UACI is, the stronger the resistance is of the algorithm to differential attacks [
The formulas used to calculate NPCR and UACI are as follows:
We calculated the correlation coefficients, NPCR, and UACI of the original and encrypted images of Lena and Shrek (Table
Correlation, NPCR, and UACI for the encrypted Lena and Shrek.
Image | Correlation | NPCR (%) | UACI (%) |
---|---|---|---|
Figures |
0.0028 | 99.7017 | 28.2970 |
Figures |
−0.0075 | 95.9316 | 26.0718 |
Zhang et al. [ |
0.0033 | 99.61 | 38 |
One of the most important problems in real-world communication technology is the robustness of a cryptosystem against noise. Signal-independent noise very often occurs between the transmitter and the receiver when an image is transmitted electronically. The error propagation phenomenon implies that errors in the encrypted image will lead to errors in the decrypted image [
Our cryptosystem is robust against noise because we first diffuse the pixels by permuting the rows and columns and then confuse each pixel according to the DNA complementary rule. Since the pixels changed by the noise will not propagate in the decrypted image during the decryption process, our algorithm is very robust against noise.
We prefer to use white Gaussian noise because it provides a reasonable assumption for the unavoidable randomness of the real physical channel, and the random numbers of this type of noise are uniformly distributed [
Figure
Correlation coefficient, NPCR, and UACI between original and decrypted images of Lena in the presence of noise.
Image | Correlation | NPCR (%) | UACI (%) |
---|---|---|---|
Figures |
0.9499 | 99.2988 | 28.4779 |
Figures |
0.9287 | 99.6154 | 28.5392 |
Figures |
0.9096 | 99.6627 | 28.7051 |
Encrypted images with noise and their corresponding decrypted images.
Mean = 0, variance = 0.001
Mean = 0, variance = 0.001
Mean = 0, variance = 0.003
Mean = 0, variance = 0.003
Mean = 0, variance = 0.005
Mean = 0, variance = 0.005
In this paper, we proposed a novel confusion/diffusion algorithm for image encryption. First, we exchanged the pixel positions of rows and columns of the digital image according to a chaotic index based on the logistic chaotic map to confuse the image pixels. Then, we encoded each of the pixels that had been confused into four nucleotides and obtained a one-dimensional nucleotide sequence after a series of iterative computations based on Chebyshev’s chaotic map. Next, we transformed each nucleotide into its corresponding base pair a random number of time(s) according to the complementary rule. Finally, we converted the two-dimensional matrix obtained into an encrypted image. Our experimental results and security analyses show that the scheme can achieve not only good encryption results, but also a sufficiently large key space to be able to repel common attacks. Therefore, the scheme is reliable enough to be applied in image encryption.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Fundamental Research Funds for the Central Universities (DL13CB04) and Nature Science Foundation of Heilongjiang Province (ZD201203/C1603), as well as Nature Science Foundation of Heilongjiang Province (LC2012C33).