# A Novel Construction of Substitution Box Involving Coset Diagram and a Bijective Map

Authors: Abdul Razaq; Awais Yousaf; Umer Shuaib; Nasir Siddiqui; Atta Ullah; Adil Waheed
Journal: Security and Communication Networks
 (2017)

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## Abstract

The substitution box is a basic tool to convert the plaintext into an enciphered format. In this paper, we use coset diagram for the action ofPSL(2,Z) on projective line over the finite field GF29 to construct proposed S-box. The vertices of the cost diagram are elements of GF29 which can be represented by powers of α, where α is the root of irreducible polynomial px=x9+x4+1 over Z2. Let GF⁎29 denote the elements of GF29 which are of the form of even powers of α. In the first step, we construct a 16×16 matrix with the elements of GF⁎29 in a specific order, determined by the coset diagram. Next, we consider h:GF⁎29⟶GF28 defined by hα2n=ωn to destroy the structure of GF28. In the last step, we apply a bijective map g on each element of the matrix to evolve proposed S-box. The ability of the proposed S-box is examined by different available algebraic and statistical analyses. The results are then compared with the familiar S-boxes. We get encouraging statistics of the proposed box after comparison.


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## Body

## 1. Introduction

In secure communication, the role of the nonlinear component for block ciphers (substitution box) is of significant importance. The concept of substitution box was given by Shannon in 1949 [1]. In order to create confusion during the process of enciphering the digital data, substitution box plays a central role [2]. If the S-box is not good, it means one has to compromise on the quality of encryption. The strength of the S-box affirms the capability of block ciphers. Several attempts have been made to increase the quality of the S-box. In order to assess the properties of well-known S-boxes, the cryptographers have drawn attention to the literature. Different techniques have been developed to inspect the statistical and algebraic structure of S-boxes. These analyses include linear approximation probability (LP) method, bit independence criterion (BIC), majority logic criterion (MLC), strict avalanche criterion (SAC), nonlinearity method, and differential approximation probability (DP) method.In this paper, we establish a novel technique to construct substitution boxes by coset diagrams and bijective maps.


## 2. Coset Diagrams for Modular Group

The modular group, denoted byPSL2,Z, has a finite presentation x,y:x2=y3=1, where x and y are linear fractional transformations which map s to -1/s and s-1/s, respectively. Coset diagrams ([3–7]) are the graphical representation of the action of PSL(2,Z) on GFpn∪∞, where p is a prime. Since the order of y is three, its three cycles are represented by triangles. The vertices of the triangles, which are elements of GFpn∪∞, are permuted anticlockwise by y. Any two of vertices of the triangles are joined by an edge which represents x. The heavy dots are used to denote fixed points of x and y, if they exist.Consider the action of modular group onGF17∪∞ (Figure 1). The permutation representations of x and y can be calculated by x:s→-1/s and y:s→s-1/s.(1)x:0∞11628311451061471291513y:0∞13128291615106114513147.The action of PSL(2,Z) is not possible on GFpn, because image of 0 under x does not belong to GFpn. Therefore, we choose GFpn∪∞ for the action of PSL(2,Z) instead of GFpn.Figure 1
The coset diagram for the action of modular group onGF(17)∪∞.


## 3. Coset Diagram Used in the Construction of Proposed Substitution Box

Consider a primitive irreducible polynomialpx=x9+x4+1 over Z2; then GF(29)=Z2x/x9+x4+1. The element of GF(29) (see Table 1) can be represented by some power of α, where α is the root of px. Consider the action of PSL(2,Z) on GF29∪{∞}. Then the permutation representations of x and y can be calculated by sx=-1/s and sy=(s-1)/s, respectively. So (2)x:α1α510α129α382α381α130α2α509α258α253α251α260α3α508α417α94α91α420α2α509α258α253α251α260α4α507α5α506α502α9α6α505α323α188α182α329α7α504α60α451α444α67α8α503α10α501α493α18α11α500α459α52α41α470α12α499α135α376α364α147α13α498α125α386α373α138α14α497α120α391α377α134α15α496α111α400α385α126α16α495α20α491α475α36α17α494α324α187α170α341α19α492α402α109α90α421α21α490α285α226α306α205α22α489α407α104α82α429α23α488α314α197α174α337α24α487α270α241α217α294α25α486α31α480α455α56α26α485α250α261α235α276α27α484α115α396α369α142α455α56α28α483α240α271α243α268α29α482α434α77α48α463α30α481α222α289α259α252α32α479α40α471α439α72α33α478α148α363α330α181α34α477α137α374α340α171α35α476α196α315α280α231α37α474α338α173α136α375α38α473α293α218α180α331α39α472α175α336α297α214α42α469α59α452α410α101α43α468α327α184α141α370α44α467α303α208α164α347α45α466α265α246α201α310α46α465α117α394α348α163α47α464α254α257α210α301α49α462α70α441α392α119α50α461α62α449α51α460α403α108α57α454α53α458α172α339α286α225α54α457α230α281α227α284α55α456α353α158α103α408α58α453α357α154α96α415α61α450α106α405α344α167α63α448α200α311α248α263α64α447α80α431α367α144α65α446α191α320α255α256α66α445α296α215α149α362α68α443α274α237α169α342α69α442α249α262α71α440α198α313α242α269α73α438α146α365α292α219α74α437α165α346α272α239α75α436α360α151α146α365α76α435α78α433α350α161α83α428α79α432α228α283α204α307α81α430α298α213α132α379α84α427α118α393α309α202α85α426α264α247α162α349α86α425α143α368α282α229α87α424α267α244α157α354α88α423α95α416α328α183α89α422α325α186α97α414α92α419α234α277α185α326α93α418α211α300α207α304α98α413α140α371α273α238α99α412α121α390α291α220α100α411α124α387α287α224α102α409α295α216α114α397α105α406α279α232α127α384α107α404α168α343α236α275α110α401α195α316α206α305α113α398α245α266α153α358α116α395α203α308α192α319α122α389α212α299α177α334α123α388α233α278α155α356α128α383α179α332α139α372α194α317α145α366α176α335α190α321α150α361α209α302α152α359α156α355α189α322α166α3450∞1y:α1α129α381α510α130α382α2α258α251α509α260α253α3α417α91α508α420α94α4α5α502α507α9α506α6α323α182α505α329α188α7α60α444α504α67α451α8α10α493α503α18α501α11α459α41α500α470α52α12α135α364α499α147α376α13α125α373α498α138α386α14α120α377α497α134α391α15α111α385α496α126α400α16α20α475α495α36α491α17α324α170α494α341α187α19α402α90α492α421α109α21α285α205α490α306α226α22α407α82α489α429α104α23α314α174α488α337α197α24α270α217α487α294α241α25α31α455α486α56α480α26α250α235α485α276α261α27α115α369α484α142α396α28α240α243α483α268α271α29α434α48α482α463α77α30α222α259α481α252α289α32α40α439α479α72α471α33α148α330α478α181α363α34α137α340α477α171α374α35α196α280α476α231α315α37α338α136α474α375α173α38α293α180α473α331α218α39α175α297α472α214α336α42α59α410α469α101α452α43α327α141α468α370α184α44α303α164α467α347α208α45α265α201α466α310α246α46α117α348α465α163α394α47α254α210α464α301α257α49α70α392α462α119α441α50α62α399α461α112α449α51α403α57α460α454α108α53α172α286α458α225α339α54α230α227α457α284α281α55α353α103α456α408α158α58α357α96α453α415α154α61α106α344α450α167α405α63α200α248α448α263α311α64α80α367α447α144α431α65α191α255α446α256α320α66α296α149α445α362α215α68α274α169α443α342α237α69α193α249α442α262α318α71α198α242α440α269α313α73α146α292α438α219α365α74α165α272α437α239α346α75α360α76α436α435α151α78α350α83α433α428α161α79α228α204α432α307α283α81α298α132α430α379α213α84α118α309α427α202α393α85α264α162α426α349α247α86α143α282α425α229α368α87α267α157α424α354α244α88α95α328α423α183α416α89α325α97α422α414α186α92α234α185α419α326α277α93α211α207α418α304α300α98α140α273α413α238α371α99α121α291α412α220α390α100α124α287α411α224α387α102α295α114α409α397α216α105α279α127α406α384α232α107α168α236α404α275α343α110α195α206α401α305α316α113α245α153α398α358α266α116α203α192α395α319α308α122α212α177α389α334α299α123α233α155α388α356α278α128α160α223α383α288α351α131α159α221α380α290α352α133α199α179α378α332α312α139α194α178α372α333α317α145α176α190α366α321α335α150α209α152α361α359α302α156α189α166α355α345α322∞10The coset diagram for this action has 86 orbits (fragments). It has one copy of λ (Figure 2) and 85 copies of γj (Figure 3). First, we find the orbit of coset diagram which contains α1 (Figure 4).Table 1
Representation of the elements ofGF29.
Binary values
G
F
2
9
Binary values
G
F
2
9
Binary values
G
F
2
9
Binary values
G
F
2
9
000000000
0
000000001
1
000000010
α
1
000000100
α
2
000001000
α
3
000010000
α
4
000100000
α
5
001000000
α
6
010000000
α
7
100000000
α
8
000010001
α
9
000100010
α
10
001000100
α
11
010001000
α
12
100010000
α
13
000110001
α
14
001100010
α
15
011000100
α
16
110001000
α
17
100000001
α
18
000010011
α
19
000100110
α
20
001001100
α
21
010011000
α
22
100110000
α
23
001110001
α
24
011100010
α
25
111000100
α
26
110011001
α
27
100100011
α
28
001010111
α
29
010101110
α
30
101011100
α
31
010101001
α
32
101010010
α
33
010110101
α
34
101101010
α
35
011000101
α
36
110001010
α
37
100000101
α
38
000011011
α
39
000110110
α
40
001101100
α
41
011011000
α
42
110110000
α
43
101110001
α
44
011110011
α
45
111100110
α
46
111011101
α
47
110101011
α
48
101000111
α
49
010011111
α
50
100111110
α
51
001101101
α
52
011011010
α
53
110110100
α
54
101111001
α
55
011100011
α
56
111000110
α
57
110011101
α
58
100101011
α
59
001000111
α
60
010001110
α
61
100011100
α
62
000101001
α
63
001010010
α
64
010100100
α
65
101001000
α
66
010000001
α
67
100000010
α
68
000010101
α
69
000101010
α
70
001010100
α
71
010101000
α
72
101010000
α
73
010110001
α
74
101100010
α
75
011010101
α
76
110101010
α
77
101000101
α
78
010011011
α
79
100110110
α
80
001111101
α
81
011111010
α
82
111110100
α
83
111111001
α
84
111100011
α
85
111010111
α
86
110111111
α
87
101101111
α
88
011001111
α
89
110011110
α
90
100101101
α
91
001001011
α
92
010010110
α
93
100101100
α
94
001001001
α
95
010010010
α
96
100100100
α
97
001011001
α
98
010110010
α
99
101100100
α
100
011011001
α
101
110110010
α
102
101110101
α
103
011111011
α
104
111110110
α
105
111111101
α
106
111101011
α
107
111000111
α
108
110011111
α
109
100101111
α
110
001001111
α
111
010011110
α
112
100111100
α
113
001101001
α
114
011010010
α
115
110100100
α
116
101011001
α
117
010100011
α
118
101000110
α
119
010011101
α
120
100111010
α
121
001100101
α
122
011001010
α
123
110010100
α
124
100111001
α
125
001100011
α
126
011000110
α
127
110001100
α
128
100001001
α
129
000000011
α
130
000000110
α
131
000001100
α
132
000011000
α
133
000110000
α
134
001100000
α
135
011000000
α
136
110000000
α
137
100010001
α
138
000110011
α
139
001100110
α
140
011001100
α
141
110011000
α
142
100100001
α
143
001010011
α
144
010100110
α
145
101001100
α
146
010001001
α
147
100010010
α
148
000110101
α
149
001101010
α
150
011010100
α
151
110101000
α
152
101000001
α
153
010010011
α
154
100100110
α
155
001011101
α
156
010111010
α
157
101110100
α
158
011111001
α
159
111110010
α
160
111110101
α
161
111111011
α
162
111100111
α
163
111011111
α
164
110101111
α
165
101001111
α
166
010001111
α
167
100011110
α
168
000101101
α
169
001011010
α
170
010110100
α
171
101101000
α
172
011000001
α
173
110000010
α
174
100010101
α
175
000111011
α
176
001110110
α
177
011101100
α
178
111011000
α
179
110100001
α
180
101010011
α
181
010110111
α
182
101101110
α
183
011001101
α
184
110011010
α
185
100100101
α
186
001011011
α
187
010110110
α
188
101101100
α
189
011001001
α
190
110010010
α
191
100110101
α
192
001111011
α
193
011110110
α
194
111101100
α
195
111001001
α
196
110000011
α
197
100010111
α
198
000111111
α
199
001111110
α
200
011111100
α
201
111111000
α
202
111100001
α
203
111010011
α
204
110110111
α
205
101111111
α
206
011101111
α
207
111011110
α
208
110101101
α
209
101001011
α
210
010000111
α
211
100001110
α
212
000001101
α
213
000011010
α
214
000110100
α
215
001101000
α
216
011010000
α
217
110100000
α
218
101010001
α
219
010110011
α
220
101100110
α
221
011011101
α
222
110111010
α
223
101100101
α
224
011011011
α
225
110110110
α
226
101111101
α
227
011101011
α
228
111010110
α
229
110111101
α
230
101101011
α
231
011000111
α
232
110001110
α
233
100001101
α
234
000001011
α
235
000010110
α
236
000101100
α
237
001011000
α
238
010110000
α
239
101100000
α
240
011010001
α
241
110100010
α
242
101010101
α
243
010111011
α
244
101110110
α
245
011111101
α
246
111111010
α
247
111100101
α
248
111011011
α
249
110100111
α
250
101011111
α
251
010101111
α
252
101011110
α
253
010101101
α
254
101011010
α
255
010100101
α
256
101001010
α
257
010000101
α
258
100001010
α
259
000000101
α
260
000001010
α
261
000010100
α
262
000101000
α
263
001010000
α
264
010100000
α
265
101000000
α
266
010010001
α
267
100100010
α
268
001010101
α
269
010101010
α
270
101010100
α
271
010111001
α
272
101110010
α
273
011110101
α
274
111101010
α
275
111000101
α
276
110011011
α
277
100100111
α
278
001011111
α
279
010111110
α
280
101111100
α
281
011101001
α
282
111010010
α
283
110110101
α
284
101111011
α
285
011100111
α
286
111001110
α
287
110001101
α
288
100001011
α
289
000000111
α
290
000001110
α
291
000011100
α
292
000111000
α
293
001110000
α
294
011100000
α
295
111000000
α
296
110010001
α
297
100110011
α
298
001110111
α
299
011101110
α
300
111011100
α
301
110101001
α
302
101000011
α
303
010010111
α
304
100101110
α
305
001001101
α
306
010011010
α
307
100110100
α
308
001111001
α
309
011110010
α
310
111100100
α
311
111011001
α
312
110100011
α
313
101010111
α
314
010111111
α
315
101111110
α
316
011101101
α
317
111011010
α
318
110100101
α
319
101011011
α
320
010100111
α
321
101001110
α
322
010001101
α
323
100011010
α
324
000100101
α
325
001001010
α
326
010010100
α
327
100101000
α
328
001000001
α
329
010000010
α
330
100000100
α
331
000011001
α
332
000110010
α
333
001100100
α
334
011001000
α
335
110010000
α
336
100110001
α
337
001110011
α
338
011100110
α
339
111001100
α
340
110001001
α
341
100000011
α
342
000010111
α
343
000101110
α
344
001011100
α
345
010111000
α
346
101110000
α
347
011110001
α
348
111100010
α
349
111010101
α
350
110111011
α
351
101100111
α
352
011011111
α
353
110111110
α
354
101101101
α
355
011001011
α
356
110010110
α
357
100111101
α
358
001101011
α
359
011010110
α
360
110101100
α
361
101001001
α
362
010000011
α
363
100000110
α
364
000011101
α
365
000111010
α
366
001110100
α
367
011101000
α
368
111010000
α
369
110110001
α
370
101110011
α
371
011110111
α
372
111101110
α
373
111001101
α
374
110001011
α
375
100000111
α
376
000011111
α
377
000111110
α
378
001111100
α
379
011111000
α
380
111110000
α
381
111110001
α
382
111110011
α
383
111110111
α
384
111111111
α
385
111101111
α
386
111001111
α
387
110001111
α
388
100001111
α
389
000001111
α
390
000011110
α
391
000111100
α
392
001111000
α
393
011110000
α
394
111100000
α
395
111010001
α
396
110110011
α
397
101110111
α
398
011111111
α
399
111111110
α
400
111101101
α
401
111001011
α
402
110000111
α
403
100011111
α
404
000101111
α
405
001011110
α
406
010111100
α
407
101111000
α
408
011100001
α
409
111000010
α
410
110010101
α
411
100111011
α
412
001100111
α
413
011001110
α
414
110011100
α
415
100101001
α
416
001000011
α
417
010000110
α
418
100001100
α
419
000001001
α
420
000010010
α
421
000100100
α
422
001001000
α
423
010010000
α
424
100100000
α
425
001010001
α
426
010100010
α
427
101000100
α
428
010011001
α
429
100110010
α
430
001110101
α
431
011101010
α
432
111010100
α
433
110111001
α
434
101100011
α
435
011010111
α
436
110101110
α
437
101001101
α
438
010001011
α
439
100010110
α
440
000111101
α
441
001111010
α
442
011110100
α
443
111101000
α
444
111000001
α
445
110010011
α
446
100110111
α
447
001111111
α
448
011111110
α
449
111111100
α
450
111101001
α
451
111000011
α
452
110010111
α
453
100111111
α
454
001101111
α
455
011011110
α
456
110111100
α
457
101101001
α
458
011000011
α
459
110000110
α
460
100011101
α
461
000101011
α
462
001010110
α
463
010101100
α
464
101011000
α
465
010100001
α
466
101000010
α
467
010010101
α
468
100101010
α
469
001000101
α
470
010001010
α
471
100010100
α
472
000111001
α
473
001110010
α
474
011100100
α
475
111001000
α
476
110000001
α
477
100010011
α
478
000110111
α
479
001101110
α
480
011011100
α
481
110111000
α
482
101100001
α
483
011010011
α
484
110100110
α
485
101011101
α
486
010101011
α
487
101010110
α
488
010111101
α
489
101111010
α
490
011100101
α
491
111001010
α
492
110000101
α
493
100011011
α
494
000100111
α
495
001001110
α
496
010011100
α
497
100111000
α
498
001100001
α
499
011000010
α
500
110000100
α
501
100011001
α
502
000100011
α
503
001000110
α
504
010001100
α
505
100011000
α
506
000100001
α
507
001000010
α
508
010000100
α
509
100001000
α
510Figure 2
The orbitλ of the coset diagram for the action of modular group on GF(29)∪∞.Figure 3
The orbitγj of the coset diagram for the action of modular group on GF(29)∪∞.Figure 4
The copy ofγj having the vertex α1.In eachγj, one can see that xyxy-1x∈PSL(2,Z) is a path from the vertex αk to the vertex αl, such that, during this journey, one has to pass each vertex of γj.


## 4. Algebraic Structure of Proposed Substitution Box

In the literature, algebraic techniques are being applied on Galois fields. In this novel technique, the initial sequence of Galois field is destroyed with the help of vertices of the coset diagram.LetGF∗29 denote the elements of GF29 which are of the form of even powers of α.Our first aim is to write the vertices of coset diagram in a16×16 matrix. For this, we choose only those vertices which belong to GF∗29.Step 1.
Our coset diagram has 86 orbits; we construct a16×16 matrix having entries from GF∗29 in the following way.
First, we find the orbit of coset diagram which containsα1. Let us denote this orbit by γ1 and apply xyxy-1x on α1, so that we reach α382. During this journey, we pass through α510, α130, α381, and α129 and at the end reach α382. Write α510,α130, and α382 as first three elements of the first row of 16×16 matrix.After writing 3 vertices of anyγk∈γj:j=1,2,3,…,85, in order to select the next copy from γj, we find a vertex v=αi1+1, where αi1, αi2, αi3, αi4, αi5, and αi6 are the vertices of γk, such that i1<ij, where j=2,3,4,5,6. If αi1+1 is already exhausted in previously chosen copies of γj, then we move to the copy of γj containing v=αi1+2 and so on. Apply xyxy-1x on v so that we pass through each vertex. Note that, in each γj, only three vertices belong to GF∗29 out of 6. Write these 3 vertices in 16×16 matrix in order. The process continues until all the 510 vertices of γj are exhausted. Next, pick 0 from λ and write it as last element of the last row.Step 2.
Considerh:GF∗29⟶GF28 defined by fα2n=ϖn; the elements of F28 can be represented by powers of ω (see Table 3), where ω is the root of irreducible polynomial px=x8+x4+x3+x2+1 over Z2. In this step, we apply h on each element of the matrix evolved in the first step. In this way, we obtain a 16×16 matrix having entries from GF28. Next, we convert each entry of the matrix into binary form and ultimately into decimal form (Table 4).Before going to Step3, let us define a map g in the following way.Letf be the linear fractional transformation, obtained by the action of PGL(2,(F28 on GF28; that is, f:PGL2,GF28×GF28→GF28.Clearlyf is of the form az+b/cz+d, where a,b,c,d∈GF28, and is a mapping from GF28→GF28. But, usually f is not a bijective map.(i) Let
there ben elements of GF28 missing in the range of f.(ii) Let
s1,s2,s3,…,sk∈GF28 be repeated in the range of f.(iii) Let
t1,t2,t3,…,tk be the smallest elements in GF28 whose images are s1,s2,s3,…,sk, respectively.Suppose thatA=a1,a2,…,an:ai<ai+1 is the set of all those elements, except t1,t2,t3,…,tk, whose images are s1,s2,s3,…,sk, and B=b1,b2,…,bn:bi>bi+1 is the set of missing elements in the range of f. Then(3)gz=fzifz∈GF28-Abiifz=ai.In this paper, we have taken fz=214z+93/124z+123.Step 3.
In this step, the bijective mapg defined by(4)gz=214z+93124z+123ifz∈GF28-Abiifz=aiis applied on the matrix obtained at the end of Step 2 to evolve proposed S-box (Figure 5 and Table 5). More details of this last step are given in Table 2.Table 2
Construction of S-box using linear fractional transformation.
z
f
(
z
)
=
214
z
+
93
124
z
+
123
g(z)
0
f
0
=
93
124
=
01011101
01111011
=
ϖ
56
ϖ
172
=
ϖ
139
=
1000010
=
66
66
1
f
1
=
51
247
=
00110011
11110111
=
ϖ
125
ϖ
232
=
ϖ
148
=
01010010
=
82
82
2
f
2
=
9
115
=
00001001
01110011
=
ϖ
223
ϖ
159
=
ϖ
64
=
01011111
=
95
95
⋮
254
f
254
=
177
131
=
10110001
10000011
=
ϖ
86
ϖ
247
=
ϖ
94
=
01110001
=
113
2
255
f
255
=
135
255
=
10000111
11111111
=
ϖ
13
ϖ
175
=
ϖ
93
=
10110110
=
182
0Table 3
Representation of the elements ofGF28.
Binary values
G
F
2
8
Binary values
G
F
2
8
Binary values
G
F
2
8
Binary values
G
F
2
8
00000000
0
00000001
1
00000010
ϖ
1
00000100
ϖ
2
00001000
ϖ
3
00010000
ϖ
4
00100000
ϖ
5
01000000
ϖ
6
10000000
ϖ
7
00011101
ϖ
8
00111010
ϖ
9
01110100
ϖ
10
11101000
ϖ
11
11001101
ϖ
12
10000111
ϖ
13
00010011
ϖ
14
00100110
ϖ
15
01001100
ϖ
16
10011000
ϖ
17
00101101
ϖ
18
01011010
ϖ
19
10110100
ϖ
20
01110101
ϖ
21
11101010
ϖ
22
11001001
ϖ
23
10001111
ϖ
24
00000011
ϖ
25
00000110
ϖ
26
00001100
ϖ
27
00011000
ϖ
28
00110000
ϖ
29
01100000
ϖ
30
11000000
ϖ
31
10011101
ϖ
32
00100111
ϖ
33
01001110
ϖ
34
10011100
ϖ
35
00100101
ϖ
36
01001010
ϖ
37
10010100
ϖ
38
00110101
ϖ
39
01101010
ϖ
40
11010100
ϖ
41
10110101
ϖ
42
01110111
ϖ
43
11101110
ϖ
44
11000001
ϖ
45
10011111
ϖ
46
00100011
ϖ
47
01000110
ϖ
48
10001100
ϖ
49
00000101
ϖ
50
00001010
ϖ
51
00010100
ϖ
52
00101000
ϖ
53
01010000
ϖ
54
10100000
ϖ
55
01011101
ϖ
56
10111010
ϖ
57
01101001
ϖ
58
11010010
ϖ
59
10111001
ϖ
60
01101111
ϖ
61
11011110
ϖ
62
10100001
ϖ
63
01011111
ϖ
64
10111110
ϖ
65
01100001
ϖ
66
11000010
ϖ
67
10011001
ϖ
68
00101111
ϖ
69
01011110
ϖ
70
10111100
ϖ
71
01100101
ϖ
72
11001010
ϖ
73
10001001
ϖ
74
00001111
ϖ
75
00011110
ϖ
76
00111100
ϖ
77
01111000
ϖ
78
11110000
ϖ
79
11111101
ϖ
80
11100111
ϖ
81
11010011
ϖ
82
10111011
ϖ
83
01101011
ϖ
84
11010110
ϖ
85
10110001
ϖ
86
01111111
ϖ
87
11111110
ϖ
88
11100001
ϖ
89
11011111
ϖ
90
10100011
ϖ
91
01011011
ϖ
92
10110110
ϖ
93
01110001
ϖ
94
11100010
ϖ
95
11011001
ϖ
96
10101111
ϖ
97
01000011
ϖ
98
10000110
ϖ
99
00010001
ϖ
100
00100010
ϖ
101
01000100
ϖ
102
10001000
ϖ
103
00001101
ϖ
104
00011010
ϖ
105
00110100
ϖ
106
01101000
ϖ
107
11010000
ϖ
108
10111101
ϖ
109
01100111
ϖ
110
11001110
ϖ
111
10000001
ϖ
112
00011111
ϖ
113
00111110
ϖ
114
01111100
ϖ
115
11111000
ϖ
116
11101101
ϖ
117
11000111
ϖ
118
10010011
ϖ
119
00111011
ϖ
120
01110110
ϖ
121
11101100
ϖ
122
11000101
ϖ
123
10010111
ϖ
124
00110011
ϖ
125
01100110
ϖ
126
11001100
ϖ
127
10000101
ϖ
128
00010111
ϖ
129
00101110
ϖ
130
01011100
ϖ
131
10111000
ϖ
132
01101101
ϖ
133
11011010
ϖ
134
10101001
ϖ
135
01001111
ϖ
136
10011110
ϖ
137
00100001
ϖ
138
01000010
ϖ
139
10000100
ϖ
140
00010101
ϖ
141
00101010
ϖ
142
01010100
ϖ
143
10101000
ϖ
144
01001101
ϖ
145
10011010
ϖ
146
00101001
ϖ
147
01010010
ϖ
148
10100100
ϖ
149
01010101
ϖ
150
10101010
ϖ
151
01001001
ϖ
152
10010010
ϖ
153
00111001
ϖ
154
01110010
ϖ
155
11100100
ϖ
156
11010101
ϖ
157
10110111
ϖ
158
01110011
ϖ
159
11100110
ϖ
160
11010001
ϖ
161
10111111
ϖ
162
01100011
ϖ
163
11000110
ϖ
164
10010001
ϖ
165
00111111
ϖ
166
01111110
ϖ
167
11111100
ϖ
168
11100101
ϖ
169
11010111
ϖ
170
10110011
ϖ
171
01111011
ϖ
172
11110110
ϖ
173
11110001
ϖ
174
11111111
ϖ
175
11100011
ϖ
176
11011011
ϖ
177
10101011
ϖ
178
01001011
ϖ
179
10010110
ϖ
180
00110001
ϖ
181
01100010
ϖ
182
11000100
ϖ
183
10010101
ϖ
184
00110111
ϖ
185
01101110
ϖ
186
11011100
ϖ
187
10100101
ϖ
188
01010111
ϖ
189
10101110
ϖ
190
01000001
ϖ
191
10000010
ϖ
192
00011001
ϖ
193
00110010
ϖ
194
01100100
ϖ
195
11001000
ϖ
196
10001101
ϖ
197
00000111
ϖ
198
00001110
ϖ
199
00011100
ϖ
200
00111000
ϖ
201
01110000
ϖ
202
11100000
ϖ
203
11011101
ϖ
204
10100111
ϖ
205
01010011
ϖ
206
10100110
ϖ
207
01010001
ϖ
208
10100010
ϖ
209
01011001
ϖ
210
10110010
ϖ
211
01111001
ϖ
212
11110010
ϖ
213
11111001
ϖ
214
11101111
ϖ
215
11000011
ϖ
216
10011011
ϖ
217
00101011
ϖ
218
01010110
ϖ
219
10101100
ϖ
220
01000101
ϖ
221
10001010
ϖ
222
00001001
ϖ
223
00010010
ϖ
224
00100100
ϖ
225
01001000
ϖ
226
10010000
ϖ
227
00111101
ϖ
228
01111010
ϖ
229
11110100
ϖ
230
11110101
ϖ
231
11110111
ϖ
232
11110011
ϖ
233
11111011
ϖ
234
11101011
ϖ
235
11001011
ϖ
236
10001011
ϖ
237
00001011
ϖ
238
00010110
ϖ
239
00101100
ϖ
240
01011000
ϖ
241
10110000
ϖ
242
01111101
ϖ
243
11111010
ϖ
244
11101001
ϖ
245
11001111
ϖ
246
10000011
ϖ
247
00011011
ϖ
248
00110110
ϖ
249
01101100
ϖ
250
11011000
ϖ
251
10101101
ϖ
252
01000111
ϖ
253
10001110
ϖ
254Table 4
16
×
16 matrix evolved after 2nd step.
1
190
65
2
23
46
142
89
35
4
216
71
8
163
113
96
138
173
16
32
58
108
235
6
64
98
165
54
47
25
128
185
194
27
161
28
29
116
45
191
214
131
56
193
207
233
146
31
232
212
20
213
127
250
205
169
41
125
24
44
135
51
33
176
188
7
19
59
218
155
143
88
38
206
102
76
180
37
137
145
22
152
215
220
67
132
11
229
153
139
90
223
189
203
104
252
117
167
72
251
55
91
234
211
13
243
114
197
201
241
141
204
26
247
156
200
245
3
192
93
244
144
80
177
84
122
12
124
42
61
221
240
48
70
60
40
123
36
17
151
18
157
106
101
9
133
230
39
82
49
78
158
179
69
92
115
134
118
172
202
154
86
74
79
246
150
148
43
53
255
249
62
68
195
164
97
239
181
210
34
184
231
242
119
21
149
121
219
236
238
198
81
178
166
182
159
237
99
162
73
85
140
94
147
83
103
100
5
222
129
10
186
208
224
130
248
107
199
112
160
136
183
14
75
109
105
217
57
111
52
126
50
171
66
95
253
168
174
77
227
87
63
228
175
225
110
254
226
196
15
30
170
120
187
209
0Table 5
Proposed S-box evolved after 3rd step.
82
114
50
95
120
28
196
91
44
242
70
153
205
147
73
40
204
136
250
48
247
166
29
183
157
237
145
243
53
122
225
121
109
65
149
253
221
144
203
112
76
219
93
110
88
33
181
63
34
79
171
78
182
7
92
140
213
227
132
160
210
52
174
133
116
226
47
234
67
162
191
151
155
90
42
233
127
246
207
185
71
175
72
61
13
217
39
37
169
202
80
57
115
97
9
5
55
142
26
6
30
74
32
81
4
17
86
105
99
19
197
94
75
12
161
100
15
46
111
235
16
188
208
130
69
163
255
229
244
164
60
21
230
49
154
223
54
212
195
176
165
159
184
102
64
214
36
252
59
117
194
158
128
240
11
45
211
198
137
98
167
168
224
239
14
177
179
1
77
0
8
87
89
108
146
248
22
126
83
193
123
35
18
186
96
178
199
62
25
23
104
206
129
143
125
152
24
20
148
27
218
200
238
180
190
232
216
228
58
222
189
119
85
56
220
10
103
101
173
150
209
124
254
107
231
31
68
170
236
251
113
245
138
215
192
3
141
135
172
41
187
201
38
134
51
131
2
43
106
241
249
139
156
118
84
66Figure 5
Flowchart for the construction of the proposed S-box.


## 5. Statistical Analysis and Simulation Results

In this section, we implement various security performance tests on newly created S-box to examine its special properties. The assessment of the characteristics of proposed S-box determines its application in different encryption methods and for security purposes. We use five different security performance tests, namely, linear approximation probability (LP), differential approximation probability (DP), nonlinearity, bit independence criterion (BIC), and strict avalanche criterion, to assess the cryptographic competence of substitution box. The results obtained from proposed S-box are then compared with the well-known S-boxes. The description of different types of tests implemented on these S-boxes is given below.


### 5.1. Nonlinearity

The concept of nonlinearity was first introduced by Pieprzyk and Finkelstein in 1988 [8]. It is the basic tool to measure the strength of the S-box. An S-box with bigger nonlinearity is more secure than that with lesser nonlinearity. The nonlinearity is expressed as(5)Nh=2n-11-2-nmax⁡Shα,where Shα=∑-1hβ⊕β·α is Walsh spectrum and α,β∈GF2n.The average value of the nonlinearity of the proposed S-box is 106.75. In Table6, nonlinearity of the proposed S-box is compared with multiple renowned substitution boxes. One can see that the nonlinearity of the proposed S-box is better than most of the familiar S-boxes.Table 6
Nonlinearity of basic functions of various substitution boxes.
S-box
f
0
f
1
f
2
f
3
f
4
f
5
f
6
f
7
Ave
Gray [13]
112
112
112
112
112
112
112
112
112
Gautam et al. [14]
108
106
104
98
102
102
98
74
99
Prime [15]
94
100
104
104
102
100
98
94
99.5
S8 AES [16]
112
112
112
112
112
112
112
112
112
Shi et al. [17]
106
104
106
106
104
106
104
106
105
AES [2]
112
112
112
112
112
112
112
112
112
Skipjac and Kea [18]
104
108
108
108
108
104
104
106
106.75
Alkhaldi et al. [19]
108
104
106
106
102
98
104
108
104
Chen et al. [20]
100
102
103
104
106
106
106
108
104.3
Tang et al. [21]
100
103
104
104
105
105
106
109
104.5
Khan et al. [22]
102
108
106
102
106
106
106
98
104.25
Belazi et al. [23]
106
106
106
104
108
102
106
104
105.25
Proposed
108
106
108
108
108
104
106
106
106.75


### 5.2. Bit Independence Criterion

According to bit independence criterion [9, 10], if any input bit i is inverted, then the output bits j and k must change independently. In other words, the avalanche variables must be pairwise independent for a given set of avalanche vectors. We have tested the nonlinearity of bit independence criterion of S-box (Table 7). We also compared the minimum and average values of bit independence criterion along with square deviation of the proposed S-box with different renowned S-boxes (Table 8).Table 7
Nonlinearity of bit independence criterion of the proposed S-box.
—
96
104
104
106
106
108
106
96
—
106
102
102
102
106
106
104
106
—
104
102
106
102
106
104
102
104
—
104
102
104
102
106
102
102
104
—
104
104
96
106
102
106
102
104
—
102
106
108
106
102
104
104
102
—
104
106
106
106
102
96
106
104
—Table 8
Bit independence criterion of various substitution boxes.
S-boxes
Minimum value
Average
Square deviation
Proposed
96
103.643
2.7283
Gray [13]
112
112
0
Gautam et al. [14]
92
103
3.5225
Prime [15]
94
101.71
3.53
S8 AES [16]
112
112
0
Hussain et al. [12]
98
103.78
2.743
AES [2]
112
112
0
Hussain et al. [15]
102
104.14
1.767


### 5.3. Strict Avalanche Criterion

Strict avalanche criterion (SAC) introduced by Tavares and Webster is founded on the ideas of the avalanche and completeness effect [9, 10]. It is a formalization of the avalanche effect. If by complementing a single input bit all the output bits are changed with a 0.5 probability, SAC is said to be satisfied. Table 9 displays the outcomes of the strict avalanche criterion.Table 9
Strict avalanche criterion of the proposed S-box.
.4375
.5000
.5000
.4844
.4844
.4844
.5312
.4375
.5000
.4375
.5000
.5625
.5000
.5469
.4688
.5000
.5312
.5469
.5156
.4687
.4531
.5312
.5156
.4375
.4844
.4375
.5469
.5312
.4688
.5781
.5781
.5000
.4688
.5156
.4531
.5625
.5781
.5469
.4844
.5156
.5000
.5156
.5312
.5000
.4688
.5156
.5488
.5000
.5156
.5313
.4844
.4687
.4531
.5000
.4688
.4531
.5312
.4844
.5625
.5469
.4844
.4531
.5625
.5000


### 5.4. Linear Approximation Probability

In linear approximation probability method, we examine the imbalance of an event [11]. The analysis is used to calculate the highest value of imbalance of the outcome of the event. The uniformity of the input bits should be similar to that of the output bits. Each ith input bit is analyzed individually and its outcomes are checked in the output bits. The masks which are applied on the parity of both input and output bits are denoted by χx and χy, respectively. Mathematically,(6)LP=maxχx,χy≠0⁡#d/d∙χx=Sd∙χy2n-12,where d represents the collection of all possible inputs and 2n is the total number of elements. The results of this important analysis obtained from our S-box and different established S-boxes are given in Table 10. The comparison shows that our S-box is strong enough to deal with different linear attacks.Table 10
Linear approximation probability analyses of different S-boxes.
S-boxes
AES
Gray
Skipjack
Prime
Proposed
Gautam et al. [14]
S8 AES
Xyi
Max value
144
144
156
162
162
164
144
168
Max LP
0.062
0.062
0.109
0.132
0.1484
0.2109
0.062
0.156


### 5.5. Differential Approximation Probability

In this analysis, differential uniformity is determined by examining the mapping from the input bits to the output. The main focus of this test is to ensure differential uniformity; that is, the input differential must be associated with an output differential in a unique way.It is represented by(7)DPsΔμ⟶Δν=#μεI/Sμ⨁Sμ⨁Δμ=Δν2n,where Δμ and Δν are the input and output differentials, respectively. We have applied differential approximation probability test on our S-box. The results are presented in Table 11.Table 11
Differential probability of the proposed S-box.
.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.01562
.02344
.03125
.02344
.02344
.03125
.03125
.03125
.03125
.03125
.02344
.02344
.03125
.02344
.02344
.02344
.02344
.02344
.02344
.01563
.02344
.02344
.03125
.03125
.03125
.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.01562
.02344
.03125
.03906
.02344
.03125
.02344
.02344
.03125
.02344
.03125
.03125
.03125
.02344
.02344
.02344
.02344
.02344
.03125
.04688
.03906
.02344
.02344
.02344
.03125
.02344
.01562
.02344
.03125
.02344
.02344
.02344
.02344
.03125
.03125
.03125
.02344
.03125
.03125
.03125
.03125
.03125
.02344
.02344
.02344
.03125
.02344
.03125
.02344
.02344
.02344
.03125
.03125
.03906
.03125
.02344
.03125
.03906
.02344
.02344
.03125
.02344
.03125
.02344
.03125
.02344
.01563
.03125
.03125
.03125
.03125
.03125
.02344
.02344
.03125
.03125
.02344
.03125
.02344
.03125
.02344
.02344
.03125
.03125
.03906
.02344
.03125
.02344
.02344
.03125
.03125
.03125
.02344
.03125
.02344
.03906
.03125
.03125
.03125
.02344
.03125
.02344
.03125
.03125
.02344
.03125
.02344
.02344
.02344
.03125
.02344
.03125
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.03125
.03125
.02344
.03125
.02344
.02344
.03125
.02344
.02344
.02344
.03125
.02344
.02344
.03125
.02344
.02344
.02344
.02344
.02344
.02344
.02344
.03125
.03125
.02344
.02344
.02344
.02344
.02344
.02344
.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.03125
.02344
.02344
.03125
.03125
.03125
.03125
.02344
.04688
.02344
.02344
.02344
.03906
.03125
.03125
.02344
.02344
.02344
.03125
.03125
.02344
.03125
.02344
.02344
.03906
.03125
.02344
.02344
.02344
.02344
.02344
.03125
.03125
.02344
.03125
.02344
.03906
.03125
.03125
.03125
.02344
.02344
.03125
.03125
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.03125
.02344
—


## 5.1. Nonlinearity

The concept of nonlinearity was first introduced by Pieprzyk and Finkelstein in 1988 [8]. It is the basic tool to measure the strength of the S-box. An S-box with bigger nonlinearity is more secure than that with lesser nonlinearity. The nonlinearity is expressed as(5)Nh=2n-11-2-nmax⁡Shα,where Shα=∑-1hβ⊕β·α is Walsh spectrum and α,β∈GF2n.The average value of the nonlinearity of the proposed S-box is 106.75. In Table6, nonlinearity of the proposed S-box is compared with multiple renowned substitution boxes. One can see that the nonlinearity of the proposed S-box is better than most of the familiar S-boxes.Table 6
Nonlinearity of basic functions of various substitution boxes.
S-box
f
0
f
1
f
2
f
3
f
4
f
5
f
6
f
7
Ave
Gray [13]
112
112
112
112
112
112
112
112
112
Gautam et al. [14]
108
106
104
98
102
102
98
74
99
Prime [15]
94
100
104
104
102
100
98
94
99.5
S8 AES [16]
112
112
112
112
112
112
112
112
112
Shi et al. [17]
106
104
106
106
104
106
104
106
105
AES [2]
112
112
112
112
112
112
112
112
112
Skipjac and Kea [18]
104
108
108
108
108
104
104
106
106.75
Alkhaldi et al. [19]
108
104
106
106
102
98
104
108
104
Chen et al. [20]
100
102
103
104
106
106
106
108
104.3
Tang et al. [21]
100
103
104
104
105
105
106
109
104.5
Khan et al. [22]
102
108
106
102
106
106
106
98
104.25
Belazi et al. [23]
106
106
106
104
108
102
106
104
105.25
Proposed
108
106
108
108
108
104
106
106
106.75


## 5.2. Bit Independence Criterion

According to bit independence criterion [9, 10], if any input bit i is inverted, then the output bits j and k must change independently. In other words, the avalanche variables must be pairwise independent for a given set of avalanche vectors. We have tested the nonlinearity of bit independence criterion of S-box (Table 7). We also compared the minimum and average values of bit independence criterion along with square deviation of the proposed S-box with different renowned S-boxes (Table 8).Table 7
Nonlinearity of bit independence criterion of the proposed S-box.
—
96
104
104
106
106
108
106
96
—
106
102
102
102
106
106
104
106
—
104
102
106
102
106
104
102
104
—
104
102
104
102
106
102
102
104
—
104
104
96
106
102
106
102
104
—
102
106
108
106
102
104
104
102
—
104
106
106
106
102
96
106
104
—Table 8
Bit independence criterion of various substitution boxes.
S-boxes
Minimum value
Average
Square deviation
Proposed
96
103.643
2.7283
Gray [13]
112
112
0
Gautam et al. [14]
92
103
3.5225
Prime [15]
94
101.71
3.53
S8 AES [16]
112
112
0
Hussain et al. [12]
98
103.78
2.743
AES [2]
112
112
0
Hussain et al. [15]
102
104.14
1.767


## 5.3. Strict Avalanche Criterion

Strict avalanche criterion (SAC) introduced by Tavares and Webster is founded on the ideas of the avalanche and completeness effect [9, 10]. It is a formalization of the avalanche effect. If by complementing a single input bit all the output bits are changed with a 0.5 probability, SAC is said to be satisfied. Table 9 displays the outcomes of the strict avalanche criterion.Table 9
Strict avalanche criterion of the proposed S-box.
.4375
.5000
.5000
.4844
.4844
.4844
.5312
.4375
.5000
.4375
.5000
.5625
.5000
.5469
.4688
.5000
.5312
.5469
.5156
.4687
.4531
.5312
.5156
.4375
.4844
.4375
.5469
.5312
.4688
.5781
.5781
.5000
.4688
.5156
.4531
.5625
.5781
.5469
.4844
.5156
.5000
.5156
.5312
.5000
.4688
.5156
.5488
.5000
.5156
.5313
.4844
.4687
.4531
.5000
.4688
.4531
.5312
.4844
.5625
.5469
.4844
.4531
.5625
.5000


## 5.4. Linear Approximation Probability

In linear approximation probability method, we examine the imbalance of an event [11]. The analysis is used to calculate the highest value of imbalance of the outcome of the event. The uniformity of the input bits should be similar to that of the output bits. Each ith input bit is analyzed individually and its outcomes are checked in the output bits. The masks which are applied on the parity of both input and output bits are denoted by χx and χy, respectively. Mathematically,(6)LP=maxχx,χy≠0⁡#d/d∙χx=Sd∙χy2n-12,where d represents the collection of all possible inputs and 2n is the total number of elements. The results of this important analysis obtained from our S-box and different established S-boxes are given in Table 10. The comparison shows that our S-box is strong enough to deal with different linear attacks.Table 10
Linear approximation probability analyses of different S-boxes.
S-boxes
AES
Gray
Skipjack
Prime
Proposed
Gautam et al. [14]
S8 AES
Xyi
Max value
144
144
156
162
162
164
144
168
Max LP
0.062
0.062
0.109
0.132
0.1484
0.2109
0.062
0.156


## 5.5. Differential Approximation Probability

In this analysis, differential uniformity is determined by examining the mapping from the input bits to the output. The main focus of this test is to ensure differential uniformity; that is, the input differential must be associated with an output differential in a unique way.It is represented by(7)DPsΔμ⟶Δν=#μεI/Sμ⨁Sμ⨁Δμ=Δν2n,where Δμ and Δν are the input and output differentials, respectively. We have applied differential approximation probability test on our S-box. The results are presented in Table 11.Table 11
Differential probability of the proposed S-box.
.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.01562
.02344
.03125
.02344
.02344
.03125
.03125
.03125
.03125
.03125
.02344
.02344
.03125
.02344
.02344
.02344
.02344
.02344
.02344
.01563
.02344
.02344
.03125
.03125
.03125
.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.01562
.02344
.03125
.03906
.02344
.03125
.02344
.02344
.03125
.02344
.03125
.03125
.03125
.02344
.02344
.02344
.02344
.02344
.03125
.04688
.03906
.02344
.02344
.02344
.03125
.02344
.01562
.02344
.03125
.02344
.02344
.02344
.02344
.03125
.03125
.03125
.02344
.03125
.03125
.03125
.03125
.03125
.02344
.02344
.02344
.03125
.02344
.03125
.02344
.02344
.02344
.03125
.03125
.03906
.03125
.02344
.03125
.03906
.02344
.02344
.03125
.02344
.03125
.02344
.03125
.02344
.01563
.03125
.03125
.03125
.03125
.03125
.02344
.02344
.03125
.03125
.02344
.03125
.02344
.03125
.02344
.02344
.03125
.03125
.03906
.02344
.03125
.02344
.02344
.03125
.03125
.03125
.02344
.03125
.02344
.03906
.03125
.03125
.03125
.02344
.03125
.02344
.03125
.03125
.02344
.03125
.02344
.02344
.02344
.03125
.02344
.03125
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.03125
.03125
.02344
.03125
.02344
.02344
.03125
.02344
.02344
.02344
.03125
.02344
.02344
.03125
.02344
.02344
.02344
.02344
.02344
.02344
.02344
.03125
.03125
.02344
.02344
.02344
.02344
.02344
.02344
.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.03125
.02344
.02344
.03125
.03125
.03125
.03125
.02344
.04688
.02344
.02344
.02344
.03906
.03125
.03125
.02344
.02344
.02344
.03125
.03125
.02344
.03125
.02344
.02344
.03906
.03125
.02344
.02344
.02344
.02344
.02344
.03125
.03125
.02344
.03125
.02344
.03906
.03125
.03125
.03125
.02344
.02344
.03125
.03125
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.03125
.02344
—


## 6. Majority Logic Criterion

The majority logic criterion [12] is helpful in finding the premier candidate S-box fit for a certain kind of encryption application. In this criterion, image encryption strength of the S-box is investigated through statistical studies. A distortion in the image is created by encryption process; therefore, it is important to study the statistical characteristics. It is achieved with the help of various analyses such as entropy, contrast, correlation, energy, and homogeneity. The proposed S-box can further be used for encryption and multimedia security. In this paper, we have used two JPEG images, Pepper and Baboon, for MLC analysis. The results of these analyses in comparison with the other well-known S-boxes are depicted in Table 12. Figure 6 shows the result of image encryption with proposed S-box. The histogram of the original image and the encrypted images of Baboon and Pepper are shown in Figure 7. These results show that our S-box fulfills all the requirements to be declared as a very suitable S-box for encryption applications. Thus, it is recommended to become a part of algorithms designed for the secure transmission of information/data.Table 12
Comparison of MLC for proposed S-box over different S-boxes.
S-boxes
Entropy
Contrast
Correlation
Energy
Homogeneity
Pepper image
Plaintext
7.5909
0.2760
0.9383
0.1288
0.9024
Proposed
7.9532
8.5155
0.0079
0.0174
0.4100
AES
7.9211
7.5509
0.0554
0.0202
0.4662
Ullah et al. [24]
7.9823
8.6727
−0.0043
0.0173
0.4076
Skipjack
7.7561
7.7058
0.1205
0.0239
0.4708
Khan et al. [25]
7.9562
8.3129
0.0103
0.0180
0.4219
Belazi
7.9233
8.1423
−0.0112
0.0286
0.4648
Baboon image
Plaintext
7.1273
0.7179
0.6782
0.1025
0.7669
Proposed
7.9551
8.5267
4.4609
e
-
004
0.0174
.4088
AES
7.2531
7.5509
0.0554
0.0202
0.4662
Prime
6.9311
7.6236
0.0855
0.0202
0.4640
Xyi
7.2531
8.3108
0.0417
0.0196
0.4533
Skipjack
7.2531
7.7058
0.1025
0.0193
0.4689
Khan et al. [25]
7.9612
8.1213
−0.0512
0.0210
0.4011
Belazi
7.9252
8.0391
0.0119 0
.02219
0.4428Figure 6
Original image and the encrypted images using two rounds of encryption: (a) Pepper and (b) Baboon.
(a)
(b)Figure 7
Histogram of the original image and the encrypted images: (a) Pepper and (b) Baboon.
(a)
(b)


## 7. Conclusion

In the present study, a strong S-box is created with the help of coset graph for the action of modular group and a bijective map. According to our information, this is the first use of coset graphs in the construction of S-box. The proposed S-box is highly secure and the results obtained from different analyses are nearly equal to the ideal ones. Therefore, it is very useful for secure communication.



---

*Source: 5101934-2017-11-20.xml*