Two Feistel rounds in image cryptography acting at the nucleotide level exploiting dna and rna property
Abstract
In recent years, a variety of chaos-based image encryption algorithm have been proposed. The majority of image encryption systems, employ the confusion-diffusion architecture and operate at the pixel level. In this paper, a new color image encryption algorithm at the nucleotide-level applied, will be proposed. The first block will be altered by confusion with a boot vector generated from the clear image. After that, two Feistel rounds will be imposed on each 12 nucleotides clear block. The first round is performed by chaotic displacement of the nucleotides, and, the second uses a confusion matrix created in preconceived designs from chaotic maps used. Ultimately, a diffusion will be implemented to maximize the avalanche impact and keep the system out of differential attacks. On output, a transition to the nucleotides complementary will be established, and a confusion with the chaos vector will be developed. Next, the RNA application can be used to synthesize amino acids, and as a result, the isolation of the exon genes from introns. Simulations made on a large database with images in a variety of sizes and formats, show that our strategy can be prevented from being attacked by any known attack.
Keywords
G_{t} = Z/tZ Chaotic map FEISTEL two-round diagram DNA RNAAbbreviations
- \(\hbox{G}_{{\rm t}}^{{*}}\)
Set of invertible elements of the G_{t}-ring
- DNA
Desoxi-ribonucleic acid
- RNA
Ribonucleic acid
- M(i:)
Line number i of matrix M
- M(:I)
Column number i of matrix M
1 Introduction
Communications has always been an important aspect in the acquisition of new knowledge and the development of humanity. The need to be able to send a message securely is probably as old as the communications themselves.
From a historical point of view, it is during conflicts between nations that this need has been most acute lively. In our modern world, where various methods of communication are used regularly, the need for confidentiality is more present than ever in a multitude of levels. For example, it is normal for a firm to want to protect its new software against piracy, that banking institutions want to ensure that transactions are and that all individuals want their personal data protected. The need for secure communications has given rise to the science we call cryptology.
The encryption operation transforms plain text into encrypted text, called a cryptogram, using a key (called the encryption key). Decryption is the processing of data that transforms encrypted text into plain text. Cryptography is the science of creating such encryption systems. Cryptanalysis is the complementary science of determining certain properties of these systems in order to reconstruct the plain text, All accepted algorithms incline to Shanon’s recommendation “confusing diffusion permutation” and also to Kerchof’s principle “the algorithm must be known, the security of the algorithm is based on the confusion of the encryption key.
Cryptology includes cryptography and cryptanalysis, often in the absence of the necessary parameters for decryption. Encryption is generally performed using a well-defined algorithm and a unique key.
With the development of innovative technologies in the field of information sciences and the digital world, all data is increasingly shared on the Internet or stored on magnetic media. On the other hand, unauthorized access to information or private information has become an issue in the virtual world. Security issues have raised many concerns not only among researchers, and people using the Internet. Encryption has become an effective way to avoid attacks. However, traditional encryption standards, such as DES and AES, are generally designed for textual information. They are considered unsuitable for information with a high correlation between data, such as image and video data.
The fundamental architecture of image encryption based on chaotic systems was first proposed by Fredrich in 1998. Within this structure, two unrelated steps are performed in a single encryption cycle. These are the phases of confusion and diffusion. First, a random permutation of all pixels leads to a large reduction in the correlation between adjacent pixels. However, confusion operations are carried out in a closed manner. Sometimes, a one-dimensional chaotic or random sequence is used to systematically modify the value of each pixel by an XOr operator and a chain with the already encrypted pixels. After this step, the basic elements of the image, the bit or pixel values, will be evenly distributed.
Since the image is introduced and processed in digital form, that its applications have continued to increase. It has become exploited by a wide public, both professionals and amateurs alike. But given the extent of the computer resources allowing the free circulation of information, the ease of transmission of confidential data, man has been pushed to improve more and more encryption algorithms to secure his confidential data. But all his techniques bow to Shanon’s recommendations [1] For permutation, some approaches use Arnold’s method [2], others use improvements to Hill’s classic method [3], and still others use static permutation matrixes [4]. For confusion, most algorithms use the operator Xor noted between the plain text and the encryption key [5]. Few color image encryption methods use diffusion to avoid differential attacks. Today, faced with the great mathematical advance of chaos theory, we are experiencing a wave of algorithms based on the construction of recurring suites with a chaotic aspect [6] that is developing more and more at high speed. Chang’e Dong [7] offers color image encryption based on the construction of a coupled chaotic map. Wanga et al. [8] proposed a crypto system based on a multitude of chaotic maps that define an effective result. All these approaches use a Lyapunov exponent calculation [5] to check the installation of chaos and sensitivity to initial conditions. Most encryption algorithms operating on blocks used the Feistel scheme with several turns. RC4, RC6, DES used more than four towers [9]. In our approach, we will apply two Feistel towers on 24-bit.
Most encryption techniques and systems have been implemented on the modular arithmetic basis. However, these techniques have been extensively exploited and are under attack. This has encouraged researchers to invent other techniques based on the use of some DNA property. DNA cryptography is a new field of instinctive cryptography that has emerged from DNA computer research. Some algorithms used for DNA cryptography have limitations in that they still use modular arithmetic cryptography at some of their stages or are based on biological laboratory experiments, which is not appropriate in the digital computing environment. To fill this gap, we describe a new algorithm for color image encryption based on chaos, DNA transformation and the biological property of the RNA.
In 1994, Adleman [10] included the first scientist who conducted DNA experiments and as a result a new generation of DNA research was developed. Many researchers have retrieved the information side of DNA to use it in computer-based investigations. At the same time, DNA use in cryptography is beginning to surface and is being used as a medium in bioinformatics [11]. Gehani et al. [12] presented an image encryption algorithm for DNA strand cryptography.
Zhang et al. [13] have proposed a color image encryption using DNA characteristics and an enforcement of DNA sequence processing to encode the pixels in the image. They used chaotic and hyper-chootic maps to effectively carry out their approach [14].
Virtually all encryption schemes use chaotic systems combined with DNA performance and fixed value allocation for nucleotides [15] (A = 0; C = 1; G = 2; T = 3), this weakens the technique. However, in our approach, the nucleotide values will be selected pseudo randomly and will be related to the chaotic maps used. Others use the properties of genetic algorithms and DNA [16].
2 The proposed method
- A.Step 1:
- (1)
Development of chaotic maps:
- (1)
- (a)
The logistics map
- (b)
The Skew Tent Map
- (c)
Writing chaotic nucleotide sequences:
\(\left( {\varvec{IV}} \right)\) | Initial values | 0 | 1 | 2 | 3 |
Nucleotide | A (adenine) | C (cytosine) | G (guanine) | T (thymine) | |
Complementary | T | G | C | A |
This allocation of values to nucleotides weaknesses the cryptographic system. To reinforce our systems against attacks, the values assigned to nucleotides will be taken in pseudo random ways and highly sensitive to the chaotic maps used.
(PI) | U(n) | U(m) | V(nm) | V(2nm) |
(IP) | V(n) | V(m) | U(nm) | U(2nm) |
\(\left( {\varvec{PI}} \right)\varvec{ }\) | 0.8858 | 0.87452 | 0.54215 | 0.92450 |
\(\left( {\varvec{IP}} \right)\varvec{ }\) | 0.31457 | 0.92450 | 0.8858 | 0.80745 |
\(\left( {\varvec{PR}} \right)\varvec{ }\) | 4 | 1 | 2 | 3 |
\(\left( {\varvec{PR}} \right)\varvec{ }\) | 1 | 4 | 3 | 2 |
\(\left( {\varvec{NT}} \right)\) | Final value | 0 | 1 | 2 | 3 | ||
Nucleotides | T | A | C | G | \(\left( {\varvec{PR}} \right)\varvec{ }\) | For logistic map | |
Complementary | T | G | C | A | \(\left( {\varvec{RP}} \right)\varvec{ }\) | For SKTM |
- (d)
Conversion of chaotic values into nucleotides.
Example:
Chaotic \((\varvec{U}_{\varvec{i}} )\) sequence | |||||||||||
0.2925 | 0.68251 | 0.9658 | 0.0325 | 0.8952 | 0.4125 | 0.3215 | 0.5214 | 0.5602 | 0.7854 | 0.2153 | 0.6582 |
Chaotic \((\varvec{V}_{\varvec{i}} )\) sequence | |||||||||||
0.2925 | 0.68251 | 0.9658 | 0.0325 | 0.8952 | 0.4125 | 0.3215 | 0.5214 | 0.5602 | 0.7854 | 0.2153 | 0.6582 |
By applying algorithms 3, 4, we obtain the following values
- (e)
Construction of the \(\left( {\varvec{CM}} \right)\varvec{ }\) confusing matrix
- (f)
Algebraic DNA operations
- B.Step 2:
- (a)
Setting the image to be encrypted
- (a)
- (a)
Translating the clear image to a vector
- (b)
Writing clear pixels in nucleotides
\(\left( {\varvec{TN}} \right)\) | Final value | 0 | 1 | 2 | 3 |
Nucleotides | G | T | C | A | |
Complementary | A | T | G | C |
To simplify the transaction of the integer values of the pixels of the vector X into nucleotides, we will use the matrix \(\left( {\varvec{MN}} \right)\varvec{ }\) described below
Later, each \(\left( {\varvec{G}_{\varvec{k}} } \right)\) block and \(\left( {\varvec{D}_{\varvec{k}} } \right).\) block will be transcribed into nucleotides values using the \(\left( {\varvec{MN}} \right)\varvec{ }\) matrix.
Example:
- (c)
Decomposition of the nucleotide image vector into two matrices
- (d)
Building The Initialization Vectors
Two boot \(\left( {\varvec{Vl}} \right)\) and \(\left( {\varvec{Vd}} \right)\) vectors are built. The first from the \(\left( {\varvec{MG}} \right)\) matrix and the second from the \(\left( {\varvec{MD}} \right)\) matrix. This process is controlled by the algorithm 9
\(\begin{aligned} & {\text{Algorithm}}\,8 \\ & \quad \left\{ {\begin{array}{*{20}l} {for \,i = 1 \,to \,6} \hfill \\ {Vl\left( i \right) = VN\left( {i + m} \right)} \hfill \\ {Vd\left( i \right) = NV\left( {i + n} \right)\,} \hfill \\ {for \,k = 2 \,to\, nm} \hfill \\ {Vl\left( \varvec{i} \right) = Vl\left( \varvec{i} \right) + MG\left( {k,i} \right)} \hfill \\ {Vd\left( \varvec{i} \right) = Vd\left( \varvec{i} \right) + MD\left( {k,i} \right)} \hfill \\ {Next \,k} \hfill \\ {Next \,i} \hfill \\ \end{array} } \right. \\ \end{aligned}\).
- A.Step 3:
- (2)
Encryption the clear image
- (2)
- (a)Analytical expression on the encryption function
- (a)
First Feistel-Round
- (a)
- (b)
Second Feistel-Round
So, in our approach the mathematical expression of two Feistel’s rounds applied to block \(\varvec{U}_{\varvec{k}} \left( {\left( {\varvec{MG}\left( {\varvec{k}:} \right),\varvec{MD}(\varvec{k}:} \right)} \right)\) is given by algorithm 14: We pose \(\left\{ {\begin{array}{*{20}c} {FEistel2 = \varvec{\varphi }_{\varvec{k}}^{2} o \varvec{\varphi }_{\varvec{k}}^{1} )} \\ \end{array} } \right\}\).
- (b)
Transition to complementary
- (3)
Applying m-RNA
The amino acid vector value \(\left( {\varvec{XR}} \right)\) is a component in \(\left( {\varvec{G}_{64} } \right)\).
Example
- (a)
Applying t-RNA
- (a)
Design the direction 5’
The transcribed gene is defined by a step shift equal to 4 nm-D5, so the first codon of the unexplained gene is the D5 codon.
- (b)
Design the direction 3′
The reading direction 3′ of the t-RNA, represents the end of the transcribed gene. In our case, it is the \(\left( {\varvec{GN}} \right)\) codon value (4 nm).
- (c)
Compute the stop codon
- (b)
Applying r-RNA
- (a)
Compute the step of the shift
L: represents a length the EXON block of position i. We note that: \(\forall \varvec{ i step}\left( \varvec{i} \right) \ne 0\).
Example: In \(\left( {\varvec{G}_{8} } \right)\)
Exon and Intron Genes Extraction is given in the table
The output genus is
- B.Step 4:
- (4)
Encrypted image decryption
- (4)
This following table gives the original image and its histogram, as well as the encrypted image and its histogram
The following table illustrates the encryption of some of the most commonly used images in encryption theory
Example:
- C.Step 5:
- (1)
Investigation of cryptosystem performance
- (1)
- (a)
Key space
- (b)
Secret key’s sensitivity Analysis
- c)
Entropy Analysis
The entropy values of the images encrypted by our algorithm are around 8, it is the maximum value for a color image encoded on 8 bits. It confirms the uniformity of the histograms. This proves that this approach is safe from entropy attack.
- (d)
Correlation analysis
- (a)
Horizontal correlation
- (b)
Vertical correlation
- (c)
Diagonal correlation
- (2)
Differential analysis
In general, an attacker can make a very small change on the clear image (for example, change only one bit, then study the change in the result obtained. In so doing, it may be able to find a relevant relationship with the original image and the encrypted image. If a small change in the clear image may cause a large change in the encrypted image, in terms of diffusion and confusion, in this case this differential attack would become ineffective and virtually useless. To test the influence of pixel change on the entire encrypted image by the proposed algorithm, two common measures were used: the pixel change rate (NPCR) and the unified average pixel change intensity (UACI). Note two encrypted images, whose corresponding free-to-air images differ by only one pixel, from \(\left( {\varvec{C}_{1} } \right)\) and \(\left( {\varvec{C}_{2} } \right)\), respectively. The expressions of these two statistical constants are given by Eqs. 12 and 13, for an image size (n, m)
All detected values are inside the confidence interval \(\left[ {99,932 99,946} \right]\). These values are largely sufficient to affirm that our crypto system is protected from known differential attacks
- (a)
Avalanche effect
The avalanche effect is a required property in virtually all cryptographic hash functions and block coding algorithms. It causes progressively more important changes as the data is propagating in the structure of the algorithm. Therefore, by perturbing a single bit at the input, we can obtain a very different output, (about 1 bit out of 2 changed) explaining the name of this phenomenon. The avalanche effect makes it more difficult to reverse the function due to its chaotic properties (if well designed).
Figure 13 depicts the evaluation of the AE score for 70 images examined by our approach.
- b)Signal-to-peak noise ratio (PSNR)
- (a)
MSE
- (a)
Mean Square Error (MSE): This is the cumulative square deviation between the original image and the additional noise image. When the MSE level is reduced, the error is reduced.
\(\left( {P\left( {i,j} \right)} \right)\); pixel of the clear image
- (b)
PSNR
- (c)
Speed analysis
Execution time (in second)
Image | Our method | DES | AES |
---|---|---|---|
Lena (256 × 256) | 0.09644 | 0.639772 | 5.687244e−002 |
Lena (512 × 512) | 0.17469 | 7.449005 | 0.347506 |
Lena (1024 × 1024) | 0.48421 | 29.11398 | 1.152980 |
- (1)Bio security
- (a)
Nucleotide statistical analyses
- (a)
- (a)
Percentage nucleotides isolated
% | A | C | T | G |
24 | 26 | 25 | 25 |
- (b)
Percentage of nucleotides used in pairs
We notice that the template is not symmetrical.
In a regrouping of two nucleotides we notice that:
- (c)
Percentage of codons
In a regrouping of nucleotides into codons, we get the following results.
We notice that the template is not symmetrical. There is nearly uniformity in the distribution of codons in the encrypted image.
- (b)
Math security
The large size of our encryption key protects the system from brutal attacks. In parallel, the different pseudo random choice of nucleotides for the transcription of chaotic maps and pixels of the original image complicates the detection and screening of nucleotides and their complementary ones. In addition, the key mask used in conjunction with (One Time Pad) OTP is the same sized as the clear image. It is widely believed that OTP is secured.
3 Conclusion
This article outlines a new method of color image encryption using several biological DNA and RNA property. Such a technique models the mathematical problem into a biological question. First, two Feistel rounds of 12 nucleotides have been applied to each block. The first is a chaotic shift of nucleotides, while the second is a confusion caused by a random matrix, followed by a diffusion with the next clear block, to increase avalanche response. In a second stage, a confusion of the output vector is established. A transition to complementary nucleotides is implemented, with amino acid formation. Finally, the 5′ and 3′ directions are computed and the stop codons are evaluated, the EXONS blocks are isolated from the INTRONS blocks. EXONS have been substituted, while INTRONS have been chaotically shifted. Simulations conducted on a significant number of color images allow us to ensure that our approach can overcome any known attack.
Notes
Compliance with ethical standards
Conflict of interest
I am the alone author of this article, and therefore no conflict. To finalize this document, I did not receive any assistance funds from any organization. This document does not contain any studies or experiments on animals.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the author.
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