-
Previous Article
A geometric characterization of minimal codes and their asymptotic performance
- AMC Home
- This Issue
-
Next Article
Binary sequences derived from differences of consecutive quadratic residues
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. |
[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.
|
[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. |
[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.
|
[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 |
[1] |
Xiao-Wen Chang, David Titley-Peloquin. An improved algorithm for generalized least squares estimation. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 451-461. doi: 10.3934/naco.2020044 |
[2] |
Guangmei Shao, Wei Xue, Gaohang Yu, Xiao Zheng. Improved SVRG for finite sum structure optimization with application to binary classification. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2253-2266. doi: 10.3934/jimo.2019052 |
[3] |
José A. Cañizo, Alexis Molino. Improved energy methods for nonlocal diffusion problems. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1405-1425. doi: 10.3934/dcds.2018057 |
[4] |
Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic and Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545 |
[5] |
Maxime Breden. Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion. Kinetic and Related Models, 2018, 11 (2) : 279-301. doi: 10.3934/krm.2018014 |
[6] |
Ihab Haidar, Alain Rapaport, Frédéric Gérard. Effects of spatial structure and diffusion on the performances of the chemostat. Mathematical Biosciences & Engineering, 2011, 8 (4) : 953-971. doi: 10.3934/mbe.2011.8.953 |
[7] |
Claude Carlet, Khoongming Khoo, Chu-Wee Lim, Chuan-Wen Loe. On an improved correlation analysis of stream ciphers using multi-output Boolean functions and the related generalized notion of nonlinearity. Advances in Mathematics of Communications, 2008, 2 (2) : 201-221. doi: 10.3934/amc.2008.2.201 |
[8] |
Guillaume Cantin, Alexandre Thorel. On a generalized diffusion problem: A complex network approach. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2345-2365. doi: 10.3934/dcdsb.2021135 |
[9] |
Hassan Belhadj, Samir Khallouq, Mohamed Rhoudaf. Parallelization of a finite volumes discretization for anisotropic diffusion problems using an improved Schur complement technique. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2075-2099. doi: 10.3934/dcdss.2020260 |
[10] |
Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191 |
[11] |
Tianran Zhang. Traveling waves for a reaction-diffusion model with a cyclic structure. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1859-1870. doi: 10.3934/dcdsb.2020006 |
[12] |
Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095 |
[13] |
Liming Wang. A passivity-based stability criterion for reaction diffusion systems with interconnected structure. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 303-323. doi: 10.3934/dcdsb.2012.17.303 |
[14] |
Hongfei Xu, Jinfeng Wang, Xuelian Xu. Dynamics and pattern formation in a cross-diffusion model with stage structure for predators. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4473-4489. doi: 10.3934/dcdsb.2021237 |
[15] |
Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic and Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159 |
[16] |
Antoine Mellet, Jean-Michel Roquejoffre, Yannick Sire. Generalized fronts for one-dimensional reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 303-312. doi: 10.3934/dcds.2010.26.303 |
[17] |
L. Cherfils, Y. Il'yasov. On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian. Communications on Pure and Applied Analysis, 2005, 4 (1) : 9-22. doi: 10.3934/cpaa.2005.4.9 |
[18] |
Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135 |
[19] |
Yukio Kan-On. Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system. Conference Publications, 2013, 2013 (special) : 427-436. doi: 10.3934/proc.2013.2013.427 |
[20] |
Shuling Yan, Shangjiang Guo. Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1559-1579. doi: 10.3934/dcdsb.2018059 |
2021 Impact Factor: 1.015
Tools
Metrics
Other articles
by authors
[Back to Top]