1 Introduction
Cryptography deals with the techniques to secure the private data. In these techniques, the data is transformed into an unreadable form by using some keys so that the adversaries cannot extract any useful information. Sbox is the only nonlinear component of many wellknown cryptosystems including AES. It is therefore the security of such cryptosystems solely depends on the cryptographic properties of their Sbox. Shannon [1], proved that a cryptosystem is secure if it can create confusion and diffusion in the data up to a certain level. An Sbox is cryptographically strong enough to create desire confusion and diffusion if it satisfies certain tests including the test of nonlinearity, approximation, strict avalanche, bit independence and algebraic complexity.
Nowadays, AES is considered to be the most secured and widely used cryptosystem, and hence many cryptographers studied its Sbox. The study in [2, 3, 4, 5] reveals that the AES Sbox is vulnerable against algebraic attacks because of its sparse polynomial representation. It is also noticed that a cryptosystem based on a single Sbox is unable to generate desirable security if the data is highly correlated [6, 7] . Furthermore, it is shown that the security of a cryptosystems can be improved by using dynamic Sboxes instead of static Sboxes [8]. Due to these reasons many researchers have proposed new Sbox generation techniques based on different mathematical structures including algebraic, and differential equations. For an Sbox design technique, it is necessary that the resultant Sbox: (a) inherits the properties of the underlying mathematical structure. This is an important requirement which leads to the efficient generation and better understanding of the cryptographic properties of the Sbox; (b) satisfies the security tests; and (c) is generated in less time and space complexity. Of course, an Sbox generation technique with high time complexity is not suitable for the cryptosystems using multiple, and dynamic Sboxes. Liu et al. [9] presented an improved AES Sbox based on an algebraic method. Cui et al. [10] used an affine function to generate an Sbox with 253 nonzero terms in its polynomial representation. Tran et al. [11] used composition of a Gray code instead of affine mapping with the AES Sbox to generate an Sbox with high algebraic complexity. Khan et al. [12, 13] proposed different methods for the generation of cryptographically strong Sboxes based on a generalization of Gray Sbox, and affine functions. Azam [6] used the later Sboxes for the encryption of confidential images. Chaotic maps including Baker, logistic, and Chebyshev maps are used to generate new Sboxes in [14, 15, 16]. Similarly, elliptic curves (ECs) are also used in the field of cryptography for the development of highly secure cryptosystems. Miller [17] presented an EC based security system which has smaller key size and higher security as compared to RSA. Jung et al. [18] developed a link between the points on hyperelliptic curves and nonlinearity of an Sbox. Hayat et al. [19] for the first time used EC over a prime field for the generation of Sbox. In this scheme, an Sbox is generated by using the coordinates of the points on the EC followed by the modulo operation . Although the technique is capable of generating cryptographically strong Sboxes, but the output is uncertain. That is, for each set of input parameters the algorithm does not necessarily output an Sbox. Furthermore, the time complexity of this scheme is , where is the prime in the underlying EC.
The purpose of this article is to develop such a novel and efficient Sbox generation technique based on a finite Mordell elliptic curve (MEC) which generates secure Sbox inheriting the properties of the underlying MEC for each set of input parameters. To achieve this, we defined some typical type of total orders on the points on the MEC, and then used coordinates instead of coordinates to obtain an Sbox. The remaining paper is organized as follows: In Section 2, some basic definitions and results related to EC are discussed. The proposed algorithm is described in Section 3. Section 4 contains the security analysis, and a comparison of the proposed Sbox design technique with some of the existing techniques. Finally conclusions are drawn in Section 5.
2 Preliminaries
For a prime , and two nonnegative integers , the EC over the prime field is defined to be the collection of infinity point
, and all ordered pairs
satisfying the equationWe call , and the parameters of the EC . An approximation for the total number of points on can be obtained by using Hasse’s formula [20]
Mordell elliptic curve (MEC) is a special kind of elliptic curve with . The significance of MEC is that some of the MECs have exactly . The following Theorem is taken from [21] gives the information about such MECs.
Theorem 1
Let be a prime such that . Then for each , the MEC has exactly distinct points, and has each integer in exactly once as a coordinate.
Henceforth, a MEC where is simply denoted by .
3 Description of the Proposed Sbox Designing Technique
In this section, we give an informal intuition of our proposed method. Our aim is to develop such an Sbox generation technique based on MEC which outputs an Sbox: (a) in constant time for each set of input parameters; (b) that inherits the properties of the underlying MEC; and (c) having high security against cryptanalysis. Note that the Sbox design technique proposed by Hayat et al. [19] does not satisfy condition (a) and (b). One of the possible way of designing such technique is to input that EC which contains all values from to without repetition. It is, therefore, the proposed algorithm takes an MEC as input, and uses coordinates to generate an Sbox instead of coordinates. Now the next task is to use these coordinates in such a way that the resultant Sbox inherits the properties of the underlying MEC. Of course, the usage of some arithmetic operations such as modulo operation on the coordinates to get an Sbox will destroy the structure of the underlying MEC. It is therefore, we used the concept of total order on the MEC to get an Sbox. Order theory is intensively used in formal methods, programming languages, logic, and statistic analysis. Now the natural question is how to define different orderings on the MEC. Note that for each value of MEC, there are two values. Thus, we can divide the orderings on MEC into two categories: (1) one is that in which the two values of each appear consecutively; and (2) the other one contains those orderings in which the two values of each do not appear consecutively. Based on this fact, we defined three different type of orderings on the MEC to generate three different Sboxes for a given MCE .
3.1 The proposed orderings on a MEC
The orderings used in the proposed method are discussed below.
(1) A natural ordering on MEC: We define a natural ordering on based on coordinates as follows
(1) 
where .
The aim of this ordering is to order the points on the MEC in such a way that the coordinates are in nondecreasing order, and the two values corresponding to each appear consecutively.
The next two orderings are introduced based on the following observation deduced from Theorem 1 to diffuse the coordinates on a MEC.
Observation: For any two distinct points and on the MEC , and either or , it holds that .
(2) A diffusion ordering on MEC: An ordering is defined on to diffuse the two values of each . Let and be any two points on , the diffusion ordering is defined to be
(2) 
Lemma 2
The relation is a total order on the MEC .
Proof. For each , we have , and therefore . This implies that is reflexive. Next, we need to show that satisfies the antisymmetric property. Thus, for , suppose that and hold. This implies that . This is because of the fact that , and are the only cases for which the supposition and are true, which eventually imply that Now if then by the supposition and the fact , we have and , which lead to the contradiction . Thus and hold, which ultimately imply that , and therefore . Now, to prove the transitivity property, suppose that , and hold, where Now if and , or and , then , and therefore Similarly, if , then and , and hence and This completes the proof.
(3) A modulo diffusion ordering on MEC: The order defined below produces diffusion in both coordinates and coordinates of the points on . Let , then
(3) 
Lemma 3
The relation is a total order on the MEC .
Lemma 3 can be proved by using the similar arguments as used in the proof of Lemma 2. The effect of these orderings and on coordinates of MEC is shown in Figure 1 by plotting them in a nondecreasing order of their points on MEC w.r.t and , respectively.
Similarly, a relation among the sets of all coordinates of MEC obtained by different proposed orderings and where is quantified by computing their correlation coefficient . The correlation results for different MECs are shown in Table 1. It is evident from the results that each ordering has different effect on the coordinates of the underlying MEC.
101  1  0.0588  0.0550  0.0497 

827  87  0.0044  0.0008  0.0027 
1013  118  0.0028  0.0059  0.0003 
2027  8  0.0007  0.0068  0.0002 
3.2 The proposed construction technique
Let be a Mordell elliptic curve (MEC), where . The lower bound on the prime is for the proposed method so that MEC has at least points. An Sbox , where is generated by selecting the coordinates on which are in the interval as
such that , and .
It is clear from Theorem 1 that is a bijection, which further implies that the proposed method generates an Sbox for each set of input parameters.
Lemma 4
For any prime such that integer , and the Sbox can be generated in constant time, and space.
Proof. The generation of requires calculation of points on the MEC with coordinates in , and then their sorting. Of course this can be done in constant time, and space complexity.
154  217  227  110  85  29  199  37  68  21  91  78  208  3  148  40 
198  52  54  2  73  7  168  201  229  184  146  6  172  28  44  67 
195  53  106  10  204  131  157  185  187  156  206  161  81  103  211  33 
96  159  72  134  164  143  140  193  145  231  237  12  221  188  197  116 
47  19  129  104  51  236  56  133  55  220  87  1  203  117  210  24 
4  174  175  113  34  213  171  255  30  43  130  191  57  137  76  234 
247  244  173  223  63  60  230  166  8  190  139  99  49  200  23  245 
58  102  226  83  122  70  241  94  127  41  194  233  97  251  107  26 
109  61  248  90  192  167  147  82  158  225  36  50  84  92  88  38 
74  136  138  232  62  176  128  189  124  118  169  14  228  0  243  181 
123  254  20  202  75  149  219  120  160  9  253  39  180  207  114  142 
183  93  101  15  238  177  132  212  35  250  239  249  179  17  65  186 
11  125  178  45  170  141  121  126  119  64  144  182  112  22  165  222 
100  69  252  216  13  27  152  235  80  5  196  59  25  151  79  155 
240  77  115  71  31  105  95  86  209  150  98  89  163  246  66  18 
162  214  218  42  242  46  111  48  215  224  135  108  153  32  16  205 
33  151  65  207  12  103  96  123  190  126  82  155  21  1  229  186 
61  224  42  179  63  178  73  153  138  168  146  41  46  9  109  184 
124  243  236  57  19  6  100  94  69  48  116  216  54  228  90  81 
47  13  88  197  247  129  206  198  221  5  78  80  150  200  145  55 
60  105  212  18  210  43  137  250  135  166  52  115  91  208  25  199 
77  170  121  122  11  254  27  157  175  34  104  201  95  222  133  176 
36  3  141  218  30  162  220  193  28  110  223  161  74  182  226  113 
0  112  234  144  241  20  156  62  49  23  26  35  148  101  233  56 
181  130  118  149  70  173  71  45  50  204  10  87  232  93  177  67 
4  120  8  40  72  125  92  114  68  83  225  246  158  143  53  196 
249  242  136  195  160  213  131  107  66  29  230  188  38  111  205  253 
171  251  102  235  31  127  217  17  183  117  37  211  164  97  119  219 
167  134  24  16  255  2  32  215  227  154  187  75  231  240  172  142 
244  89  14  98  76  85  147  79  64  180  214  139  152  238  51  185 
22  44  194  99  39  169  203  189  108  86  132  237  163  239  209  245 
59  202  15  58  248  128  174  140  192  191  106  165  159  84  7  252 
15  13  247  249  167  183  179  173  101  204  105  210  214  205  199  19 
164  38  85  72  98  90  113  12  239  217  165  228  123  195  26  216 
207  30  182  219  14  215  232  135  241  145  17  244  223  114  29  70 
104  81  71  99  191  128  227  86  172  185  5  75  197  184  109  248 
162  250  25  110  125  230  129  35  102  234  54  171  194  16  33  73 
155  246  154  84  149  134  238  18  240  67  200  253  61  31  170  180 
55  20  224  187  10  147  92  133  196  242  146  27  34  140  28  192 
63  127  143  203  137  2  74  193  65  4  124  51  107  24  42  122 
103  22  41  226  235  252  116  212  77  49  48  201  148  221  251  80 
229  115  93  139  181  52  97  119  189  166  21  45  53  100  32  131 
112  94  59  142  117  36  153  254  66  158  79  121  8  130  132  60 
245  231  126  152  151  89  0  39  160  136  37  78  236  56  206  157 
222  174  82  69  6  83  220  3  57  111  208  47  141  87  168  176 
11  118  169  58  243  120  150  91  190  23  178  44  7  43  177  76 
161  144  163  68  88  138  218  108  159  186  40  237  175  46  198  96 
202  9  62  50  64  233  255  209  188  1  106  225  95  213  156  211 
4 Security Analysis and Comparison
Several standard tests are applied on the Sboxes obtained by the proposed method to test their cryptographic strength. A brief introduction to these security tests, and their results for some of the newly generated Sboxes , , , , , , and are discussed in this section.
4.1 NonLinearity (NL)
It is important for an Sbox to create confusion in the data up to a certain level to keep the data secure from the adversaries. The confusion creation capability of an Sbox over the Galois Field is measured by its nonlinearity , which is defined below
where , , and “” represents dot product over
An Sbox with high NL is capable of generating high confusion in the data. However, it is also shown in [22] that an Sbox with high NL may not satisfy other cryptographic properties. The NL of some of the newly constructed Sboxes is listed in Table 5. Note that each listed Sbox has NL 106, which is large enough to create high confusion.
Sboxes  

NL  106  106  106  106  106  106  106  106  106 
4.2 Approximation Attacks
A cryptographically strong Sbox must have high resistance against approximation attacks. The approximation attacks can be divided into two categories namely linear approximation attacks, and differential approximation attacks which are explained below.
4.2.1 Linear Approximation Probability (LAP)
The resistance of an Sbox against linear approximation attacks is measured by calculating its maximum number of coincident input bits with the output bits. The mathematical expression of is as follows
where and .
An Sbox is said to be highly resistive against linear approximation attacks if it has low value of . The LAP of the newly generated Sboxes is listed in Table 6. The average LAP of all of the listed Sboxes is which is very low, and hence the proposed technique is capable of generating Sboxes with high resistance against linear approximation attacks.
Sboxes  

LAP  0.1328  0.1328  0.1406  0.1484  0.1328  0.1406  0.1328  0.1328  0.1406 
4.2.2 Differential Approximation Probability (DAP)
The strength of an Sbox against differential approximation attacks is measured by calculating its DAP. For an Sbox , the DAP
is the maximum probability of a specific change
in the output bits when the input bits are changed to i.e.,where and “” is bitwise addition over .
The smaller is the value of DAP, the higher is the security of the Sbox against differential approximation attacks. The experimental results of DAP on the newly generated Sboxes are presented in Table 7. It is evident from Table 7 that the newly generated Sboxes have high resistance against differential attacks.
Sboxes  

DAP  0.0391  0.0391  0.0391  0.0391  0.0391  0.0391  0.0391  0.0391  0.0391 
4.3 Strict Avalanche Criterion (SAC)
The diffusion creation capability of an Sbox is calculated by SAC. The SAC of an Sbox is the measure of change in output bits when a single input bit is changed. The SAC of an Sbox with boolean functions where , is computed by calculating an eight dimensional square matrix by using each of the eight elements with only one nonzero bit as
where
denotes the number of nonzero bits in the vector
.SAC test is fulfilled, if all entries of are close to . The entries of SAC matrix corresponding to each newly generated Sboxes , and are plotted in a linear order in Figure 2. The average of minimum, and maximum values of corresponding to each of the newly generated Sboxes are and , respectively. Table 8 clearly shows that the Sboxes generated by the proposed method based on MEC is capable of generating high diffusion in the data.
Sboxes  

SAC(max)  0.5938  0.625  0.6563  0.6406  0.6094  0.6094  0.5938  0.6094  0.625 
SAC(min)  0.4531  0.4219  0.4219  0.4063  0.4219  0.4063  0.375  0.3906  0.3594 
4.4 Bit Independence Criterion (BIC)
BIC is also an important test to measure the diffusion creation strength of an Sbox. The main idea of this test is to investigate the dependence of a pair of output bits when an input bit is reversed. The BIC of an Sbox over with boolean functions is also calculated by computing a square matrix of dimension eight as follows
Of course . An Sbox is said to be good if all offdiagonal values of its BIC matrix are near to . The experimental results of this test on the newly generated Sboxes , and excluding the value are shown in a linear order in Figure 3. The minimum, and maximum values of BIC matrix of each of the newly generated Sboxes are listed in Table 9. It is evident from Figure 3 and Table 9 that the Sboxes generated by the proposed methods are strong enough to generate high diffusion in the data.
Sboxes  

BIC(max)  0.5273  0.5293  0.5313  0.5371  0.5273  0.5254  0.5254  0.5313  0.5449 
BIC(min)  0.4648  0.4629  0.4707  0.4707  0.4844  0.4746  0.4688  0.4766  0.4727 
4.5 Algebraic Complexity(AC)
The resistance of an Sbox against algebraic attacks is measured by computing its linear polynomial. The AC of an Sbox is the number of nonzero terms in its linear polynomial. The greater is the AC, the greater is the security of the Sbox against algebraic attacks. The AC of the newly generated Sboxes is computed, and is presented in Table 10. The minimum, and maximum values of AC of the newly generated Sboxes are and , respectively, which are very close to the optimal value . Thus, the proposed method is able to generate Sboxes with good AC based on MEC.
Sboxes  

AC  254  254  255  255  254  255  253  253  255 
5 Comparison and Discussion
A comparison of the security efficiency of the proposed Sbox design technique with some of the existing techniques [14, 15, 16, 23, 24, 25, 26, 27, 28, 29, 30] is presented in this section by comparing the cryptographic properties of their Sboxes. The cryptographic properties of the Sboxes used in this comparison are listed in Table 11. Note that the nonlinearity (NL) of the Sboxes , and is greater than that of the Sboxes in [14, 15, 19, 25, 28, 29, 30], and hence the newly generated Sboxes create better confusion in the data as compared to the later Sboxes. This implies that the proposed technique is capable of generating Sboxes with good NL as compared to some of the other existing techniques. Moreover, the linear approximation probability (LAP) of the newly generated Sboxes is better than the LAP of the Sboxes in [14, 15, 16, 28, 29, 30], while their differential approximation probability (DAP) is at most the DAP of the Sboxes in [14, 15, 16, 19, 25, 28, 29, 30]. Hence, the Sboxes generated by the proposed technique have same or better security against approximation attacks as compared to the other Sboxes. Similarly, the SAC, BIC and AC test results of the newly generated Sboxes are comparable with the Sboxes listed in Table 11. Hence, the proposed Sbox generation technique based on MEC is capable of generating cryptographically strong Sboxes as compared to some of the existing Sbox construction techniques based on different mathematical structures. Furthermore, the proposed algorithm takes constant time for the generation of an Sbox, while the method based on EC in [19] takes time, where is the prime of the underlying EC. This implies that the proposed algorithm is fast as compared to the method in [19].
Sboxes  NL  LAP  DAP  SAC(Max)  SAC(Min)  BIC(Max)  BIC(Min)  AC 

Ref. [14]  103  0.1328  0.0391  0.5703  0.4414  0.5039  0.4961  255 
Ref. [15]  102  0.1484  0.0391  0.6094  0.375  0.5215  0.4707  254 
Ref. [16]  106  0.1406  0.0391  0.5938  0.4375  0.5313  0.4648  251 
Ref. [19]  104  0.0391  0.0391  0.625  0.3906  0.53125  0.4707  255 
Ref. [25]  104  0.109  0.0469  0.593  0.39  0.499  0.454  255 
Ref. [27]  112  0.062  0.0156  0.562  0.453  0.504  0.480  9 
Ref. [28]  74  0.2109  0.0547  0.6875  0.1094  0.5508  0.4023  253 
Ref. [29]  100  0.1328  0.0547  0.6094  0.4219  0.5313  0.4746  255 
Ref. [30]  103  0.1328  0.0391  0.5703  0.3984  0.5352  0.4727  255 
106  0.1328  0.0391  0.5938  0.4531  0.5273  0.4648  254  
106  0.1484  0.0391  0.6406  0.4063  0.5371  0.4707  255  
106  0.1328  0.0391  0.5938  0.375  0.5254  0.4688  253 
6 Conclusion
In this article, we presented an Sbox design technique based on coordinates of a finite Mordell elliptic curve (MEC) where prime is congruent to modulo . The technique uses some special type of total orders on the points of the MEC, and generates an Sbox in constant time. Several standard security tests are performed on the Sboxes generated by the proposed method to analyze its cryptographic efficiency. Experimental results show that the newly generated Sboxes are cryptographically strong. Furthermore, a comparison of some of the newly generated Sboxes with Sboxes generated by some of the existing techniques is also performed. It is evident from the comparison that the proposed method is capable of generating more secure Sboxes as compared to some of the existing Sbox design techniques.
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