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Fermat Encoding in S-Box for Secured Encryption

Md. Hasanujjaman

Md. Hasanujjaman

Department of Computer Software Technology, Sheikhpara ARM PolytechnicWest Bengal, India
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Hasanujjaman, Md.
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Partha Sarathi Goswami

Partha Sarathi Goswami

Department of Cyber Forensics and Information Security, Behala Government PolytechnicKolkata, India
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Goswami, Partha Sarathi
  
21 feb 2025
Fermat Encoding in S-Box for Secured Encryption's Cover Image
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Publicado en línea: 21 feb 2025
Páginas: 266 - 283
DOI: https://doi.org/10.2478/ias-2024-0018
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© 2024 Md. Hasanujjaman et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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Figure 1.

Encryption of the S-box
Encryption of the S-box

Figure 2.

Decryption of the S-box
Decryption of the S-box
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Comparison of S-boxes

Performance index AES S-Box Affine-Power-Affine Proposed Scheme
Balance criteria balance balance balance
Differential uniformity(F) 4 4 4
Non-zero linear structure none none none
Number of terms in S-box algebraic expression 9 255 256
Iterative period less than 88 less than 88 256

Decrypted S-Box

198 55 37 212 1 240 226 19 73
184 170 91 142 127 109 156 217 40
58 203 30 239 253 12 86 167 181
68 145 96 114 131 248 9 27 234
63 206 220 45 119 134 148 101 176
65 83 162 231 22 4 245 32 209
195 50 104 153 139 122 175 94 76
189 186 75 89 168 125 140 158 111
53 196 214 39 242 3 17 224 165
84 70 183 98 147 129 112 42 219
201 56 237 28 14 255 132 117 103
150 67 178 160 81 11 250 232 25
204 61 47 222 155 106 120 137 92
173 191 78 20 229 247 6 211 34
48 193 62 207 221 44 249 8 26
235 177 64 82 163 118 135 149 100
33 208 194 51 230 23 5 244 174
95 77 188 105 152 138 123 0 241
227 18 199 54 36 213 143 126 108
157 72 185 171 90 31 238 252 13
216 41 59 202 144 97 115 130 87
166 180 69 66 179 161 80 133 116
102 151 205 60 46 223 10 251 233
24 93 172 190 79 154 107 121 136
210 35 49 192 21 228 246 7 124
141 159 110 187 74 88 169 243 2
16 225 52 197 215 38 99 146 128
113 164 85 71 182 236 29 15 254
43 218 200 57

Final Decrypted S-Box

11000110, 00110111, 00100101, 11010100, 00000001,
11110000, 11100010, 00010011, 01001001, 10111000,
10101010, 01011011, 10001110, 01111111, 01101101,
10011100, 11011001, 00101000, 00111010, 11001011,
00011110, 11101111, 11111101, 00001100, 01010110,
10100111, 10110101, 01000100, 10010001, 01100000,
01110010, 10000011, 11111000, 00001001, 00011011,
11101010, 00111111, 11001110, 11011100, 00101101,
01110111, 10000110, 10010100, 01100101, 10110000,
01000001, 01010011, 10100010, 11100111, 00010110,
00000100, 11110101, 00100000, 11010001, 11000011,
00110010, 01101000, 10011001, 10001011, 01111010,
10101111, 01011110, 01001100, 10111101, 10111010,
01001011, 01011001, 10101000, 01111101, 10001100,
10011110, 01101111, 00110101, 11000100, 11010110,
00100111, 11110010, 00000011, 00010001, 11100000,
10100101, 01010100, 01000110, 10110111, 01100010,
10010011, 10000001, 01110000, 00101010, 11011011,
11001001, 00111000, 11101101, 00011100, 00001110,
11111111, 10000100, 01110101, 01100111, 10010110,
01000011, 10110010, 10100000, 01010001, 00001011,
11111010, 11101000, 00011001, 11001100, 00111101,
00101111, 11011110, 10011011, 01101010, 01111000,
10001001, 01011100, 10101101, 10111111, 01001110,
00010100, 11100101, 11110111, 00000110, 11010011,
00100010, 00110000, 11000001, 00111110, 11001111,
11011101, 00101100, 11111001, 00001000, 00011010,
11101011, 10110001, 01000000, 01010010, 10100011,
01110110, 10000111, 10010101, 01100100, 00100001,
11010000, 11000010, 00110011, 11100110, 00010111,
00000101, 11110100, 10101110, 01011111, 01001101,
10111100, 01101001, 10011000, 10001010, 01111011,
00000000, 11110001, 11100011, 00010010, 11000111,
00110110, 00100100, 11010101, 10001111, 01111110,
01101100, 10011101, 01001000, 10111001, 10101011,
01011010, 00011111, 11101110, 11111100, 00001101,
11011000, 00101001, 00111011, 11001010, 10010000,
01100001, 01110011, 10000010, 01010111, 10100110,
10110100, 01000101, 01000010, 10110011, 10100001,
01010000, 10000101, 01110100, 01100110, 10010111,
11001101, 00111100, 00101110, 11011111, 00001010,
11111011, 11101001, 00011000, 01011101, 10101100,
10111110, 01001111, 10011010, 01101011, 01111001,
10001000, 11010010, 00100011, 00110001, 11000000,
00010101, 11100100, 11110110, 00000111, 01111100,
10001101, 10011111, 01101110, 10111011, 01001010,
01011000, 10101001, 11110011, 00000010, 00010000,
11100001, 00110100, 11000101, 11010111, 00100110,
01100011, 10010010, 10000000, 01110001, 10100100,
01010101, 01000111, 10110110, 11101100, 00011101,
00001111, 11111110, 00101011, 11011010, 11001000,
00111001

Fermat encoding using backward method

Index = 0 Index = 1 Index = 2
241 - 3 = 238 18 - 3 = 15 241 - 3 = 238
238 - 5 = 233 15 - 5 = 10 238 - 5 = 233
233 - 17 = 216 10 - 17 = 249 233 - 17 = 216
216 - 1 = 215 249 - 1 = 248 216 - 1 = 215
215 - 1 = 214 248 - 1 = 247 215 - 1 = 214
214 - 1 = 213 247 - 1 = 246 214 - 1 = 213
213 - 1 = 212 246 - 1 = 245 213 - 1 = 212
212 - 1 = 211 245 - 1 = 244 212 - 1 = 211

Decryption using Fermat encoding in forward method

Index = 0 Index = 1 Index = 2
211 + 3 = 214 244 + 3 = 247 211 + 3 = 214
214 + 5 = 219 247 + 5 = 252 214 + 5 = 219
219 + 17 = 236 252 + 17 = 13 219 + 17 = 236
236 + 1 = 237 13 + 1 = 14 236 + 1 = 237
237 + 1 = 238 14 + 1 = 15 237 + 1 = 238
238 + 1 = 239 15 + 1 = 16 238 + 1 = 239
239 + 1 = 240 16 + 1 = 17 239 + 1 = 240
240 + 1 = 241 17 + 1 = 18 240 + 1 = 241

S-box Generation

11110001, 00010010, 11110001, 11010101, 11110001,
00010010, 11110001, 01011010, 11110001, 00010010,
11110001, 11010101, 11110001, 00010010, 11110001,
01000101, 11110001, 00010010, 11110001, 11010101,
11110001, 00010010, 11110001, 01011010, 11110001,
00010010, 11110001, 11010101, 11110001, 00010010,
11110001, 01111011, 11110001, 00010010, 11110001,
11010101, 11110001, 00010010, 11110001, 01011010,
11110001, 00010010, 11110001, 11010101, 11110001,
00010010, 11110001, 01000101, 11110001, 00010010,
11110001, 11010101, 11110001, 00010010, 11110001,
01011010, 11110001, 00010010, 11110001, 11010101,
11110001, 00010010, 11110001, 00000111, 11110001,
00010010, 11110001, 11010101, 11110001, 00010010,
11110001, 01011010, 11110001, 00010010, 11110001,
11010101, 11110001, 00010010, 11110001, 01000101,
11110001, 00010010, 11110001, 11010101, 11110001,
00010010, 11110001, 01011010, 11110001, 00010010,
11110001, 11010101, 11110001, 00010010, 11110001,
01111011, 11110001, 00010010, 11110001, 11010101,
11110001, 00010010, 11110001, 01011010, 11110001,
00010010, 11110001, 11010101, 11110001, 00010010,
11110001, 01000101, 11110001, 00010010, 11110001,
11010101, 11110001, 00010010, 11110001, 01011010,
11110001, 00010010, 11110001, 11010101, 11110001,
00010010, 11110001, 11111111, 11110001, 00010010,
11110001, 11010101, 11110001, 00010010, 11110001,
01011010, 11110001, 00010010, 11110001, 11010101,
11110001, 00010010, 11110001, 01000101, 11110001,
00010010, 11110001, 11010101, 11110001, 00010010,
11110001, 01011010, 11110001, 00010010, 11110001,
11010101, 11110001, 00010010, 11110001, 01111011,
11110001, 00010010, 11110001, 11010101, 11110001,
00010010, 11110001, 01011010, 11110001, 00010010,
11110001, 11010101, 11110001, 00010010, 11110001,
01000101, 11110001, 00010010, 11110001, 11010101,
11110001, 00010010, 11110001, 01011010, 11110001,
00010010, 11110001, 11010101, 11110001, 00010010,
11110001, 00000111, 11110001, 00010010, 11110001,
11010101, 11110001, 00010010, 11110001, 01011010,
11110001, 00010010, 11110001, 11010101, 11110001,
00010010, 11110001, 01000101, 11110001, 00010010,
11110001, 11010101, 11110001, 00010010, 11110001,
01011010, 11110001, 00010010, 11110001, 11010101,
11110001, 00010010, 11110001, 01111011, 11110001,
00010010, 11110001, 11010101, 11110001, 00010010,
11110001, 01011010, 11110001, 00010010, 11110001,
11010101, 11110001, 00010010, 11110001, 01000101,
11110001, 00010010, 11110001, 11010101, 11110001,
00010010, 11110001, 01011010, 11110001, 00010010,
11110001, 11010101, 11110001, 00010010, 11110001,
11001000

Encrypted S-Box

211 244 211 183 211 244 211 60 211
244 211 183 211 244 211 39 211 244
211 183 211 244 211 60 211 244 211
183 211 244 211 93 211 244 211 183
211 244 211 60 211 244 211 183 211
244 211 39 211 244 211 183 211 244
211 60 211 244 211 183 211 244 211
233 211 244 211 183 211 244 211 60
211 244 211 183 211 244 211 39 211
244 211 183 211 244 211 60 211 244
211 183 211 244 211 93 211 244 211
183 211 244 211 60 211 244 211 183
211 244 211 39 211 244 211 183 211
244 211 60 211 244 211 183 211 244
211 225 211 244 211 183 211 244 211
60 211 244 211 183 211 244 211 39
211 244 211 183 211 244 211 60 211
244 211 183 211 244 211 93 211 244
211 183 211 244 211 60 211 244 211
183 211 244 211 39 211 244 211 183
211 244 211 60 211 244 211 183 211
244 211 233 211 244 211 183 211 244
211 60 211 244 211 183 211 244 211
39 211 244 211 183 211 244 211 60
211 244 211 183 211 244 211 93 211
244 211 183 211 244 211 60 211 244
211 183 211 244 211 39 211 244 211
183 211 244 211 60 211 244 211 183
211 244 211 170
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