cp's OEIS Frontend

A354500 The Rijndael S-box used in the Advanced Encryption Standard (AES).

%I A354500 #17 Aug 15 2022 23:32:43
%S A354500 99,124,119,123,242,107,111,197,48,1,103,43,254,215,171,118,202,130,
%T A354500 201,125,250,89,71,240,173,212,162,175,156,164,114,192,183,253,147,38,
%U A354500 54,63,247,204,52,165,229,241,113,216,49,21,4,199,35,195,24,150,5,154,7,18,128,226
%N A354500 The Rijndael S-box used in the Advanced Encryption Standard (AES).
%C A354500 A permutation of {0, 1, ..., 255}.
%C A354500 The permutation can be decomposed into 5 cycles: 4, 242, 137, 167, ..., 48 (length 87); 1, 124, 16, 202, ..., 9 (length 81); 0, 99, 251, 15, ..., 82 (length 59); 11, 43, 241, 161, ..., 158 (length 27); 115, 143 (length 2).
%H A354500 Jianing Song, <a href="/A354500/b354500.txt">Table of n, a(n) for n = 0..255</a>
%H A354500 Institute of Telematics, <a href="http://www.tm.uka.de/software/pktanon/documentation/sourcedoc/html/rijndael_8cpp.html">Rijndael.cpp File Reference</a> (contains the table of the Rijndael S-box written in decimal)
%H A354500 Wikipedia, <a href="https://en.wikipedia.org/wiki/Rijndael_S-box">Rijndael S-box</a>
%F A354500 a(n) = ivgenpoly(((x^4+x^3+x^2+x+1)*(genpoly(n)^254 mod (x^8+x^4+x^3+x+1)) + x^6 + x^5 + x + 1) mod (x^8+1)), where ivgenpoly and genpoly are the notations introduced in A355891. Beware that all the operations are done in GF(2)[x].
%F A354500 To be more concretely, to obtain a(n):
%F A354500 - Write the binary expansion of n and view it as a polynomial p(x) in GF(2)[x]; (E.g., 10 = 1010_2 => x^3 + x)
%F A354500 - Compute q(x) = p(x)^254 mod (x^8+x^4+x^3+x+1) in GF(2)[x]; (E.g., x^3 + x => x^5 + x^3 + 1)
%F A354500 - Compute r(x) = ((x^4+x^3+x^2+x+1)*q(x) + x^6 + x^5 + x + 1) mod (x^8+1) in GF(2)[x]; (E.g., x^5 + x^3 + 1 => x^6 + x^5 + x^2 + x + 1)
%F A354500 - To get a(n), view r(x) as a number. (E.g., x^6 + x^5 + x^2 + x + 1 => 2^6 + 2^5 + 2^2 + 2 + 1 = 103)
%e A354500 The Rijndael S-box written in hexadecimal:
%e A354500      +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +A +B +C +D +E +F
%e A354500   00 63 7C 77 7B F2 6B 6F C5 30 01 67 2B FE D7 AB 76
%e A354500   10 CA 82 C9 7D FA 59 47 F0 AD D4 A2 AF 9C A4 72 C0
%e A354500   20 B7 FD 93 26 36 3F F7 CC 34 A5 E5 F1 71 D8 31 15
%e A354500   30 04 C7 23 C3 18 96 05 9A 07 12 80 E2 EB 27 B2 75
%e A354500   40 09 83 2C 1A 1B 6E 5A A0 52 3B D6 B3 29 E3 2F 84
%e A354500   50 53 D1 00 ED 20 FC B1 5B 6A CB BE 39 4A 4C 58 CF
%e A354500   60 D0 EF AA FB 43 4D 33 85 45 F9 02 7F 50 3C 9F A8
%e A354500   70 51 A3 40 8F 92 9D 38 F5 BC B6 DA 21 10 FF F3 D2
%e A354500   80 CD 0C 13 EC 5F 97 44 17 C4 A7 7E 3D 64 5D 19 73
%e A354500   90 60 81 4F DC 22 2A 90 88 46 EE B8 14 DE 5E 0B DB
%e A354500   A0 E0 32 3A 0A 49 06 24 5C C2 D3 AC 62 91 95 E4 79
%e A354500   B0 E7 C8 37 6D 8D D5 4E A9 6C 56 F4 EA 65 7A AE 08
%e A354500   C0 BA 78 25 2E 1C A6 B4 C6 E8 DD 74 1F 4B BD 8B 8A
%e A354500   D0 70 3E B5 66 48 03 F6 0E 61 35 57 B9 86 C1 1D 9E
%e A354500   E0 E1 F8 98 11 69 D9 8E 94 9B 1E 87 E9 CE 55 28 DF
%e A354500   F0 8C A1 89 0D BF E6 42 68 41 99 2D 0F B0 54 BB 16
%e A354500 The Rijndael S-box written in decimal:
%e A354500        +0  +1  +2  +3  +4  +5  +6  +7  +8  +9 +10 +11 +12 +13 +14 +15
%e A354500     0  99 124 119 123 242 107 111 197  48   1 103  43 254 215 171 118
%e A354500    16 202 130 201 125 250  89  71 240 173 212 162 175 156 164 114 192
%e A354500    32 183 253 147  38  54  63 247 204  52 165 229 241 113 216  49  21
%e A354500    48   4 199  35 195  24 150   5 154   7  18 128 226 235  39 178 117
%e A354500    64   9 131  44  26  27 110  90 160  82  59 214 179  41 227  47 132
%e A354500    80  83 209   0 237  32 252 177  91 106 203 190  57  74  76  88 207
%e A354500    96 208 239 170 251  67  77  51 133  69 249   2 127  80  60 159 168
%e A354500   112  81 163  64 143 146 157  56 245 188 182 218  33  16 255 243 210
%e A354500   128 205  12  19 236  95 151  68  23 196 167 126  61 100  93  25 115
%e A354500   144  96 129  79 220  34  42 144 136  70 238 184  20 222  94  11 219
%e A354500   160 224  50  58  10  73   6  36  92 194 211 172  98 145 149 228 121
%e A354500   176 231 200  55 109 141 213  78 169 108  86 244 234 101 122 174   8
%e A354500   192 186 120  37  46  28 166 180 198 232 221 116  31  75 189 139 138
%e A354500   208 112  62 181 102  72   3 246  14  97  53  87 185 134 193  29 158
%e A354500   224 225 248 152  17 105 217 142 148 155  30 135 233 206  85  40 223
%e A354500   240 140 161 137  13 191 230  66 104  65 153  45  15 176  84 187  22
%o A354500 (PARI) m(P) = Mod(P, 2);
%o A354500 A354500(n) = subst(lift(lift(Mod(m(x^4+x^3+x^2+x+1)*lift(Mod(m(Pol(binary(n))), m(x^8+x^4+x^3+x+1))^254)+m(x^6+x^5+x+1), m(x^8+1)))), x, 2)
%Y A354500 Cf. A354501 (the inverse permutation), A355891.
%K A354500 nonn,easy,fini,full
%O A354500 0,1
%A A354500 _Jianing Song_, Aug 15 2022