One can download
here the computer results described in the
paper “On the diffusion of the Improved
Generalized Feistel”, to appear in Advances in Mathematics of
Communications.
The paper considers the
Improved Generalized Feistel Structure (IGFS) suggested by Suzaki and Minematsu
(LNCS, 2010). It is a generalization of the classical Feistel cipher. The message
is divided into k subblocks, a Feistel transformation is applied to each pair
of successive subblocks, and then a permutation of the subblocks follows. This
permutation affects the diffusion property of the cypher. IGFS with relatively
big k
and good diffusion are of particular interest for light weight applications. Suzaki
and Minematsu (LNCS, 2010) study the case when one and the same permutation is
applied at each round, while we consider IGFS with possibly different
permutations at the different rounds. In this case we present permutation
sequences yielding IGFS with the best known by now diffusion for all even k ≤
2048. For k ≤ 16 they are
found by a computer-aided search, while for 18 ≤ k ≤ 2048
we first consider several recursive constructions of a permutation sequence for
k
subblocks from two permutation sequences for ka< k and kb<
k subblocks respectively. Using computer, we apply these constructions
to obtain permutation sequences with good diffusion for each even k ≤
2048. Finally we obtain infinite families of permutation sequences for k>2048.
The results for
different k are in different files. The file DiffeeK_R_B.txt contains the results for k=K. Here R is the diffusion round of IGFS with the permutation sequence
presented in the file, and B is the
lower bound for the diffusion round of IGFS with k subblocks for even-odd
permutations. Our results attain this bound for k = 2, 4, 6, 8, 10, 12, 14, 16,
18, 20, 28, 30, 32, 48, 50, 70, 80, 112, 128, and are quite close to it for the
other values of k ≤ 2048. For each even k ≤ 2048 we present
here information for the results in the file DiffeeK_R_B.txt (explanations are given in the file too). Consider,
for instance, the row for k=22 below: 9 – 8 means that the diffusion round of IGFS
with the permutation sequence presented in Diffee22_9_8.txt
is 9, while 8 is the lower bound, and the permutation sequence is obtained
by construction 5 using the sequences for k=10 and k=12 subblocks.
k R B Construction Remark
2 2
- 2 computer computer
4 4
- 4 computer computer
6 5
- 5 computer computer
8 6
- 6 computer computer
10 6
- 6 computer computer
12 7
- 7 computer computer
14 7
- 7 computer computer
16 7
- 7 computer computer
18 8
- 8 2 2.3.3
20 8
- 8 1 2.10
22 9
- 8 5 10+12
24 9
- 8 1 2.12
26 10
- 8 3 12+14
28 9
- 9 1 2.14
30 9
- 9 2 2.3.5
32 9
- 9 1 2.16
34 10
- 9 4 16+18
36 10
- 9 1 2.18
38 11
- 9 3 18+20
40 10
- 9 1 2.20
42 10
- 9 2 2.3.7
44 11
- 10 1 2.22
46 12
- 10 3 22+24
48 10
- 10 2 2.3.8
50 10
- 10 2 2.5.5
52 12
- 10 1 2.26
54 11
- 10 2 2.3.9
56 11
- 10 1 2.28
58 12
- 10 3 28+30
60 11
- 10 1 2.30
62 12
- 10 3 30+32
64 11
- 10 1 2.32
66 12
- 10 2 2.3.11
68 12
- 10 1 2.34
70 11
- 11 2 2.5.7
72 12
- 11 1 2.36
74 13
- 11 4 36+38
76 13
- 11 1 2.38
78 13
- 11 2 2.3.13
80 11
- 11 2 2.5.8
82 13
- 11 3 40+42
84 12
- 11 1 2.42
86 13
- 11 5 42+44
88 13
- 11 1 2.44
90 12
- 11 2 2.3.15
92 14
- 11 1 2.46
94 14
- 11 6 46+48
96 12
- 11 1 2.48
98 12
- 11 2 2.7.7
100 12
- 11 1 2.50
102 13
- 11 2 2.3.17
104 14
- 11 1 2.52
106 15
- 11 3 52+54
108 13
- 11 1 2.54
110 13
- 11 2 2.5.11
112 12
- 12 2 2.7.8
114 14
- 12 2 2.3.19
116 14
- 12 1 2.58
118 14
- 12 6 58+60
120 13
- 12 1 2.60
122 14
- 12 4 60+62
124 14
- 12 1 2.62
126 13
- 12 2 2.3.21
128 12
- 12 2 2.8.8
130 14
- 12 2 2.5.13
132 14
- 12 1 2.66
134 15
- 12 3 66+68
136 14
- 12 1 2.68
138 15
- 12 3 68+70
140 13
- 12 1 2.70
142 14
- 12 5 70+72
144 13
- 12 2 2.3.24
146 15
- 12 4 72+74
148 15
- 12 1 2.74
150 13
- 12 2 2.3.25
152 15
- 12 1 2.76
154 14
- 12 2 2.7.11
156 15
- 12 1 2.78
158 15
- 12 6 78+80
160 13
- 12 1 2.80
162 14
- 12 2 2.3.27
164 15
- 12 1 2.82
166 15
- 12 6 82+84
168 14
- 12 1 2.84
170 14
- 12 2 2.5.17
172 15
- 12 1 2.86
174 15
- 12 2 2.3.29
176 14
- 12 2 2.8.11
178 16
- 12 3 88+90
180 14
- 13 1 2.90
182 15
- 13 2 2.7.13
184 16
- 13 1 2.92
186 15
- 13 2 2.3.31
188 16
- 13 1 2.94
190 15
- 13 2 2.5.19
192 14
- 13 1 2.96
194 15
- 13 3 96+98
196 14
- 13 1 2.98
198 15
- 13 3 98+100
200 14
- 13 1 2.100
202 15
- 13 4 100+102
204 15
- 13 1 2.102
206 16
- 13 5 102+104
208 15
- 13 2 2.8.13
210 14
- 13 2 2.3.35
212 17
- 13 1 2.106
214 17
- 13 6 106+108
216 15
- 13 1 2.108
218 16
- 13 3 108+110
220 15
- 13 1 2.110
222 16
- 13 3 110+112
224 14
- 13 1 2.112
226 16
- 13 4 112+114
228 16
- 13 1 2.114
230 16
- 13 2 2.5.23
232 16
- 13 1 2.116
234 16
- 13 2 2.3.39
236 16
- 13 1 2.118
238 15
- 13 2 2.7.17
240 14
- 13 2 2.3.40
242 16
- 13 2 2.11.11
244 16
- 13 1 2.122
246 16
- 13 2 2.3.41
248 16
- 13 1 2.124
250 14
- 13 2 2.5.25
252 15
- 13 1 2.126
254 16
- 13 3 126+128
256 14
- 13 1 2.128
258 16
- 13 2 2.3.43
260 16
- 13 1 2.130
262 16
- 13 7 130+132
264 16
- 13 1 2.132
266 16
- 13 2 2.7.19
268 17
- 13 1 2.134
270 15
- 13 2 2.3.45
272 15
- 13 2 2.8.17
274 17
- 13 4 136+138
276 17
- 13 1 2.138
278 17
- 13 6 138+140
280 15
- 13 1 2.140
282 16
- 13 7 140+142
284 16
- 13 1 2.142
286 17
- 13 3 142+144
288 15
- 13 1 2.144
290 16
- 14 2 2.5.29
292 17
- 14 1 2.146
294 15
- 14 2 2.3.49
296 17
- 14 1 2.148
298 18
- 14 3 148+150
300 15
- 14 1 2.150
302 17
- 14 5 150+152
304 16
- 14 2 2.8.19
306 16
- 14 2 2.3.51
308 16
- 14 1 2.154
310 16
- 14 2 2.5.31
312 17
- 14 1 2.156
314 18
- 14 3 156+158
316 17
- 14 1 2.158
318 17
- 14 7 158+160
320 15
- 14 1 2.160
322 16
- 14 4 160+162
324 16
- 14 1 2.162
326 17
- 14 5 162+164
328 17
- 14 1 2.164
330 16
- 14 2 2.3.55
332 17
- 14 1 2.166
334 17
- 14 7 166+168
336 15
- 14 2 2.3.56
338 17
- 14 3 168+170
340 16
- 14 1 2.170
342 17
- 14 2 2.3.57
344 17
- 14 1 2.172
346 18
- 14 3 172+174
348 17
- 14 1 2.174
350 15
- 14 2 2.5.35
352 16
- 14 1 2.176
354 17
- 14 2 2.3.59
356 18
- 14 1 2.178
358 18
- 14 6 178+180
360 16
- 14 1 2.180
362 17
- 14 4 180+182
364 17
- 14 1 2.182
366 17
- 14 2 2.3.61
368 17
- 14 2 2.8.23
370 17
- 14 2 2.5.37
372 17
- 14 1 2.186
374 17
- 14 2 2.11.17
376 18
- 14 1 2.188
378 16
- 14 2 2.3.63
380 17
- 14 1 2.190
382 17
- 14 6 190+192
384 15
- 14 2 2.3.64
386 17
- 14 4 192+194
388 17
- 14 1 2.194
390 17
- 14 2 2.3.65
392 16
- 14 1 2.196
394 17
- 14 4 196+198
396 17
- 14 1 2.198
398 17
- 14 6 198+200
400 15
- 14 2 2.5.40
402 17
- 14 4 200+202
404 17
- 14 1 2.202
406 17
- 14 2 2.7.29
408 17
- 14 1 2.204
410 17
- 14 2 2.5.41
412 18
- 14 1 2.206
414 18
- 14 2 2.3.69
416 17
- 14 1 2.208
418 17
- 14 6 208+210
420 16
- 14 1 2.210
422 19
- 14 5 210+212
424 19
- 14 1 2.212
426 17
- 14 2 2.3.71
428 19
- 14 1 2.214
430 17
- 14 2 2.5.43
432 16
- 14 2 2.3.72
434 17
- 14 2 2.7.31
436 18
- 14 1 2.218
438 18
- 14 2 2.3.73
440 17
- 14 1 2.220
442 18
- 14 2 2.13.17
444 18
- 14 1 2.222
446 18
- 14 6 222+224
448 16
- 14 1 2.224
450 16
- 14 2 2.3.75
452 18
- 14 1 2.226
454 18
- 14 7 226+228
456 18
- 14 1 2.228
458 19
- 14 3 228+230
460 18
- 14 1 2.230
462 17
- 14 2 2.3.77
464 17
- 14 2 2.8.29
466 19
- 14 3 232+234
468 18
- 15 1 2.234
470 18
- 15 2 2.5.47
472 18
- 15 1 2.236
474 18
- 15 2 2.3.79
476 17
- 15 1 2.238
478 18
- 15 3 238+240
480 16
- 15 1 2.240
482 18
- 15 7 240+242
484 18
- 15 1 2.242
486 17
- 15 2 2.3.81
488 18
- 15 1 2.244
490 16
- 15 2 2.5.49
492 18
- 15 1 2.246
494 18
- 15 7 246+248
496 17
- 15 2 2.8.31
498 18
- 15 2 2.3.83
500 16
- 15 1 2.250
502 17
- 15 5 250+252
504 17
- 15 1 2.252
506 18
- 15 4 252+254
508 18
- 15 1 2.254
510 17
- 15 2 2.3.85
512 16
- 15 1 2.256
514 18
- 15 7 256+258
516 18
- 15 1 2.258
518 18
- 15 2 2.7.37
520 18
- 15 1 2.260
522 18
- 15 2 2.3.87
524 18
- 15 1 2.262
526 19
- 15 3 262+264
528 17
- 15 2 2.3.88
530 19
- 15 3 264+266
532 18
- 15 1 2.266
534 19
- 15 2 2.3.89
536 19
- 15 1 2.268
538 20
- 15 3 268+270
540 17
- 15 1 2.270
542 18
- 15 3 270+272
544 17
- 15 1 2.272
546 18
- 15 2 2.3.91
548 19
- 15 1 2.274
550 17
- 15 2 2.5.55
552 19
- 15 1 2.276
554 20
- 15 3 276+278
556 19
- 15 1 2.278
558 18
- 15 2 2.3.93
560 16
- 15 2 2.5.56
562 18
- 15 7 280+282
564 18
- 15 1 2.282
566 18
- 15 7 282+284
568 18
- 15 1 2.284
570 18
- 15 2 2.3.95
572 19
- 15 1 2.286
574 18
- 15 2 2.7.41
576 17
- 15 1 2.288
578 18
- 15 2 2.17.17
580 18
- 15 1 2.290
582 18
- 15 2 2.3.97
584 19
- 15 1 2.292
586 20
- 15 3 292+294
588 17
- 15 1 2.294
590 18
- 15 2 2.5.59
592 18
- 15 2 2.8.37
594 18
- 15 2 2.3.99
596 20
- 15 1 2.298
598 20
- 15 2 2.13.23
600 17
- 15 1 2.300
602 18
- 15 2 2.7.43
604 19
- 15 1 2.302
606 18
- 15 2 2.3.101
608 18
- 15 1 2.304
610 18
- 15 2 2.5.61
612 18
- 15 1 2.306
614 19
- 15 3 306+308
616 18
- 15 1 2.308
618 19
- 15 3 308+310
620 18
- 15 1 2.310
622 19
- 15 5 310+312
624 18
- 15 2 2.3.104
626 20
- 15 4 312+314
628 20
- 15 1 2.314
630 17
- 15 2 2.3.105
632 19
- 15 1 2.316
634 20
- 15 3 316+318
636 19
- 15 1 2.318
638 19
- 15 2 2.11.29
640 16
- 15 2 2.5.64
642 18
- 15 4 320+322
644 18
- 15 1 2.322
646 19
- 15 3 322+324
648 18
- 15 1 2.324
650 18
- 15 2 2.5.65
652 19
- 15 1 2.326
654 19
- 15 2 2.3.109
656 18
- 15 2 2.8.41
658 19
- 15 2 2.7.47
660 18
- 15 1 2.330
662 19
- 15 5 330+332
664 19
- 15 1 2.332
666 19
- 15 2 2.3.111
668 19
- 15 1 2.334
670 19
- 15 2 2.5.67
672 17
- 15 1 2.336
674 19
- 15 4 336+338
676 19
- 15 1 2.338
678 19
- 15 2 2.3.113
680 18
- 15 1 2.340
682 19
- 15 2 2.11.31
684 19
- 15 1 2.342
686 17
- 15 2 2.7.49
688 18
- 15 2 2.8.43
690 19
- 15 2 2.3.115
692 20
- 15 1 2.346
694 20
- 15 6 346+348
696 19
- 15 1 2.348
698 20
- 15 3 348+350
700 17
- 15 1 2.350
702 18
- 15 5 350+352
704 18
- 15 1 2.352
706 19
- 15 7 352+354
708 19
- 15 1 2.354
710 18
- 15 2 2.5.71
712 20
- 15 1 2.356
714 18
- 15 2 2.3.119
716 20
- 15 1 2.358
718 20
- 15 7 358+360
720 17
- 15 2 2.3.120
722 19
- 15 4 360+362
724 19
- 15 1 2.362
726 19
- 15 2 2.3.121
728 19
- 15 1 2.364
730 19
- 15 2 2.5.73
732 19
- 15 1 2.366
734 19
- 15 7 366+368
736 19
- 15 1 2.368
738 19
- 15 2 2.3.123
740 19
- 15 1 2.370
742 19
- 15 7 370+372
744 19
- 15 1 2.372
746 20
- 15 3 372+374
748 19
- 15 1 2.374
750 17
- 15 2 2.3.125
752 19
- 15 2 2.8.47
754 20
- 15 2 2.13.29
756 18
- 16 1 2.378
758 19
- 16 5 378+380
760 19
- 16 1 2.380
762 19
- 16 2 2.3.127
764 19
- 16 1 2.382
766 19
- 16 7 382+384
768 17
- 16 1 2.384
770 18
- 16 2 2.5.77
772 19
- 16 1 2.386
774 19
- 16 2 2.3.129
776 19
- 16 1 2.388
778 20
- 16 3 388+390
780 19
- 16 1 2.390
782 19
- 16 6 390+392
784 17
- 16 2 2.7.56
786 19
- 16 2 2.3.131
788 19
- 16 1 2.394
790 19
- 16 2 2.5.79
792 19
- 16 1 2.396
794 20
- 16 3 396+398
796 19
- 16 1 2.398
798 19
- 16 2 2.3.133
800 17
- 16 1 2.400
802 19
- 16 4 400+402
804 19
- 16 1 2.402
806 20
- 16 3 402+404
808 19
- 16 1 2.404
810 18
- 16 2 2.3.135
812 19
- 16 1 2.406
814 19
- 16 7 406+408
816 18
- 16 2 2.3.136
818 20
- 16 3 408+410
820 19
- 16 1 2.410
822 20
- 16 2 2.3.137
824 20
- 16 1 2.412
826 19
- 16 2 2.7.59
828 20
- 16 1 2.414
830 19
- 16 2 2.5.83
832 19
- 16 1 2.416
834 20
- 16 3 416+418
836 19
- 16 1 2.418
838 19
- 16 7 418+420
840 18
- 16 1 2.420
842 21
- 16 7 420+422
844 21
- 16 1 2.422
846 19
- 16 2 2.3.141
848 20
- 16 2 2.8.53
850 18
- 16 2 2.5.85
852 19
- 16 1 2.426
854 19
- 16 2 2.7.61
856 21
- 16 1 2.428
858 20
- 16 2 2.3.143
860 19
- 16 1 2.430
862 20
- 16 3 430+432
864 18
- 16 1 2.432
866 19
- 16 4 432+434
868 19
- 16 1 2.434
870 19
- 16 2 2.3.145
872 20
- 16 1 2.436
874 21
- 16 3 436+438
876 20
- 16 1 2.438
878 20
- 16 6 438+440
880 18
- 16 2 2.5.88
882 18
- 16 2 2.3.147
884 20
- 16 1 2.442
886 21
- 16 3 442+444
888 20
- 16 1 2.444
890 20
- 16 2 2.5.89
892 20
- 16 1 2.446
894 20
- 16 7 446+448
896 17
- 16 2 2.7.64
898 19
- 16 3 448+450
900 18
- 16 1 2.450
902 20
- 16 2 2.11.41
904 20
- 16 1 2.452
906 20
- 16 2 2.3.151
908 20
- 16 1 2.454
910 19
- 16 2 2.5.91
912 19
- 16 2 2.3.152
914 21
- 16 4 456+458
916 21
- 16 1 2.458
918 19
- 16 2 2.3.153
920 20
- 16 1 2.460
922 21
- 16 3 460+462
924 19
- 16 1 2.462
926 20
- 16 3 462+464
928 19
- 16 1 2.464
930 19
- 16 2 2.3.155
932 21
- 16 1 2.466
934 21
- 16 6 466+468
936 20
- 16 1 2.468
938 20
- 16 2 2.7.67
940 20
- 16 1 2.470
942 20
- 16 7 470+472
944 19
- 16 2 2.8.59
946 20
- 16 2 2.11.43
948 20
- 16 1 2.474
950 19
- 16 2 2.5.95
952 19
- 16 1 2.476
954 20
- 16 2 2.3.159
956 20
- 16 1 2.478
958 20
- 16 6 478+480
960 18
- 16 1 2.480
962 20
- 16 7 480+482
964 20
- 16 1 2.482
966 19
- 16 2 2.3.161
968 20
- 16 1 2.484
970 19
- 16 2 2.5.97
972 19
- 16 1 2.486
974 20
- 16 5 486+488
976 19
- 16 2 2.8.61
978 20
- 16 2 2.3.163
980 18
- 16 1 2.490
982 20
- 16 5 490+492
984 20
- 16 1 2.492
986 20
- 16 2 2.17.29
988 20
- 16 1 2.494
990 19
- 16 2 2.3.165
992 19
- 16 1 2.496
994 19
- 16 2 2.7.71
996 20
- 16 1 2.498
998 20
- 16 7 498+500
1000 18
- 16 1 2.500
1002 19
- 16 7 500+502
1004 19
- 16 1 2.502
1006 20
- 16 3 502+504
1008 18
- 16 2 2.3.168
1010 19
- 16 2 2.5.101
1012 20
- 16 1 2.506
1014 20
- 16 2 2.3.169
1016 20
- 16 1 2.508
1018 21
- 16 3 508+510
1020 19
- 16 1 2.510
1022 20
- 16 3 510+512
1024 17
- 16 2 2.8.64
1026 20
- 16 2 2.3.171
1028 20
- 16 1 2.514
1030 20
- 16 2 2.5.103
1032 20
- 16 1 2.516
1034 21
- 16 3 516+518
1036 20
- 16 1 2.518
1038 20
- 16 7 518+520
1040 19
- 16 2 2.5.104
1042 21
- 16 3 520+522
1044 20
- 16 1 2.522
1046 20
- 16 7 522+524
1048 20
- 16 1 2.524
1050 18
- 16 2 2.3.175
1052 21
- 16 1 2.526
1054 20
- 16 2 2.17.31
1056 19
- 16 1 2.528
1058 21
- 16 4 528+530
1060 21
- 16 1 2.530
1062 20
- 16 2 2.3.177
1064 20
- 16 1 2.532
1066 21
- 16 2 2.13.41
1068 21
- 16 1 2.534
1070 21
- 16 2 2.5.107
1072 20
- 16 2 2.8.67
1074 21
- 16 2 2.3.179
1076 22
- 16 1 2.538
1078 19
- 16 2 2.7.77
1080 19
- 16 1 2.540
1082 20
- 16 4 540+542
1084 20
- 16 1 2.542
1086 20
- 16 2 2.3.181
1088 19
- 16 1 2.544
1090 20
- 16 2 2.5.109
1092 20
- 16 1 2.546
1094 21
- 16 5 546+548
1096 21
- 16 1 2.548
1098 20
- 16 2 2.3.183
1100 19
- 16 1 2.550
1102 21
- 16 2 2.19.29
1104 20
- 16 2 2.3.184
1106 20
- 16 2 2.7.79
1108 22
- 16 1 2.554
1110 20
- 16 2 2.3.185
1112 21
- 16 1 2.556
1114 22
- 16 3 556+558
1116 20
- 16 1 2.558
1118 20
- 16 6 558+560
1120 18
- 16 1 2.560
1122 20
- 16 2 2.3.187
1124 20
- 16 1 2.562
1126 21
- 16 3 562+564
1128 20
- 16 1 2.564
1130 20
- 16 2 2.5.113
1132 20
- 16 1 2.566
1134 19
- 16 2 2.3.189
1136 19
- 16 2 2.8.71
1138 21
- 16 3 568+570
1140 20
- 16 1 2.570
1142 21
- 16 5 570+572
1144 21
- 16 1 2.572
1146 20
- 16 2 2.3.191
1148 20
- 16 1 2.574
1150 20
- 16 2 2.5.115
1152 18
- 16 2 2.3.192
1154 20
- 16 4 576+578
1156 20
- 16 1 2.578
1158 20
- 16 2 2.3.193
1160 20
- 16 1 2.580
1162 20
- 16 2 2.7.83
1164 20
- 16 1 2.582
1166 21
- 16 5 582+584
1168 20
- 16 2 2.8.73
1170 20
- 16 2 2.3.195
1172 22
- 16 1 2.586
1174 22
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