Difference between revisions of "CCZ-invariants for all known APN functions in dimension 7"

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We consider the 490 APN functions in dimension 7 constructed by the matrix method <ref name="yu">Yu, Yuyin, Mingsheng Wang, and Yongqiang Li. "A Matrix Approach for Constructing Quadratic APN Functions."</ref> We enumerate the functions in the order in which they appear in <ref name="yu" />.
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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, 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.
 
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.

Revision as of 20:36, 9 July 2020

We consider the known instances of APN functions over GF(2^7). All of these 490 functions, with the exception of the inverse function , have the Gold-like Walsh spectrum . The inverse function has the Walsh spectrum .

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 -rank. The following table lists the indices of the functions having each of the given -rank.

Γ-rank Indices
3610 1,2
3708 3
4026 9
4034 8
4038 9
4040 10,11,12
4042 113, 119, 340
4044 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
4046 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
4048 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
4050 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
4270 4
4704 5
8128 6

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.

Δ-rank Indices
198 1,2,3
210 8, 10, 14, 15, 16, 18, 45, 260, 470
212 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
338 4
436 5
4928 6