1 Introduction
The exchange of confidential images via internet is usual in today’s life, even though the internet is an open source that is unsafe and unauthorised persons can steal useful or sensitive information. Therefore it is essential to be able to share images in a secure way. This goal is achieved by cryptography. Traditional cryptographic techniques such as data encryption standard (DES) and advanced encryption standard (AES) are not suitable for image transmission because image pixels are usually highly correlated
Mahmud (2020); Zhang (2014). In contrast, DES and AES are ideal techniques for text encryption ElLatif (2013), so researchers are trying to develop such techniques to meet the demand for reliable image delivery.A number of image encryption schemes have been developed using different approaches Yang (2015); Zhong (2019); Li (2017); Hua (2018); Xie (2017); Alzaidi (2018); Azam (2017); Luo (2018); Li (2015). One of the dominant trends in encryption techniques is the chaos based encryption Ismaila (2020); Tang (2019); Abdelfatah (2019); Yu (2020); Zhu (2018); ElKamchouchi (2020). The reason is that the chaos based encryption schemes are highly sensitive to the initial parameters. The chaos based algorithms normally use pseudorandom numbers and substitution boxes (Sboxes) to create confusion and diffusion Cheng (2015); Belazi (2016). “Confusion” means to hide the relation between input image, secret keys and the corresponding cipherimage, and “diffusion” means to alter the value of each pixel in an input image Mahmud (2020). Cheng et al. Cheng (2015) proposed an image encryption algorithm based on pseudorandom numbers and AES Sbox. The pseudorandom numbers are generated using AES Sbox and chaotic tent maps. The scheme is optimised by combining the permutation and diffusion phases, but the image is encrypted in rounds which is time consuming. Belazi et al. Belazi (2016) suggested an image encryption algorithm using a new chaotic map and logistic map. The new chaotic map is used to generate a sequence of pseudorandom numbers for a masking phase. Then eight dynamic Sboxes are generated. The masked image is substituted in blocks via aforementioned Sboxes. The substituted image is again masked by another pseudorandom sequence generated by the logistic map. Finally the encrypted image is obtained by permuting the masked image. The permutation is done by a sequence generated by the map function. This algorithm fulfils the security analysis but performs slowly due to the four cryptographic phases. In Rehman (2016) an image encryption method based on chaotic maps and dynamic Sboxes is proposed. The chaotic maps are used to generate the pseudorandom sequences and Sboxes. To break the correlation, pixels of an input image are permuted by the pseudorandom sequences. In a second phase the permuted image is decomposed into blocks. Then blocks are encrypted by the generated Sboxes to get the cipherimage. From histogram analysis it follows that the suggested technique generates cipherimages with a nonuniform distribution.
Similar to the chaotic maps, Elliptic Curves (ECs) are sensitive to input parameters, but EC based cryptosystems are more secure than those of chaos Jia (2016). Toughi et al. Toughi (2017) developed a hybrid encryption algorithm using elliptic curve cryptography (ECC) and AES. The points of an EC are used to generate pseudorandom numbers and keys for encryption are acquired by applying AES to the pseudorandom numbers. The proposed algorithm gets the promising security but pseudorandom numbers are generated via the group law, which is time consuming. In ElLatif (2013) a cyclic EC and a chaotic map are combined to design an encryption algorithm. The developed scheme overcomes the drawbacks of small key space but is unsafe to the knownplaintext/chosenplaintext attack Liu (2014). Similarly Hayat et al. Hayat (2019) proposed an EC based encryption technique. The stated scheme generates pseudorandom numbers and dynamic Sboxes in two phases, where the construction of Sbox is not guaranteed for each input EC. Therefore, changing of ECs to generate an Sbox is a timeconsuming work. Furthermore, the generation of ECs for each input image makes it insufficient.
Based on the above discussion, we propose an improved image encryption algorithm, based on quasiresonant Rossby/drift wave triads Bustamante (2013); Hayat (2016) (triads, for short) and Mordell Elliptic Curves (MECs). The triads are utilised in the generation of pseudorandom numbers and MECs are employed to create dynamic Sboxes using the technique introduced in Azam1 (2019). The proposed scheme is novel in that it introduces the technique of pseudorandom numbers generation using triads, which is faster than generating pseudorandom numbers by ECs. Moreover, the scheme does not require to separately generate triads for each input image of the same size. In the present scheme MECs are used opposite to Hayat (2019) in the sense that now, for each input image, the generation of a dynamic Sbox is guaranteed Azam (2018). Finally, extensive performance analyses and comparisons reveal the efficiency of the proposed scheme.
This paper is organised as follows. Preliminaries are described in Section 2. In Section 3, the proposed encryption algorithm is explained in detail. Section 4 provides the experimental results of the newly developed scheme. A comparison is drawn in Section 5 between the proposed method and existing popular schemes. Lastly, conclusions are presented in Section 6.
2 Preliminaries
Barotropic vorticity equation: The barotropic vorticity equation (in the socalled
plane approximation) is a partial differential equation of the form
(1) 
where represents the stream function, is a real constant and is a nonnegative real constant. We assume periodic boundary conditions: for all . In literature Eq. (1) is also known as the CharneyHasegawaMima equation (CHM) Charney (1948); Hasegawa (1978); Connaughton (2010); Harris (2013); Galperin (2019). This equation accepts many special solutions, including travelling wave solutions. Of central importance is the socalled Rossby wave, which is a solution of both the linearised form and the whole (nonlinear) form of Eq. (1). This solution is given explicitly by the parameterised function , where is called the angular frequency and
is called the wave vector. For simplicity we take
and in what follows Bustamante (2013); Hayat (2016).Resonant triad, quasiresonant triad and detuning level: A linear combination of Rossby waves parameterised by wave vectors that are not collinear, is again a solution of the linearised form of Eq. (1), but not a solution of the whole equation. However, to the lowest order of nonlinearity in Eq. (1), approximate solutions known as resonant triad solutions can be constructed via linear combinations. Any set of three wave vectors and satisfying the equations:
(2) 
for is called a resonant triad. If the equation is replaced by the inequality , for a large positive number , then the triad becomes a quasiresonant triad and is known as the detuning level of the quasiresonant triad. For simplicity, in what follows we call a quasiresonant triad simply a triad and denote it by . Finally, to avoid overcounting of triads we will impose the condition .
Rational transformation: In Bustamante (2013) wave vectors are explicitly expressed in terms of rational variables and as follows:
(3) 
In the case the rational variables lie on an elliptic surface. The transformation is bijective and its inverse mapping is given by:
(4) 
New parameterisation: In Kopp (2017), Kopp parameterised the resonant triads and in terms of parameters and it follows by (Kopp, 2017, Eq. (1.22)) that:
(5)  
(6)  
(7) 
In , Hayat et al. Hayat (2016) found a new parameterisation of and in terms of auxiliary parameters and hence and are given by:
(8)  
(9)  
(10) 
Elliptic curve: Let be a finite field for any prime , then an EC over is defined by
(11) 
where . The integers and are called parameters of an EC. The number of all satisfying the congruence (11) is denoted by . For , the type of ECs is known as Mordell elliptic curve (MEC). If points on are ordered according to some total order then is said to be an ordered EC. Recall that total order is a binary relation which possesses the reflexive, antisymmetric and transitive properties. Azam et al. Azam1 (2019) introduced a total order known as a natural ordering on MECs given by
and generated efficient Sboxes using the aforesaid ordering. We will use natural ordering to generate Sboxes. Thus from here on stands for a naturally ordered MEC unless it is specified otherwise.
3 The Proposed encryption scheme
The proposed encryption scheme is based on pseudorandom numbers and Sboxes. The pseudorandom numbers are generated using quasiresonant triads.
To get an appropriate level of diffusion we need to properly order the s. For this purpose we define a binary relation as follows.
3.1 Ordering on quasiresonant triads
Let represent the triads respectively, then
where and are the corresponding auxilary parameters of and respectively.
Lemma .
If denotes the set of s in a box of size , then is a total order on .
Proof.
The reflexivity of follows from and and hence As for antisymmetry we suppose and . Then, by definition and , which imply . So we are left with two results: and , which imply . Thus, we obtain the results and , which ultimately give . Solving Eqs. (8)–(10) for the obtained values we get and from Eq. (2) it follows that . Consequently and is antisymmetric. As for transitivity, let us assume and . Then and , implying . If then transitivity follows. If , then too. Thus, and , so . If then transitivity follows. If , then too. Thus, and , implying and hence transitivity follows: . ∎
Let stand for the set of s ordered with respect to the order . The main steps of the proposed scheme are explained as follows.
3.2 Encryption
A. Public parameters: In order to exchange the useful information the sender and receiver should agree on the public parameters described as below:

Three sets: Choose three sets of consecutive numbers with unknown step sizes, where the end points are rational numbers.

A total order: Select a total order so that the triads generated by the above mentioned sets may be arranged with respect to that order.
Suppose that represents an image of size to be encrypted, and the pixels of are arranged in column wise linear ordering. Thus, for positive integer , represents the th pixel value in linear ordering. Assume that is the sum of all pixel values of image . Then the proposed scheme chooses the secret keys in the following ways.
B. Secret keys: To generate confusion and diffusion in an image, the sender chooses the secret keys as follows.

Step size: Select positive integers to construct the step sizes of . Also choose a nonnegative integer as a step size of in such a way that , where represents the number of elements in .

Detuning level: Fix some posive integer to find the detuning level allowed for the triads.

Bound: Select a positive integer such that for This condition is imposed in order to bound the components of the triad wave vectors. Furthermore, choose an integer to find , where gives the nearest integer when is divided by . The reason for choosing such a is to generate key dependent Sboxes and the integer is used to diffuse the components of triads.

A prime: Select a prime such that as a secret key for computing to generate an Sbox on the .
The positive integers and are secret keys. Here it is mentioned that the parameters and are used to generate triads in a box of size . The generation of triads is explained step by step in Algorithm 1. These triads along with keys and are used to generate the sequence of pseudorandom numbers.
Thus represents the th triad in ordered set . Moreover, are the components of . In Algorithm 2, the generation of is explained.
The proposed sequence is cryptographically a good source of pseudorandomness because triads are highly sensitive to the auxiliary parameters Hayat (2016) and inverse detuning level . It is shown in Bustamante (2013) that the intricate structure of clusters formed by triads depends on the chosen , and the size of the clusters increases as the inverse detuning level increases. Moreover, the generation of triads is rapid due to the absence of modular operation.
C. Performing diffusion. To change the statistical properties of an input image diffusion process is performed. While performing the diffusion, the pixel values are changed using the sequence . Let denote the diffused image for a plainimage . The proposed scheme alters the pixels of according to the Eq. (12):
(12) 
D. Performing confusion. Nonlinear function causes confusion in a cryptosystem, and nonlinear components are necessary for a secure data encryption scheme. The current scheme uses the dynamic Sboxes to produce the confusion in an encrypted image. If stands for the encrypted image of , then confusion is performed by the Eq. (13) as follows:
(13) 
Lemma .
If and is a prime chosen for the generation of an Sbox, then the time complexity of the proposed encryption scheme is max.
Proof.
The computation of all possible values of and in Algorithm (1) takes time. Similarly the time complexity for generating is but executes times. Thus the time required by and hence by is . Also Algorithm 1 in Azam1 (2019) shows that the proposed Sbox can be constructed in time. Thus the time complexity of the proposed scheme is max. ∎
3.3 Decryption
In our scheme the decryption process can take place by reversing the operations of the encryption process. One should know the inverse Sbox and the pseudorandom numbers . Assume the situation when the secret keys , and are transmitted by a secure channel, so that the set is obtained using keys and , and hence the Sbox and the pseudorandom numbers can be computed by and . Finally, the receiver gets the original image by applying the following equations:
(14)  
(15) 
4 Security analysis
In this section the security of the proposed scheme is analysed. For this purpose the current scheme is implemented on all gray images of USCSIPI Image Database Dbase . The USCSIPI database contains images of size , Furthermore, some security analyses which are explained one by one in the associated subsections are presented. To validate the quality of the proposed scheme, the experimental results are compared with some other encryption schemes. The parameters used for the experiments are for resp., and varies for each . The experiments are performed using Matlab Ra on a personal computer with GHz Processor and GB RAM. All encrypted images of the database along with histograms are available at https://github.com/ikram702314/Results. Some plainimages House, Stream, Boat and Male and their cipherimages are displayed in Fig. 1.
4.1 Statistical attack
A cryptosystem is said to be secure if it has high resistance against statistical attacks. The strength of resistance against statistical attacks is measured by entropy, correlation and histogram tests. All these tests are applied to evaluate the performance of the discussed scheme.

Histogram. Histogram is a graphical way to display the frequency distribution of pixel values of an image. A secure cryptosystem generates cipherimages with uniform histograms. The histograms of the encrypted images using the proposed method are available at https://github.com/ikram702314/Results. However, the respective histograms for images in Fig. 1 are shown in Fig. 2. The histograms of the encrypted images are almost uniform. Moreover, the histogram of an encrypted image is totally different from that of the respective plainimage, so that it does not allow useful information to the adversaries, and the proposed algorithm can resist any statistical attack.

Entropy. Entropy is a standout feature to measure the disorder. Let be a source of information over a set of symbols , then entropy of is calculated by:
(16) where
is the probability of occurrence of symbol
The ideal value of is , if all symbols of occur in with the same probability. Thus, an image emanating gray levels is absolutely pseudorandom if The entropy results for all images encrypted by the suggested technique are shown in Fig. 3, where the minimum, average and maximum values are and respectively. These results are close to , and hence the developed mechanism is secure against entropy attacks. 
Pixel correlation. A meaningful image has strong correlation among the adjacent pixels. In fact, a good cryptosystem has the ability to break the pixel correlation and bring it close to zero. For any two gray values and , the pixel correlation can be computed as:
(17) where and
denote expectation and variance of
respectively. The range of is to . The gray values and are in low correlation if is close to zero. As the pixels may be adjacent in horizontal, diagonal and vertical directions, the correlation coefficients of all encrypted images along all the three directions are shown in Fig. 3, where the respective ranges of are [, ], [,] and [,]. These results show that the presented method is capable of reducing the pixel correlation near to zero.Figure 3: (a)(c) The horizontal, diagonal and vertical correlations among pixels of each image in USCSIPI database; (d) The entropy of each image in USCSIPI database. In addtion, pairs of adjacent pixels of the plainimage and cipherimage of Lena are pseudorandomly selected. Then correlation distributions of the adjacent pixels in all the three directions are shown in Fig. 4, which reveals the strong pixel correlation in the plainimage but weak pixel correlation in the cipherimage generated by the current scheme.
Figure 4: (b)(d) The distribution of pixels of planeimage in horizontal, diagonal and vertical directions; (f)(h) The distribution of pixels of cipherimage in horizontal, diagonal and vertical directions.
4.2 Differential attack
In differential attacks the opponents try to get the secret keys by studying the relation between the plainimage and cipherimage. Normally attackers encrypt two images by applying a small change to these images, then compare the properties of the corresponding cipherimages. If a minor change in original image can cause the significant change in encrypted image, then the cryptosystem has high security level. The two tests NPCR (number of pixels change rate) and UACI (unified average changing intensity) are usually used to describe the security level against differential attacks. For two plainimages and different at only one pixel value, let and are the cipherimages of and respectively, then NPCR and UACI are calculated as:
(18)  
(19) 
where if and otherwise. The ideal values of NPCR and UACI are and respectively. We applied the above two tests on each image of the database by pseudorandomly changing the pixel value of each image. The experimental results are shown in Fig. 5, where the average values of NPCR and UACI are and respectively. It follows from the obtained results that our scheme is capable of resisting a differential attack.
4.3 Key analysis
For a secure cryptosystem it is essential to perform well against key attacks. A cryptosystem is highly secure against key attacks if it has key sensitivity, large key space and strongly opposes the knownplaintext/chosenplaintext attack. The proposed scheme is analysed against key attacks as follows.

Key sensitivity. Attackers usually use slightly different keys to encrypt a plainimage and then compare the obtained cipherimage with the original cipherimage to get the actual keys. Thus, high key sensitivity is essential for higher security. That is, cipherimages of a plainimage generated by slightly two different keys should be entirely different. The difference of the cipherimages is quantified by Eqs. (18) and (19). In experiments we encrypted the whole database by changing only one key, while other keys are remain unchanged. The key sensitivity results are shown in Table 1, where the average values of NPCR and UACI are and respectively, which specify the remarkable difference in the cipherimages. Moreover, our cryptosytem is based on pseudorandom numbers and Sboxes. The sensitivity of pseudorandom numbers sequences and and Sboxes and for Lena is shown in Fig. 5.
Image NPCR(%) UACI(%) Image NPCR(%) UACI(%) Image NPCR(%) UACI(%) Female 99.62 33.39 House 99.62 33.23 Couple 99.56 33.30 Tree 99.59 33.35 Beans 99.64 33.23 Splash 99.60 33.97 Table 1: Difference between two encrypted images when key is changed to . 
Key space. In order to resist a brute force attack, key space should be sufficiently large. For any cryptosystem key space represents the set of all possible keys required for encryption process. Generally, the size of key space should be greater than In present scheme the parameters and are used as secret keys, and we store each of them in bits. So the key space of the proposed cryptosystem is which is larger than and hence capable to resist the brute force attack.

Knownplaintext/chosenplaintext attack. In knownplaintext attack, the attacker has partial knowledge about the plainimage and cipherimage, and tries to break the cryptosystem, while in chosenplaintext attack the attacker encrypts an arbitrary image to get the encryption keys. An allwhite/black image is usually encrypted to test the performance of a scheme against these powerful attacks. We analysed our scheme by encrypting an allwhite/black image of size . The results are shown in Fig. 6 and Table 2, revealing that the encrypted images are significantly pseudorandomised. Thus the proposed system is capable of preventing the above mentioned attacks.
Figure 6: (a) Allwhite; (b) Allblack; (cd) cipherimages of (ab); histograms of (cd).
Plainimage Entropy Correlation of plainimage NPCR(%) UACI(%) Hori. Diag. Ver. Allwhite 7.9969 0.0027 0.0020 0.0090 99.60 33.45 Allblack 7.9969 0.0080 0.0035 0.0057 99.62 33.41 Table 2: Security analysis of allwhite/black encrypted images by the proposed encryption technique.
5 Comparison and discussion
Apart from security analyses, the proposed scheme is compared with some well known image encryption techniques. The gray scale images of Lena and Lena are encrypted using the presented method, and experimental results are listed in Table 3.
Size  Algorithm  Entropy  Correlation  NPCR (%)  UACI(%)  Dynamic  
Hori.  Diag.  Ver.  Sboxes  Sboxes  
Ours  7.9974  0.0001  0.0007  0.0001  99.91  33.27  1  Yes  
Ref. Hayat (2019)  7.9993  0.0012  0.0003  0.0010  99.60  33.50  1  Yes  
Ref. ElLatif (2013)  7.9973        99.50  33.30  0    
Ref. Rehman (2016)  7.9046  0.0164  0.0098  0.0324  98.92  32.79  >1<50  Yes  
Ref. Belazi (2016)  7.9963  0.0048  0.0045  0.0112  99.62  33.70  8  Yes  
Ref. Wu (2017)  7.9912  0.0001  0.0091  0.0089  100  33.47  0    
Ref. Wan (2020)  7.9974  0.0020  0.0020  0.0105  99.59  33.52  0    
Ours  7.9993  0.0001  0.0042  0.0021  99.61  33.36  1  Yes  
Ref. Cheng (2015)  7.9992  0.0075  0.0016  0.0057  99.61  33.38  1  No  
Ref. Toughi (2017)  7.9993  0.0004  0.0018  0.0001  99.60  33.48  1  No  
  Ref. Tong (2016)  7.9970  0.0029  0.0135  0.0126  99.60  33.48  0   
Ref. Zhang (2014)  7.9994  0.0018  0.0012  0.0011  99.62  33.44  >1  Yes  
Ref. Zhang (2014)  7.9993  0.0032  0.0011  0.0002  99.60  33.47  >1  Yes 
It is deduced that our scheme generates cipherimages with comparable security. Furthermore, we remark that the scheme in Toughi (2017) generates pseudorandom numbers using group law on EC, while the proposed method generates pseudorandom numbers by constructing triads using auxiliary parameters of elliptic surfaces. Group law consists of many operations, which makes the pseudorandom number generation process slower than the one we present here. Moreover, the Sbox used in Toughi (2017) is a static one which is vulnerable Rosenthal (2003), while our scheme generates a dynamic Sbox for each image which is more secure Kazlauskas (2009). The scheme in Belazi (2016) decomposes an image to eight blocks and uses eight dynamic Sboxes for encryption purpose. The computation of multiple Sboxes takes more time than computing only one Sbox. Similarly the techniques in Zhang (2014); Rehman (2016) use a set of Sboxes and encrypt an image in blocks, while our newly developed scheme encrypts the whole image using only one dynamic Sbox. Thus, our scheme is faster than the schemes in Zhang (2014); Rehman (2016). The security system in Tong (2016) uses a chaotic system to encrypt blocks of an image. The results in Table 3 reveal that our proposed system is cryptographically stronger than the scheme in Tong (2016). The algorithms in Wu (2017); ElLatif (2013) combine chaotic systems and different ECs to encrypt images. It follows from Table 3 that the security level of our scheme is comparable to that of the schemes in Wu (2017); ElLatif (2013). The technique in Wan (2020) uses double chaos along with DNA coding to get good results as shown in Table 3, but the results obtained by the new scheme are better than that of Wan (2020). Similarly the technique in Hayat (2019) encrypts images using ECs, the disadvantage of that scheme is that it does not generate an Sbox for each input image and the generation of an Sbox needs trials, while our scheme generates a dynamic Sbox for each input image, thus making our scheme faster and more robust than the scheme developed in Hayat (2019).
6 Conclusion
An image encryption scheme based on quasiresonant triads and MECs is introduced. The proposed technique constructs triads to generate pseudorandom numbers and computes a MEC to construct an Sbox for each input image. The pseudorandom numbers and Sbox are then used for altering and scrambling the pixels of the plainimage respectively. As for the advantages of our proposed method, firstly triads are based on auxiliary parameters of elliptic surfaces, and thus pseudorandom numbers and Sboxes generated by our method are highly sensitive to the plainimage, which prevents adversaries from initiating any successful attack. Secondly, generation of triads using auxiliary parameters of elliptic surfaces consumes less time than computing points on ECs (we find a 4x speed increase for a range of image resolutions ), which makes the new encryption system relatively faster. Thirdly, our algorithm generates the cipherimages with an appropriate security level.
In summary, all of the above analyses imply that the presented scheme is able to resist all attacks. It has high encryption efficiency and less time complexity than some of the existing techniques. In the future, the current scheme will be further optimised by means of new ideas to construct the Sboxes using the constructed triads, so that for each input image we will not need to compute a MEC.
All authors contributed equally to this work.
This research is funded through HEC project NRPU7433.
Acknowledgements.
We thank Gene Kopp for useful comments and suggestions. The authors declare no conflict of interest. The funding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results. The following abbreviations are used in this manuscript:MEC  Mordell elliptic curve 
Sbox  Substitution box 
EC  Elliptic curves 
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