View source for CCZ-invariants for all known APN functions in dimension 7

We consider the [[known instances of APN functions over GF(2^7)]]. All of these 490 functions, with the exception of the inverse function <math>x \mapsto x^{126}</math>, have the Gold-like Walsh spectrum <math>-16^{3556}, 0^{8128}, 16^{4572}</math>. The inverse function has the Walsh spectrum <math>-20^{889}, -16^{889}, -12^{1016}, -8^{2667}, -4^{889}, 0^{1905}, 4^{2667}, 8^{889}, 12^{1778}, 16^{1778}, 20^{889}</math>.

All of these 490 functions have one of the 14 values 3610, 3708, 4026, 4034, 4038, 4040, 4042, 4044, 4046, 4048, 4050, 4270, 4704, 8128 as their <math>\Gamma</math>-rank. The following table lists the indices of the functions having each of the given <math>\Gamma</math>-rank.

<table>
<tr>
<th>Γ-rank</th>
<th>Indices</th>
</tr>

<tr>
<td>3610</td>
<td>1,2</td>
</tr>

<tr>
<td>3708</td>
<td>3</td>
</tr>

<tr>
<td>4026</td>
<td>7</td>
</tr>

<tr>
<td>4034</td>
<td>8</td>
</tr>

<tr>
<td>4038</td>
<td>9</td>
</tr>

<tr>
<td>4040</td>
<td>10,11,12</td>
</tr>

<tr>
<td>4042</td>
<td>113, 119, 340</td>
</tr>

<tr>
<td>4044</td>
<td>13, 27, 36, 37, 41, 52, 54, 57, 91, 120, 146, 163, 194, 204, 212, 248, 277, 291, 300, 307, 318, 323, 337, 367, 392, 399, 401, 417, 421, 423, 432, 436, 488</td>
</tr>

<tr>
<td>4046</td>
<td>19, 21, 22, 24, 25, 29, 32, 34, 38, 40, 43, 47, 48, 49, 50, 51, 53, 55, 56, 60, 61, 63, 64, 65, 66, 70, 71, 74, 75, 76, 78, 80, 85, 90, 100, 102, 104, 105, 107, 109, 110, 111, 122, 126, 129, 131, 133, 136, 142, 148, 151, 153, 160, 161, 169, 170, 173, 174, 180, 182, 183, 187, 188, 191, 193, 197, 200, 202, 211, 213, 221, 231, 232, 233, 235, 236, 239, 241, 242, 243, 244, 247, 252, 256, 257, 258, 259, 262, 263, 264,268, 279, 283, 287, 290, 292, 298, 301, 302, 303, 304, 309, 312, 314, 315, 319, 326, 328, 333, 336, 341, 343, 344, 345, 347, 349, 351, 353, 357, 358, 359, 360, 363, 365, 366, 368, 370, 371, 373, 374, 375, 376, 379, 381, 384, 385, 389, 395, 396, 400, 404, 405, 406, 409, 411, 413, 419, 426, 429, 431, 435, 438, 441, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 460, 465, 472, 475, 476, 478, 479, 485, 487, 490</td>
</tr>

<tr>
<td>4048</td>
<td>14, 15, 16, 17, 20, 23, 26, 28, 31, 33, 39, 42, 44, 45, 46, 58, 59, 62, 67, 68, 69, 72, 73, 77, 79, 82, 83, 84, 86, 87, 88, 92, 93, 95, 96, 97, 99, 103, 106, 108, 112, 114, 115, 117, 118, 121, 123, 124, 125, 127, 128, 130, 132, 134, 135, 137, 138, 139, 140, 141, 143, 144, 145, 147, 149, 150, 152, 155, 156, 157, 158, 159, 162, 164, 166, 167, 168, 171, 175, 176, 177, 178, 179, 184, 186, 190, 192, 195, 196, 199, 201, 203, 205, 206, 207, 208, 210, 214, 215, 216, 217, 218, 219, 220, 222, 223, 224, 225, 226, 227, 228, 229, 234, 238, 240, 245, 249, 251, 255, 260, 261, 265, 266, 270, 271, 273, 275, 276, 278, 280, 281, 282, 285, 286, 288, 289, 293, 294, 295, 296, 297, 299, 310, 311, 313, 316, 317, 320, 321, 322, 324, 327, 329, 330, 332, 334, 335, 338, 339, 342, 346, 350, 354, 356, 361, 369, 372, 377, 382, 387, 390, 391, 393, 394, 398, 402, 407, 408, 412, 416, 418, 420, 424, 425, 427, 428, 434, 437, 440, 443, 444, 455, 456, 457, 458, 461, 462, 463, 464, 466, 468, 470, 471, 473, 477, 481, 482, 489</td>
</tr>

<tr>
<td>4050</td>
<td>18, 30, 35, 81, 89, 94, 98, 101, 116, 154, 165, 172, 181, 185, 189, 198, 209, 230, 237, 246, 250, 253, 254, 267, 269, 272, 274, 284, 305, 306, 308, 325, 331, 348, 352, 355, 362, 364, 378, 380, 383, 386, 388, 397, 403, 410, 414, 415, 422, 430, 433, 439, 442, 459, 467, 469, 474, 480, 483, 484, 486</td>
</tr>

<tr>
<td>4270</td>
<td>4</td>
</tr>

<tr>
<td>4704</td>
<td>5</td>
</tr>

<tr>
<td>8128</td>
<td>6</td>
</tr>

</table>

The 490 functions have six distinct Δ-ranks, viz. 198, 210, 212, 338, 436, 4928. The indices of the functions corresponding to each rank are given in the table below.

<table>
<tr>
<th>Δ-rank</th>
<th>Indices</th>
</tr>

<tr>
<td>198</td>
<td>1,2,3</td>
</tr>

<tr>
<td>210</td>
<td>8, 10, 14, 15, 16, 18, 45, 260, 470</td>
</tr>

<tr>
<td>212</td>
<td>7, 9, 11, 12, 13, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490</td>
</tr>

<tr>
<td>338</td>
<td>4</td>
</tr>

<tr>
<td>436</td>
<td>5</td>
</tr>

<tr>
<td>4928</td>
<td>6</td>
</tr>

</table>

The 490 functions have five distinct values of the order of the multiplier group , viz. 128, 889, 896, 1778, 113792. The indices of the functions corresponding to each order are given in the table below.

<table>
<tr>
<th>Multiplier group order</th>
<th>Indices</th>
</tr>

<tr>
<td>128</td>
<td>19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103,  104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490
</td>
</tr>

<tr>
<td>889</td>
<td>4, 5</td>
</tr>

<tr>
<td>896</td>
<td>7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 </td>
</tr>

<tr>
<td>1778</td>
<td>6</td>
</tr>

<tr>
<td>113792</td>
<td>1,2,3</td>
</tr>