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A New Construction of odd-variable Rotation symmetric Boolean functions with good cryptographic properties
On the diffusion of the Improved Generalized Feistel
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, P.O.Box 323, 5000 Veliko Tarnovo, Bulgaria |
We consider 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 cipher. 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\le 2048 $. For $ k\le 16 $ they are found by a computer-aided search, while for $ 18\le k\le 2048 $ we first consider several recursive constructions of a permutation sequence for $ k $ subblocks from two permutation sequences for $ k_a< k $ and $ k_b< k $ subblocks respectively. Using computer, we apply these constructions to obtain permutation sequences with good diffusion for each even $ k\le 2048 $. Finally we obtain infinite families of permutation sequences for $ k>2048 $.
References:
[1] |
T. Baicheva and S. Topalova, On the diffusion property of the Improved Generalized Feistel with different permutations for each round, in Algebraic Informatics, CAI 2019 (eds. M. Ćirić, M. Droste and J.É. Pin), Lecture Notes in Computer Science, 11545 (2019), 38–49.
doi: 10.1007/978-3-030-21363-3_4. |
[2] |
T. Berger, M. Minier and G. Thomas, Extended generalized Feistel networks using matrix representation, Selected Areas in Cryptography–SAC 2013, Lecture Notes in Comput. Sci., Springer, Heidelberg, 8282 (2014), 289–305.
doi: 10.1007/978-3-662-43414-7_15. |
[3] |
T. Berger, J. Francq, M. Minier and G. Thomas,
Extended generalized Feistel networks using matrix representation to propose a new lightweight block cipher: Lilliput, IEEE Transactions on Computers, 65 (2016), 2074-2089.
doi: 10.1109/TC.2015.2468218. |
[4] |
D. Hong, J. Sung, S. Hong, J. Lim, S. Lee, B. Koo, C. Lee, D. Chang, J. Lee, K. Jeong, H. Kim, J. Kim and S. Chee,
HIGHT: A new block cipher suitable for low-resource device, Lecture Notes in Computer Science - CHES, 4249 (2006), 46-59.
doi: 10.1007/11894063_4. |
[5] |
K. Nyberg, Generalized Feistel networks, in Advances in Cryptology - ASIACRYPT '96 (eds. K. Kim and T. Matsumoto), Lecture Notes in Computer Science, 1163 (1996), 90–104.
doi: 10.1007/BFb0034838. |
[6] |
R. L. Rivest, M. J. B. Robshaw, R. Sidney and Y. L. Yin, The RC6 block cipher, August 1998. Available from: http://people.csail.mit.edu/rivest/pubs/RRSY98.pdf. Google Scholar |
[7] |
C. E. Shannon,
Communication theory of secrecy systems, Bell System Technical Journal, 28 (1949), 656-715.
doi: 10.1002/j.1538-7305.1949.tb00928.x. |
[8] |
T. Shirai, K. Shibutani, T. Akishita, S. Moriai and T. Iwata, The 128-bit block cipher CLEFIA (Extended abstract), Lecture Notes in Computer Science–FSE, 4593 (2007), 181-195. Google Scholar |
[9] |
T. Suzaki and K. Minematsu,
Improving the generalized Feistel, Lecture Notes in Computer Science–FSE, 6147 (2010), 19-39.
doi: 10.1007/978-3-642-13858-4_2. |
[10] |
L. Zhang and W. Wu,
Analysis of permutation choices for enhanced generalised Feistel structure with SP-type round function, IET Information Security, 11 (2017), 121-128.
doi: 10.1049/iet-ifs.2015.0433. |
[11] |
Y. Zheng, T. Matsumoto and H. Imai, On the construction of block ciphers provably secure and not relying on any unproved hypothesis, Advances in Cryptology - CRYPTO'89, Lecture Notes in Computer Science, 435 (1990), 461–480.
doi: 10.1007/0-387-34805-0_42. |
[12] |
Y. Wang and W. Wu,
New criterion for diffusion property and applications to improved GFS and EGFN, Designs Codes and Cryptography, 81 (2016), 393-412.
doi: 10.1007/s10623-015-0161-8. |
show all references
References:
[1] |
T. Baicheva and S. Topalova, On the diffusion property of the Improved Generalized Feistel with different permutations for each round, in Algebraic Informatics, CAI 2019 (eds. M. Ćirić, M. Droste and J.É. Pin), Lecture Notes in Computer Science, 11545 (2019), 38–49.
doi: 10.1007/978-3-030-21363-3_4. |
[2] |
T. Berger, M. Minier and G. Thomas, Extended generalized Feistel networks using matrix representation, Selected Areas in Cryptography–SAC 2013, Lecture Notes in Comput. Sci., Springer, Heidelberg, 8282 (2014), 289–305.
doi: 10.1007/978-3-662-43414-7_15. |
[3] |
T. Berger, J. Francq, M. Minier and G. Thomas,
Extended generalized Feistel networks using matrix representation to propose a new lightweight block cipher: Lilliput, IEEE Transactions on Computers, 65 (2016), 2074-2089.
doi: 10.1109/TC.2015.2468218. |
[4] |
D. Hong, J. Sung, S. Hong, J. Lim, S. Lee, B. Koo, C. Lee, D. Chang, J. Lee, K. Jeong, H. Kim, J. Kim and S. Chee,
HIGHT: A new block cipher suitable for low-resource device, Lecture Notes in Computer Science - CHES, 4249 (2006), 46-59.
doi: 10.1007/11894063_4. |
[5] |
K. Nyberg, Generalized Feistel networks, in Advances in Cryptology - ASIACRYPT '96 (eds. K. Kim and T. Matsumoto), Lecture Notes in Computer Science, 1163 (1996), 90–104.
doi: 10.1007/BFb0034838. |
[6] |
R. L. Rivest, M. J. B. Robshaw, R. Sidney and Y. L. Yin, The RC6 block cipher, August 1998. Available from: http://people.csail.mit.edu/rivest/pubs/RRSY98.pdf. Google Scholar |
[7] |
C. E. Shannon,
Communication theory of secrecy systems, Bell System Technical Journal, 28 (1949), 656-715.
doi: 10.1002/j.1538-7305.1949.tb00928.x. |
[8] |
T. Shirai, K. Shibutani, T. Akishita, S. Moriai and T. Iwata, The 128-bit block cipher CLEFIA (Extended abstract), Lecture Notes in Computer Science–FSE, 4593 (2007), 181-195. Google Scholar |
[9] |
T. Suzaki and K. Minematsu,
Improving the generalized Feistel, Lecture Notes in Computer Science–FSE, 6147 (2010), 19-39.
doi: 10.1007/978-3-642-13858-4_2. |
[10] |
L. Zhang and W. Wu,
Analysis of permutation choices for enhanced generalised Feistel structure with SP-type round function, IET Information Security, 11 (2017), 121-128.
doi: 10.1049/iet-ifs.2015.0433. |
[11] |
Y. Zheng, T. Matsumoto and H. Imai, On the construction of block ciphers provably secure and not relying on any unproved hypothesis, Advances in Cryptology - CRYPTO'89, Lecture Notes in Computer Science, 435 (1990), 461–480.
doi: 10.1007/0-387-34805-0_42. |
[12] |
Y. Wang and W. Wu,
New criterion for diffusion property and applications to improved GFS and EGFN, Designs Codes and Cryptography, 81 (2016), 393-412.
doi: 10.1007/s10623-015-0161-8. |
C | Remark | |||||
2 | 2 | 2 | c | - | 2 | |
4 | 4 | 4 | c | - | 4 | |
6 | 5 | 5 | c | - | 5 | |
8 | 6 | 6 | c | - | 6 | |
10 | 6 | 6 | c | - | 7 | |
12 | 7 | 7 | c | - | 8 | |
14 | 7 | 7 | c | - | 8 | |
16 | 7 | 7 | c | - | 8 | |
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 | 10 | |
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 | 12 | |
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 | 14 |
C | Remark | |||||
2 | 2 | 2 | c | - | 2 | |
4 | 4 | 4 | c | - | 4 | |
6 | 5 | 5 | c | - | 5 | |
8 | 6 | 6 | c | - | 6 | |
10 | 6 | 6 | c | - | 7 | |
12 | 7 | 7 | c | - | 8 | |
14 | 7 | 7 | c | - | 8 | |
16 | 7 | 7 | c | - | 8 | |
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 | 10 | |
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 | 12 | |
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 | 14 |
C | Remark | ||||
140 | 13 | 12 | 1 | 2.70 | - |
144 | 13 | 12 | 2 | 2.3.24 | - |
150 | 13 | 12 | 2 | 2.3.25 | - |
160 | 13 | 12 | 1 | 2.80 | - |
180 | 14 | 13 | 1 | 2.90 | - |
192 | 14 | 13 | 1 | 2.96 | - |
196 | 14 | 13 | 1 | 2.98 | - |
200 | 14 | 13 | 1 | 2.100 | - |
210 | 14 | 13 | 2 | 2.3.35 | - |
224 | 14 | 13 | 1 | 2.112 | - |
240 | 14 | 13 | 2 | 2.3.40 | - |
250 | 14 | 13 | 2 | 2.5.25 | - |
256 | 14 | 13 | 1 | 2.128 | 16 |
294 | 15 | 14 | 2 | 2.3.49 | - |
300 | 15 | 14 | 1 | 2.150 | - |
320 | 15 | 14 | 1 | 2.160 | - |
336 | 15 | 14 | 2 | 2.3.56 | - |
350 | 15 | 14 | 2 | 2.5.35 | - |
384 | 15 | 14 | 2 | 2.3.64 | - |
400 | 15 | 14 | 2 | 2.5.40 | - |
480 | 16 | 15 | 1 | 2.240 | - |
490 | 16 | 15 | 2 | 2.5.49 | - |
500 | 16 | 15 | 1 | 2.250 | - |
512 | 16 | 15 | 1 | 2.256 | 18 |
560 | 16 | 15 | 2 | 2.5.56 | - |
640 | 16 | 15 | 2 | 2.5.64 | - |
768 | 17 | 16 | 1 | 2.384 | - |
784 | 17 | 16 | 2 | 2.7.56 | - |
800 | 17 | 16 | 1 | 2.400 | - |
896 | 17 | 16 | 2 | 2.7.64 | - |
1024 | 17 | 16 | 2 | 2.8.64 | 20 |
1250 | 18 | 17 | 2 | 2.5.125 | - |
1280 | 18 | 17 | 1 | 2.640 | - |
2000 | 19 | 18 | 2 | 2.5.200 | - |
2048 | 19 | 18 | 1 | 2.1024 | 22 |
C | Remark | ||||
140 | 13 | 12 | 1 | 2.70 | - |
144 | 13 | 12 | 2 | 2.3.24 | - |
150 | 13 | 12 | 2 | 2.3.25 | - |
160 | 13 | 12 | 1 | 2.80 | - |
180 | 14 | 13 | 1 | 2.90 | - |
192 | 14 | 13 | 1 | 2.96 | - |
196 | 14 | 13 | 1 | 2.98 | - |
200 | 14 | 13 | 1 | 2.100 | - |
210 | 14 | 13 | 2 | 2.3.35 | - |
224 | 14 | 13 | 1 | 2.112 | - |
240 | 14 | 13 | 2 | 2.3.40 | - |
250 | 14 | 13 | 2 | 2.5.25 | - |
256 | 14 | 13 | 1 | 2.128 | 16 |
294 | 15 | 14 | 2 | 2.3.49 | - |
300 | 15 | 14 | 1 | 2.150 | - |
320 | 15 | 14 | 1 | 2.160 | - |
336 | 15 | 14 | 2 | 2.3.56 | - |
350 | 15 | 14 | 2 | 2.5.35 | - |
384 | 15 | 14 | 2 | 2.3.64 | - |
400 | 15 | 14 | 2 | 2.5.40 | - |
480 | 16 | 15 | 1 | 2.240 | - |
490 | 16 | 15 | 2 | 2.5.49 | - |
500 | 16 | 15 | 1 | 2.250 | - |
512 | 16 | 15 | 1 | 2.256 | 18 |
560 | 16 | 15 | 2 | 2.5.56 | - |
640 | 16 | 15 | 2 | 2.5.64 | - |
768 | 17 | 16 | 1 | 2.384 | - |
784 | 17 | 16 | 2 | 2.7.56 | - |
800 | 17 | 16 | 1 | 2.400 | - |
896 | 17 | 16 | 2 | 2.7.64 | - |
1024 | 17 | 16 | 2 | 2.8.64 | 20 |
1250 | 18 | 17 | 2 | 2.5.125 | - |
1280 | 18 | 17 | 1 | 2.640 | - |
2000 | 19 | 18 | 2 | 2.5.200 | - |
2048 | 19 | 18 | 1 | 2.1024 | 22 |
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