Literature DB >> 34956343

A Novel Image Encryption Technique Based on Mobius Transformation.

Muhammad Asif1, Sibgha Mairaj2, Zafar Saeed3, M Usman Ashraf4, Kamal Jambi5, Rana Muhammad Zulqarnain1.   

Abstract

The nonlinear transformation concedes as S-box which is responsible for the certainty of contemporary block ciphers. Many kinds of S-boxes are planned by various authors in the literature. Construction of S-box with a powerful cryptographic analysis is the vital step in scheming block cipher. Through this paper, we give more powerful and worthy S-boxes and compare their characteristics with some previous S-boxes employed in cryptography. The algorithm program planned in this paper applies the action of projective general linear group PGL(2, GF(28)) on Galois field GF(28). The proposed S-boxes are constructed by using Mobius transformation and elements of Galois field. By using this approach, we will encrypt an image which is the preeminent application of S-boxes. These S-boxes offer a strong algebraic quality and powerful confusion capability. We have tested the strength of the proposed S-boxes by using different tests, BIC, SAC, DP, LP, and nonlinearity. Furthermore, we have applied these S-boxes in image encryption scheme. To check the strength of image encryption scheme, we have calculated contrast, entropy, correlation, energy, and homogeneity. The results assured that the proposed scheme is better. The advantage of this scheme is that we can secure our confidential image data during transmission.
Copyright © 2021 Muhammad Asif et al.

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Year:  2021        PMID： 34956343      PMCID： PMC8709777          DOI： 10.1155/2021/1912859

Source DB:  PubMed          Journal:  Comput Intell Neurosci


1. Introduction

The notion of S-box was first introduced by applied scientist Claude Shannon in 1949, and afterward this notion has attracted the attention of many researchers. With the quick evolution of the network communication and massive data application, the security of the data has become more popular topic. The scholars have proposed a spread of the information encryption, privacy protection. In symmetric cryptography, the block encryption algorithm is used customarily, for example, in encryption (DES), AES, and other systems. In block cipher system, there is a predominant nonlinear component called substitution box. S-box plays a crucial role in the security of symmetric cryptosystem. AES is taken into account to be an efficient cryptosystem to a large extent. One of the important components of the AES is its prime S-box which is predicted on the inversion and transformation because of recognition of AES in the communication system; substitution box captivates traditional attention. However, the substitution box which is employed in AES is predetermined. The S-box is a nonlinear component of block cipher which creates confusion. The maintenance of information security has become an excellent challenge for the cryptography. Substitution boxes have been employed in many cryptosystems including encryption standard (DES), international data encryption algorithm (IDEA), and advanced encryption standard (AES). The security strength of substitution box determines the safety of the entire cryptosystem. It is therefore established that the substitution box is the important nonlinear component of the cryptographic system. Cryptography has unprecedented ways of the utilization of encryption capabilities to produce security of the data. Many image encryption algorithms with S-box have been presented [1-8]. Liu et al. [6] explained the image encryption scheme using one-time S-boxes. Hussain and Gondal [5] gave an extended image encryption using chaotic coupled map and S-box transformation, the confusion-diffusion structure was assumed, the places of the pixels of the plain image were mixed up by a chaotic tent map, and after that delayed coupled map lattices and S-box transformation were used to puzzle the association between the original image and the cipher image. Zhang et al. [9] introduced an efficient chaotic image encryption based on alternate circular S-boxes, and a set of S-boxes were constructed by Chen chaotic system. Liu et al. [10, 11] developed the adaptive controller design and fuzzy synchronization for uncertain fractional order nonlinear system and fractional order chaotic systems. The scrutiny of AES is explained considering the high throughput, area efficiency, and elevated performance [12]. Khan et al. [13] introduced an efficient image encryption scheme based on double affine substitution box and chaotic system. Asif and Shah [14] explained the image encryption scheme using BCH codes. Alanazi et al. [15] explained cryptanalysis of novel image encryption scheme based on multiple chaotic substitution boxes. An approach to increasing multimedia security employing 3D mixed chaotic map and hybrid permutation substitution is explained by Naseer et al. [16, 17]. Khalid et al. [18] defined elliptic curve based image encryption scheme by using S-boxes. Cryptanalysis on S-box based on encryption method is clarified by Munir et al. [19]. Nonlinear component based on elliptic curve and power associative loop structure is defined by Haider et al. [20] and Hussain et al. [21], respectively. The above presented studies are not enough to secure data in communication channel. To overcome this drawback, we proposed a novel approach using Mobius transformation. Existing studies deal only with one S-box for AES algorithm, but our proposed scheme is utilized to encrypt image using ten S-boxes. The rest of the paper is organized as follows: The elements of S-box are constructed by using elements of Galois field in Section 2, and the elements of Galois field are utilized in linear fractional transformation for S-boxes. In Section 3, analysis of S-boxes is carried out, and comparison with other S-boxes is also made. In Section 4, image encryption scheme is proposed by utilizing S-boxes, different tests are applied on encrypted image, and comparison of image encryption scheme with existing techniques is provided. The conclusion of the paper is presented in Section 5.

2. Construction of S-Box Using Galois Field

2.1. Galois Field

Any finite field is called Galois field. Nowadays, Galois field is used in many cryptographic algorithms for data security. A Galois filed extension is defined aswhere f(x) is primitive irreducible polynomial of degree m.

2.2. Scheme for Construction of S-Box

A 16 × 16 S-box is constructed using the elements of Galois field. The total elements of the proposed S-box are 256, which are constructed by the action of PGL(2, GF(28)) on GF(28) [22]. Now, we have the Mobius transformation:where a, b, c, d, and t ∈ GF(28). Here, T(t) are the values of GF(28) to construct the new S-box. This algorithm will stop working when a.d − b.c ≠ 0 does not exist. Moreover, after the change of the values in dividend and divisor, there is also a scenario where results of divisor are equal to zero. We additionally check this worth of unassisted degree rule that makes divisor zero; to overcome this type of error zero divisor, we assign the associated remaining value to conclude the values of substitution box. To find the elements, we substitute the value of t from 0 to 255 and convert t,  a,  b,  c,  d to binary form. Before control on the binary type, simply ones tend to delineate the values in form of polynomials. The terms from dividend and divisor of the unit being modified with the corresponding binary values “m” are interpreted as a particular primitive polynomial Here, P(x) is utilized for the construction of the elements of the GF(28) [23]. The mathematical methodology for GF(28) will be used in our further process. We can define GF(28)=ℤ2[x]/〈P(x)〉, where ℤ2={0,1} and the polynomial P(x)=x8+x4+x3+x2+1 is the primitive irreducible polynomial. Now, we construct the values of the transformed S-box by using Mobius transformation and elements of Galois field from Table 1. Here, we consider a=220, b=30, c=90,  and d=200,  where t=0 to 255.
Table 1

Elements of Galois field GF(28).

Exp.DecimalPolynomial
x 0 01 x 0
x 1 02 x 1
x 2 04 x 2
x 3 08 x 3
x 4 16 x 4
x 5 32 x 5
x 6 64 x 6
x 7 128 x 7
x 8 29 x 4+x3+x2+1
x 9 58 x 5+x4+x3+x
x 10 116 x 6+x5+x4+x2
x 11 232 x 7+x6+x5+x3
x 12 205 x 7+x6+x3+x2+1
x 13 135 x 7+x2+x+1
x 14 19 x 4+x+1
x 15 38 x 5+x2+x
x 16 76 x 6+x3+x2
x 17 152 x 7+x4+x3
x 18 45 x 5+x3 + x2+1
x 19 90 x 6+x4+x3+x
x 20 180 x 7+x5+x4+x2
x 21 117 x 6+x5+x4+x2+1
x 22 234 x 7+x6+x5+x3+x
x 23 201 x 7+x6+x3+1
x 24 143 x 7+x3+x2+x+1
x 25 3 x+1
x 26 6 x 2+x
x 27 12 x 3+x2
x 28 24 x 4+x3
x 29 48 x 5+x4
x 30 96 x 6+x5
x 31 192 x 7+x6
x 32 157 x 8+x7
x 33 39 x 5+x2+x+1
x 34 78 x 6+x3+x2+x
x 35 156 x 7+x4+x3+x2
x 36 37 x 5+x2+1
x 37 74 x 6+x3+x
x 38 148 x 7+x4+x2
x 39 53 x 5+x4+x2+1
x 40 106 x 6+x5+x3+x
x 41 212 x 7+x6+x4+x2
x 42 181 x 7+x5+x4+x2+1
x 43 119 x 6+x5+x4+x2+x+1
x 44 238 x 7+x6+x5+x3+x2+x
x 45 193 x 7+x6+1
x 46 159 x 7+x4+x3+x2+x+1
x 47 35 x 5+x+1
x 48 70 x 6+x2+x
x 49 140 x 7+x3+x2
x 50 5 x 2+1
x 51 10 x 3+x
x 52 20 x 4+x2
x 53 40 x 5+x3
x 54 80 x 6+x4
x 55 160 x 7+x5
x 56 93 x 6+x4+x3+x2+1
x 57 186 x 7+x5+x4+x3+x
x 58 105 x 6+x5+x3+1
x 59 210 x 7+x6+x4+x
x 60 185 x 7+x5+x4+x3+1
x 61 111 x 6+x5+x3+x2+x+1
x 62 223 x 7+x6+x4+x3+x2+x
x 63 161 x 7+x5+1
x 64 95 x 6+x4+x3+x2+x+1
x 65 190 x 7+x5+x4+x3+x2+x
x 66 97 x 6+x5+1
x 67 194 x 7+x6+x
x 68 153 x 7+x4+x3+1
x 69 47 x 5+x4+x2+x+1
x 70 94 x 6+x4+x3+x2+x
x 71 188 x 7+x5+x4+x3+x2
x 72 101 x 6+x5+x2+1
x 73 202 x 7+x6+x3+x
x 74 137 x 7+x3+1
x 75 15 x 3+x2+x+1
x 76 30 x 4+x3+x2+x
x 77 60 x 5+x4+x3+x2
x 78 120 x 6+x5+x4+x3
x 79 240 x 7+x6+x5+x4
x 80 253 x 7+x6+x5+x4+x3+x2+1
x 81 231 x 7+x6+x5+x2+x+1
x 82 211 x 7+x6+x2+x+1
x 83 187 x 7+x5+x4+x3+x+1
x 84 107 x 6+x5+x3+x+1
x 85 214 x 7+x6+x4+x2+x
x 86 177 x 7+x5+x4+1
x 87 127 x 6+x5+x4+x3+x2+x+1
x 88 254 x 7+x6+x5+x4+x3+x2+x
x 89 225 x 7+x6+x5+1
x 90 223 x 7+x6+x4+x3+x2+x+1
x 91 163 x 7+x5+x+1
x 92 91 x 6+x4+x3+x3+x+1
x 93 182 x 7+x5+x4+x2+x
x 94 113 x 6+x5+x4+1
x 95 226 x 7+x6+x3+x
x 96 217 x 7+x6+x4+x3+1
x 97 175 x 7+x5+x3+x2+x+1
x 98 67 x 6+x+1
x 99 134 x 7+x2+x
x 100 17 x 4+1
x 101 34 x 5+x
x 102 68 x 6+x2
x 103 136 x 7+x3
x 104 13 x 3+x2+1
x 105 26 x 4+x3+x
x 106 152 x 5+x4+x2
x 107 104 x 6+x5+x3
x 108 208 x 7+x6+x4
x 109 189 x 8+x7+x5
x 110 103 x 6+x5+x2+x+1
x 111 206 x 7+x6+x3+x2+x
x 112 129 x 7+1
x 113 31 x 4+x3+x2+x+1
x 114 62 x 5+x4+x3+x2+x
x 115 124 x 6+x5+x4+x3+x2
x 116 248 x 7+x6+x5+x4+x3
x 117 237 x 7+x6+x5+x3+x2+1
x 118 199 x 7+x6+x2+x+1
x 119 147 x 7+x4+x+1
x 120 59 x 5+x4+x3+x+1
x 121 118 x 6+x5+x4+x2+x
x 122 236 x 7+x6+x5+x3+x2
x 123 197 x 7+x6+x2+1
x 124 151 x 7+x4+x2+x+1
 x12551 x 5+x4+x+1
 x126102 x 6+x5+x2+x
x 127 204 x 7+x6+x3+x2
x 128 133 x 7+x2++1
x 129 23 x 4+x2+x+1
x 130 46 x 5+x3+x2+x
x 131 92 x 6+x4+x3+x2
x 132 184 x 7+x5+x4+x3
  x133109 x 6+x5+x3+x2+1
  x134218 x 7+x6+x4+x3+x
 x135169 x 7+x5+x3+1
 x13647 x 6+x3+x2+x+1
 x13794 x 7+x4+x3+x2+x
x 138 188 x 5+1
x 139 66 x 6+x
x 140 132 x 7+x2
x 141 21 x 4+x2+1
x 142 42 x 5+x3+x
x 143 84 x 6+x4+x2
x 144 168 x 7+x5+x3
x 145 77 x 6+x3+x2+1
x 146 154 x 7+x4+x3+x
x 147 41 x 5+x3+1
x 148 82 x 6+x4+x
x 149 164 x 7+x5+x2
x 150 85 x 6+x4+x2
x 151 170 x 7+x5+x3+x
x 152 73 x 6+x3+1
  x153146 x 7+x4+x
x 154 57 x 5+x4+x3+1
x 155 114 x 6+x5+x4+x
x 156 228 x 7+x6+x5+x2
x 157 213 x 7+x6+x4+x2+1
x 158 183 x 7+x5+x4+x2+x+1
x 159 115 x 6+x5+x4+x+1
x 160 230 x 7+x6+x5+x2+x
x 161 209 x 7+x6+x4+1
x 162 191 x 7+x5+4+x3+x2+x+1
x 163 99 x 6+x5+x+1
x 164 198 x 7+x6+x2+x
x 165 145 x 7+x4+1
x 166 63 x 5+x4+x3+x2+x+1
x 167 126 x 6+x5+x4+x3+x2+x
x 168 252 x 7+x6+x5+x4+x3+x2
x 169 229 x 7+x6+x5+x2
x 170 215 x 7+x6+x4+x4+x+1
x 171 179 x 7+x5+x4+x+1
x 172 123 x 6+x5+x4+x3+x+1
x 173 246 x 7+x6+x5+x4+x2+x
x 174 241 x 7+x6+x5+x4+1
x 175 255 x 7+x6+x5+x4+x3+x2+x+1
x 176 227 x 7+x6+x5+x+1
x 177 219 x 7+x6+x4+x3+x+1
x 178 171 x 7+x5+x3+x+1
x 179 75 x 6+x3+x+1
x 180 150 x 7+x4+x2+x
x 181 49 x 5+x4+1
x 182 98 x 6+x5+x
x 183 196 x 7+x6+x2
x 184 149 x 7+x4+x2+1
x 185 55 x 5+x4+x2+x+1
x 186 110 x 6+x5+x3+x2+x
   x187220 x 7+x6+x4+x3+x2
x 188 165 x 7+x5+x2+1
x 189 87 x 6+x4+x2+x+1
x 190 174 x 7+x5+x3+x2+x
x 191 65 x 6+1
x 192 130 x 7+x
x 193 25 x 4+x3+1
x 194 50 x 5+x4+x
x 195 100 x 6+x5+x2
x 196 200 x 7+x6+x3
x 197 141 x 7+x3+x2+1
x 198 7 x 2+x+1
x 199 14 x 3+x2+x
x 200 28 x 4+x3+x2
x 201 56 x 5+x4+x3
x 202 112 x 6+x5+x4
x 203 224 x 7+x6+x5
x 204 221 x 7+x6+x4+x3+x2+1
x 205 167 x 7+x5+x2+x+1
x 206 83 x 6+x4+x+1
x 207 166 x 7+x5+x2+x
x 208 81 x 6+x4+1
x 209 162 x 7+x5+x
x 210 89 x 6+x4+x3+1
x 211 178 x 7+x5+x4+x
x 212 121 x 6+x5+x4+x3+1
x 213 242 x 7+x6+x5+x4+x
x 214 249 x 7+x6+x5+x4+x3+1
x 215 239 x 7+x6+x5+x3+x2+x+1
x 216 195 x 7+x6+x+1
x 217 155 x 7+x4+x3+x+1
x 218 43 x 5+x3+x+1
x 219 86 x 6+x4+x2+x
x 220 172 x 7+x5+x3+x2
x 221 69 x 6+x2+1
x 222 138 x 7+x3+x
x 223 9 x 3+1
x 224 18 x 4+x
x 225 36 x 5+x2
x 226 72 x 6+x3
x 227 144 x 7+x4
x 228 61 x 5+x4+x3+x2+1
x 229 122 x 6+x5+x4+x3+x
x 230 244 x 7+x6+x5+x4+x2
x 231 245 x 7+x6+x5+x4+x2+1
x 232 247 x 7+x6+x5+x4+x2+x+1
x 233 243 x 7+x6+x5+x4+x+1
x 234 251 x 7+x6+x5+x4+x3+x+1
x 235 235 x 7+x6+x5+x3+x+1
x 236 203 x 7+x6+x3++x+1
x 237 139 x 7+x3++x+1
x 238 11 x 3+x+1
x 239 22 x 4+x2+x
x 240 44 x 5+x3+x2
x 241 88 x 6+x4+x3
x 242 176 x 7+x5+x4
x 243 125 x 6+x5+x4+x3+x2+1
x 244 250 x 7+x6+x5+x4+x3+x
x 245 233 x 7+x6+x5+x3+1
x 246 207 x 7+x6+x3+x2+x+1
x 247 131 x 7+x+1
x 248 27 x 4+x3+x+1
x 249 54 x 5+x4+x2+x+1
x 250 108 x 6+x5+x3+x2
x 251 216 x 7+x6+x4+x3
x 252 173 x 7+x5+x3+x2+1
x 253 71 x 6+x2+x+1
x 254 142 x 7+x3+x2+x
x 255 11
Here, t=0 to 255. We consider t=0; then, Converting each value into binary form, we will get the polynomial of the corresponding values. We can see the corresponding value of this polynomial in Table 1 in form of “x.” Therefore, the first value of the transformed S-box is 169; by following the same procedure, we will compute the remaining elements of the S-box.

3. Analysis of Transformed S-Boxes and Their Comparison Support

Nonlinearity constitutes the quantity of bits which are necessarily altered to succeed in affine at the lowest distance. Therefore, for an outsize “m,” that calculation is going to be difficult. Now, we will mention the series of the function on Fⁿ with α, so the nonlinearity is defined as The nonlinearity of these S-boxes should satisfy the following relation [24]:120 is considered as an absolute nonlinearity value. From Tables 2–11, we observe that from Table 12 the maximum nonlinearity of transformed S-box is equal to 107.3, which is better when compared to other S-boxes.
Table 2

S-box in the form of 16 × 16 matrix.

135225225712101341051356218813918116024234194
17911918212610712922223257126147011918134105
624216475204221181111222191114519735132210
1081867344181171511347718863107468111947
1928813417224894119242240221174571724225317
165135237215982482087413419221023917172142
177158247147246148127125125410718151213121164
143132881602532233634215252132443312015179
73332321253622312680217471722251200235
171160254801442522462002322354652120246106219
1517395141082459574160240251203325272208
142396413814182121231061971252042523235
2252481081206017242250903617347742237194
81148177108212522128124710064243824477174
235252602102132081391294521933981382017141
3818818116520812121261401711318818236250143
Table 3

S-box in the form of 16 × 16 matrix.

1693636188892182616922216566492301767850
7514798102104231382471866410147497826
222181198152269492061389020619314115618489
2081120223849179102186016516110415923114735
13025421812327113147176446924118615224771152
145169139239671432981137218130892217910142
21918313141233822051882161421044910242118198
84184254230719377823917313525039593875
202392475137910210241523512336528235
17923012531681732072824723515920592075286
3824622619208233226137230443593917310181
1922953321291212059952141512213247156
36272085918515217610822337152786018113950
2318221920812120121231131179514314825060241
23517318589242816623193863967335610021
1481654914581118410213217913565983710884
Table 4

S-box in the form of 16 × 16 matrix.

69222621613450526994322221016716888146
1866524989213978329244461742346513910852
1621571991891021188712283691221915219819116
1823425217387102635013631591591718014966
21898166109192861496120118213244336112228
176135190111141220611151711742181154602208
20286249174191761772161132021432106344135199
22819982031221757881111811425017617994186
451768923020321789904233661906121669
102167202901591914961752081711851557146193
94183205170181920517120320156179192181220115
17032102721616111517712118815223010215689198
182192106155129113168233189203228108136157190146
184176202106160185160801137102220254250173213
2081911291344419822297191931921417259148161
254137139176198159424637604413724957233230
Table 5

S-box in the form of 16 × 16 matrix.

471386419521852047113813889126252254154
11019054225242175187482501592412511906620820
191213148768199127236187472369073790248
4525117324612768161579811511517925316497
43671631741301771641111801992422503911120561
22716917420621172111124179241432571856929
112177542419022721919531112848916123816914
619067224205155186254206491910822775113110
1932272252442241552252231813997174641186447
6812611222311565133642538117955114915425
11319616721545901671792241802287513049172124
215157681012091111242191189873241682282257
98130521142331252243872416120879213174154
19622711252230552302532327468172142108246242
8113023218238713817590251302110121082209
16615822771151811597418523815854186243244
Table 6

S-box in the form of 16 × 16 matrix.

14131198210310222714219187162182068816375
234130852421382326160157122170171306570227
21917218218222192723561362313711196160253
21726707472352498618710451602127232
24311754237146242717558215541571617518755
3957316130130841877454143253652621875
42417517017733152251213138182351245182
591601171881871521451631611241601421315519320
2613123015952152242115451623223719816913336
472063211551661771336043160153617725445
1932140191197701147418858242559124218187
1916514515721284751521692541111592224223096
1311461346183518817118252551308617220275
233413425153372117519145130671427054
431661831037225162242374514130157183183212
67326539225104101714626160328514517159
Table 7

S-box in the form of 16 × 16 matrix.

19927413668144198622019145832549915
25146214176332471111852132362151524619094144
861239898234130128235111372457420621723071
41526941283523554177220133223011712247
12523780139154143122551052398021376255220160
5332202209964610722013780847119064315
16143255215219397341024233452351513298
210230237165220737799209151230429216025180
692244115267317612419376247139722910937
3583157115516617713360431601532136177254
451934132651419462137165105176160015143
220651907721312110730732291206115234176244
217921542186419610254179982016046177123112
15439218501464911725527746194429480
119631961361283619117674193696213196196121
194157190533613116179159623015721477179210
Table 8

S-box in the form of 16 × 16 matrix.

94143242595520784180238152161741792412149
44100118621119013185637413281193955749
2441891581942371416912123212199172905120320
1282455825211985101221483051254111112188100
16215037145271967509412316821528121679
454240196351822442147022316921021319662121
211196827215821520222948189217636248137192
5319698651612238534194222581765917094203
3083631523103175110998720587292516294
1052081481098614743246345491842516924249
121243151632411141962526620405813612180217
169910617519621121079921872291623310720787
1262512371972161312202391832051832482996217209
23031126462521122151416219621219012022243128
11311589131114502411865342161411422220
31831981081721697120888748101479910
Table 9

S-box in the form of 16 × 16 matrix.

113841762101601661071501138195241758824958
23817199222991741355516113718423125226186140
250871835013921471182472051412322310224180
332331051731472143469709610120612916517
191857477129019451131972522392420512658
19316442001569825024994922989242200222118
178200211101183239112122708715516122270158130
402006719020992147850138105227210215113224
96187161738136255103134127167127483191113
2681821891774111920778321401492164714354
118125170161886220017397221110579205150155
7613452255200178892409122012219139104166127
10221613914119592172221961671962748217155162
2441921021591731292391919120012117459234125133
311242259262588110401812092121944172
19218726720812322918818025412770343513480
Table 10

S-box in the form of 16 × 16 matrix.

3024517981790131143253100175140189538245
213252192120233138148895511249230180123187172
9112719722826243180811162103619518620232104
161301861351631731604824775240718618138201
186249152220223794170217303992206162160163
43010319516320418411915932077510913127207
135231246801012727972117397238761887077
17471792491871772274419771297215923413634
118631096168701752174175342382147494157
16615529171341201419091196268162181132
33159213115116015311819589140200232563174
826319021023218148171124215820315853228
176156122235213162169219172125761304158
13720012622741529611132362433818736231
4418622310721811214171176251519484171134151
2151321861225415591861413110510014423614519
Table 11

S-box in the form of 16 × 16 matrix.

96233752917919284711725513245226148233
24917313059243338222516029108244150197220123
163204141616125150231248893710011018024713
6396110169992462307013115881281104914856
110547317291281132151559653918319123099
1191136100992211491471158166151899212166
1692452072531162041217517820217511301659460
152357554220219144238141188481011152517978
19916118911115394255155212157811249137113213
63114481797821801917416320062919149184
39115492170230146199100225132282013161241
2111611748920143821792051811832243821415724
22722820542352421612298612351304616183158
281021442125217206157371251482208222192238
11091044312931211882272163250107179218170
2391841104914211450110199226171682037723
Table 12

Assessment of nonlinearity.

S-boxesNonlinearity
Transformed S-box107.3
APA S-box [25]112
S 8 Liu J S-box [26]104.87
Hussain et al. [27]104.75
Residue prime [28]99.5

3.1. Strict Avalanche Criterion (SAC)

A median consequence of the resulting bits should be modified to 0.5. Once one input bit is executed, then the given alteration shows associated avalanche result. The given operate clutch an effective avalanche result if the method is replicate for all input bits also almost 50%  avalanche variable attain value 1. S-box fulfills the SAC if only 1 input bit is modified so that in the result 0.5 quantities of output bits are changed. For the function expression, f(x) ⊕ f(x ⊕ α) is safe for the sequence α such that the weight of the α=1, so the function f :  F2⟶F2 fulfills SAC [29]. By considering the maximum values and minimum values, we observe that the average value of SAC from Table 13 is comparatively better and ∼0.5.
Table 13

Comparison of strict avalanche criterion.

S-boxesMax. valueMin. value
Transformed S-boxes0.610.57
APA S-box [25]0.560.437
S8 Liu J S-box [26]0.590.429
Hussain et al. [27]0.590.391
Residue prime [28]0.670.343

3.2. Bit Independence Criterion (BIC)

This is another style of criterion for the S-box to calculate the worth outlined because the output Y and Z should be altered separately. A bit independence criterion is an appropriate property for each crypt analytical scheme, which was introduced by Webster and Tavares. It has been argued that the Boolean functions f, f(y ≠ z) are two different output bits of the S-box. If S-box encounters bit independence criterion, f ⊕ f(y ≠ z, 1 ≤ y, z ≤ n) should be exceptionally nonlinear and are available as near as possible in order to satisfy strict avalanche criterion. We are able to conjointly attest the bit independence by evaluating nonlinearity and strict avalanche criterion of f ⊕ f [30]. In Table 14 for comparison of BIC, we take minimum value; our minimum value is 101.3, which is better compared to S8 Liu J, Hussain et al., and residue prime S-boxes.
Table 14

Assessment of BIC.

S-boxesMin. value
Transformed S-box101.3
APA S-box [25]112
S 8 Liu J S-box [26]99
Hussain et al. [27]100
Residue prime [28]94

3.3. Linear Approximation Probability (LP)

It is determined as the highest worth of inequality of a happening. The uniformity of input bits should be the image of the uniformity of the output bits. At the level of input ith, input bit is evaluated severally and also its consequences part discovered within the output bits formula where 2 shows the quantity of pats belong to the constructed S-box and also the assortment of every feasible input bits to S-box, part is denoted by X, where Φ(x) and Φ(y) show input/output [15]. Table 15 shows that our transformed S-box against linear attacks is better when compared to residue prime S-box and identical to Hussain S-box.
Table 15

Analysis of LP.

S-boxesMax. value
Transformed S-box0.15
APA S-box [25]0.062
S 8 Liu J S-box [26]0.105
Hussain et al. [27]0.125
Residue prime [28]0.132

3.4. Differential Approximation Probability (DP)

The S-box is considered because it is a nonlinear component of block cipher. In the perfect situation, S-box shows the different consistency. Δx is considered as the input differential whereas Δy indicates the output differential. During the technique of immigrant, it has been noticed, what quantity chance that differential of the input bits is is separately mapped on differential at output bits. Associate degree input differential associated degree should separately map to output Δy. To calculate the differential uniformity, the DP of specified S-box can be explicit as follows: Here, x represents a set of the possible feasible input values and their number of components are denoted by 2. Table 16 shows that S-box maintains its maximum differential probability at 0.06 which is acceptable value for resistance against differential attacks.
Table 16

Analysis of DP.

S-boxesMax. DP
Transformed S-box0.06
APA S-box [25]0.0156
S 8 Liu J S-box [26]0.0390
Hussain et al. [27]0.125
Residue prime [28]0.281

4. Application of the Proposed S-Boxes

The most advanced encryption standard algorithm is used for image encryption data. We can encrypt any image by using AES in MATLAB. We will not get any information of the original image when the image is encrypted. From this, we can see that AES encryption algorithm can get the results of image encryption. The AES encryption system is symmetric; it has three types of key length of encryption: 128, 196, and 256 bits, with a packet size of 128 bits for all; and the algorithm has fantastic flexibility. Therefore, it is being used in software and also in hardware. In this 3-key length of AES algorithm, 128 bits' key length is commonly used. Under this key length, 10-time iterative computation is done in the internal algorithm. Additionally, in the final round, every round contains five portions: Sub Bytes, S-box, Shift Rows, Mix Columns, and Add Round Key. Here, we perform the digital image encryption and will get the date which uses encryption algorithm of AES. Then, digital image encrypted by using AES algorithm is realized in the MATLAB simulation. From Figure 1, we can see that the host image is unpredictable when performing 1st round of AES, and this disorder in the picture increases as we perform other rounds. The feature of the image can also be described through gray histogram of image, which shows the number of occurrences of different pixel values. If the image contains a low contrast, then histogram will be narrow and will be focused in the middle of gray scale. From the result, it has been clear that AES algorithm has excellent effect for the encrypted image.
Figure 1

Image encryption with histogram analysis.

4.1. Majority Logic Criterion for S-Boxes

The majority logic criterion is applicable within the evaluation procedure of S-boxes, employed in AES (advanced encryption standard). The strength of the proposed S-boxes is checked by statistical analyses. The essential component of statistical analysis used for the sake of majority logic criterion is derived from the results of the following: Contrast Correlation Energy Homogeneity Entropy In the process of substitution, firstly data is altered into the form of encrypted data. On the other side, within the permutation process, the order of data material or contents is changed, which results in a different arrangement of the bits. The process of the substitution depends on the quantity of bits' n which makes the number of keys equal to 2ⁿ. The amalgamation of the permutation and the substitution of the data bit at the level of input make the encryption of the data stronger.

4.1.1. Contrast

The bulk of the contrast within the picture allows the viewer to brightly identify objects in a picture. Because the picture is encrypted, the amount of disorderness increases; as a result, it elevates the level of contrast to a really high value. Contrast is actually associated with the quantity of confusion which is created by the S-box within the original image. The mathematical depiction of contrast analysis is Here i, j denotes the pixels of the image. Figure 2 shows the illustration of contrast.
Figure 2

Illustration of contrast.

4.1.2. Correlation

Correlation elaborates the relation between the pixels in the image data. Correlation analysis is split into three different parts. It is performed on the following: Vertical and horizontal Diagonal formats General correlation Additionally, for analysis on a partial region, the complete image is additionally included within the processing. This analysis calculates the correlation of the pixel to its neighbor by taking into consideration the pixels of complete image data. If M, N identifies two matrix and M identifies the mean of the matrix elements, the for correlation is The correlation of the same image is one bit; if the correlations are equal, this does not mean that photographs are the same. Two different pictures may have the same correlation, but distribution of the pixel colors might be completely distinct as shown in Figure 3.
Figure 3

Correlation of encrypted image.

4.1.3. Energy

The analysis of energy is employed to measure the encrypted image. The gray-level cooccurrence matrix is employed to conduct energy. The performance of previous substitution box is healthier than the previous S-box utilized in analysis. The mathematical representation of the energy is

4.1.4. Homogeneity

The data of image contains a natural distribution that is related to the contents of the corresponding image. We execute the homogeneity which calculates the closeness of the distributed components. This is often called gray-tone spatial dependency matrix. The GLCM represents the combination of the pixel brightness values or the gray-levels that are formed in a table. The frequency of gray levels is often illuminated from the table GLCM. The homogeneity is often determined as Here, gray level cooccurrence matrices in the GLCM are mentioned by P(i, j).

4.1.5. Entropy

Entropy is often defined because of the randomness in the picture. The entropy of encrypted image is denoted aswhere p(x) have the histogram count. Figure 4 shows the comparison of higher and lower entropy. A superbly random image entropy has the value 8. Because the image gets foreseeable, entropy decreases. Therefore, in order to get good encrypted image, entropy must be closest to 8.
Figure 4

Higher vs. lower entropy.

Here, Table 17 shows the majority logic criterion of S-boxes which satisfy all the criteria up to standard that can be used for the sake of communication.
Table 17

Comparison of MLC.

S-boxesEntropyContrastCorrelationEnergyHomogeneity
Host image [23]7.60620.48960.90750.07850.8009
Proposed S-box7.997211.2629−0.00390.01590.3855
AES7.730187.3220850.0879040.0244770.483523
APA [29]7.6883837.7368590.2168160.0229420.486265
Prime [26]7.659556.3683670.0996340.0260990.49848
Skipjack [31]7.6738536.8051010.1958490.0261310.495087
The entropy measures the strength of image encryption scheme. If entropy is nearly equal to 8, this means that our image encryption scheme is good. If contrast is high, the strength of the encrypted image is more beneficial. Correlation close to zero as much as possible shows better encrypted image quality. If energy and homogeneity decrease and approximately equal zero, then the proposed image encryption scheme is better. Table 17 shows that entropy of encrypted image is close to 8, contrast is very high, correlation and energy are close to zero, and homogeneity also decreases. From these tests and comparisons, we can say that the proposed image encryption scheme using 10 different S-boxes in different round is very good.

5. Conclusion

S-box is the consequential component in the algorithm of encryption melded into SPN that plays a crucial part. In this work, we utilize a technique for the construction of worthy S-box which is established because of the action of PGL(2, GF(28)) on a Galois field GF(28). This new constructed S-box relates to a special sort of Mobius transformation. It has been observed from the appraisal that representation scheme of new S-box is undemanding and straightforward for the software and hardware application. Furthermore, to analyze the capability of S-box, we have applied different tests, nonlinearity, BIC, SAC, LP, and DP. Then, we have used these S-boxes in image encryption and checked the strength of image encryption by applying different tests, contrast, correlation, entropy, energy, and homogeneity. We have compared the results with others; hence, we conclude that the proposed scheme produces efficient results compared to other ones. The comparison of LP, DP, strict avalanche criterion, and bit independence criterion with existing techniques assured that the proposed scheme for image encryption is better. In future, proposed S-boxes will be used for audio, video, and text encryption scheme. Proposed image encryption scheme is used for data security of different military intelligence agencies.
  1 in total

1.  Human Psychological Disorder towards Cryptography: True Random Number Generator from EEG of Schizophrenics and Its Application in Block Encryption's Substitution Box.

Authors:  Muhammad Fahad Khan; Khalid Saleem; Mohammad Mazyad Hazzazi; Mohammed Alotaibi; Piyush Kumar Shukla; Muhammad Aqeel; Seda Arslan Tuncer
Journal:  Comput Intell Neurosci       Date:  2022-06-21
  1 in total

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