February  2020, 14(1): 127-136. doi: 10.3934/amc.2020010

Highly nonlinear (vectorial) Boolean functions that are symmetric under some permutations

1. 

Department of Computer Engineering, Faculty of Engineering, Balıkesir University, 10145 Balıkesir, Turkey

2. 

Department of Mathematics, Faculty of Arts and Science, Balıkesir University, 10145 Balıkesir, Turkey

*Corresponding author: Selçuk Kavut

Received  November 2018 Published  August 2019

Fund Project: This work is supported financially by Balıkesir University under grant BAP 2015/23

We first give a brief survey of the results on highly nonlinear single-output Boolean functions and bijective S-boxes that are symmetric under some permutations. After that, we perform a heuristic search for the symmetric (and involution) S-boxes which are bijective in dimension 8 and identify corresponding permutations yielding rich classes in terms of cryptographically desirable properties.

Citation: SelÇuk Kavut, Seher Tutdere. Highly nonlinear (vectorial) Boolean functions that are symmetric under some permutations. Advances in Mathematics of Communications, 2020, 14 (1) : 127-136. doi: 10.3934/amc.2020010
References:
[1]

M. Bartholomew-Biggs, Chapter 5: The steepest descent method, in Nonlinear Optimization with Financial Applications. Springer US, (2005), 51–64. Google Scholar

[2]

E. Biham and A. Shamir, Differential cryptanalysis of DES-like cryptosystems, Journal of Cryptology, 4 (1991), 3-72. doi: 10.1007/BF00630563. Google Scholar

[3]

K. A. BrowningJ. F. DillonM. T. McQuistan and A. J. Wolfe, An APN permutation in dimension six, Contemporary Mathematics, 518 (2010), 33-42. doi: 10.1090/conm/518/10194. Google Scholar

[4]

C. Ding, G. Xiao and W. Shan, The Stability Theory of Stream Ciphers, Lecture Notes in Computer Science, 561. Springer-Verlag, Berlin Heidelberg, 1991. doi: 10.1007/3-540-54973-0. Google Scholar

[5]

H. Dobbertin, Construction of bent functions and balanced Boolean functions with high nonlinearity, Lecture Notes in Computer Science, 1008 (1994), 61-74. doi: 10.1007/3-540-60590-8_5. Google Scholar

[6]

E. Filiol and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity, Lecture Notes in Computer Science, 1403 (1998), 475-488. doi: 10.1007/BFb0054147. Google Scholar

[7]

C. Fontaine, On some cosets of the first-order Reed-Muller code with high minimum weight, IEEE Transactions on Information Theory, 45 (1999), 1237-1243. doi: 10.1109/18.761276. Google Scholar

[8]

X.-D. Hou, On the norm and covering radius of first-order Reed-Muller codes, IEEE Transactions on Information Theory, 43 (1997), 1025-1027. doi: 10.1109/18.568715. Google Scholar

[9]

S. Kavut, Results on rotation-symmetric S-boxes, Information Sciences, 201 (2012), 93-113. doi: 10.1016/j.ins.2012.02.030. Google Scholar

[10]

S. Kavut and Sevdenur Baloǧlu, Results on symmetric S-boxes constructed by concatenation of RSSBs, Cryptography and Communications, 11 (2019), 641–660, http://dx.doi.org/10.1007/s12095-018-0318-1. doi: 10.1007/s12095-018-0318-1. Google Scholar

[11]

S. KavutS. MaitraS. Sarkar and M. D. Yücel, Enumeration of 9-variable rotation symmetric Boolean functions having nonlinearity $>240$, Lecture Notes in Computer Science, 4329 (2006), 266-279. doi: 10.1007/11941378_19. Google Scholar

[12]

S. KavutS. Maitra and M. D. Yücel, Search for Boolean functions with excellent profiles in the rotation symmetric class, IEEE Transactions on Information Theory, 53 (2007), 1743-1751. doi: 10.1109/TIT.2007.894696. Google Scholar

[13]

S. Kavut and M. D. Yücel, 9-variable Boolean functions with nonlinearity 242 in the generalized rotation symmetric class, Information and Computation, 208 (2010), 341-350. doi: 10.1016/j.ic.2009.12.002. Google Scholar

[14]

S. Maitra, Balanced Boolean function on 13-variables having nonlinearity strictly greater than the bent concatenation bound, Boolean Functions in Cryptology and Information Security, 173–182, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 18, IOS, Amsterdam, 2008. Available from: https://eprint.iacr.org/2007/309.pdf.Google Scholar

[15]

S. Maitra, S. Kavut and M. D. Yücel, Balanced Boolean function on 13-variables having nonlinearity greater than the bent concatenation bound, Proceedings of Boolean Functions: Cryptography and Applications, (2008), 109–118.Google Scholar

[16]

M. Matsui, Linear cryptanalysis method for DES cipher, Lecture Notes in Computer Science, 765 (1993), 386-397. doi: 10.1007/3-540-48285-7_33. Google Scholar

[17]

K. Nyberg, Differentially uniform mappings for cryptography, Lecture Notes in Computer Science, 765 (1994), 55-64. doi: 10.1007/3-540-48285-7_6. Google Scholar

[18]

N. J. Patterson and D. H. Wiedemann, The covering radius of the $(2^15, 16)$ Reed-Muller code is at least 16276, IEEE Transactions on Information Theory, 29 (1983), 354-356. doi: 10.1109/TIT.1983.1056679. Google Scholar

[19]

V. RijmenP. S. L. M. Barreto and D. L. Gazzoni Filho, Rotation symmetry in algebraically generated cryptographic substitution tables, Information Processing Letters, 106 (2008), 246-250. doi: 10.1016/j.ipl.2007.09.012. Google Scholar

[20]

S. Sarkar and S. Maitra, Idempotents in the neighbourhood of Patterson-Wiedemann functions having Walsh spectra zeros, Designs, Codes and Cryptography, 49 (2008), 95-103. doi: 10.1007/s10623-008-9181-y. Google Scholar

[21]

P. Stǎnicǎ and S. Maitra, Rotation symmetric Boolean functions - count and cryptographic properties, Discrete Applied Mathematics, 156 (2008), 1567-1580. doi: 10.1016/j.dam.2007.04.029. Google Scholar

[22]

P. StǎnicǎS. Maitra and J. Clark, Results on rotation symmetric bent and correlation immune Boolean functions, Lecture Notes in Computer Science, 3017 (2004), 161-177. Google Scholar

show all references

References:
[1]

M. Bartholomew-Biggs, Chapter 5: The steepest descent method, in Nonlinear Optimization with Financial Applications. Springer US, (2005), 51–64. Google Scholar

[2]

E. Biham and A. Shamir, Differential cryptanalysis of DES-like cryptosystems, Journal of Cryptology, 4 (1991), 3-72. doi: 10.1007/BF00630563. Google Scholar

[3]

K. A. BrowningJ. F. DillonM. T. McQuistan and A. J. Wolfe, An APN permutation in dimension six, Contemporary Mathematics, 518 (2010), 33-42. doi: 10.1090/conm/518/10194. Google Scholar

[4]

C. Ding, G. Xiao and W. Shan, The Stability Theory of Stream Ciphers, Lecture Notes in Computer Science, 561. Springer-Verlag, Berlin Heidelberg, 1991. doi: 10.1007/3-540-54973-0. Google Scholar

[5]

H. Dobbertin, Construction of bent functions and balanced Boolean functions with high nonlinearity, Lecture Notes in Computer Science, 1008 (1994), 61-74. doi: 10.1007/3-540-60590-8_5. Google Scholar

[6]

E. Filiol and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity, Lecture Notes in Computer Science, 1403 (1998), 475-488. doi: 10.1007/BFb0054147. Google Scholar

[7]

C. Fontaine, On some cosets of the first-order Reed-Muller code with high minimum weight, IEEE Transactions on Information Theory, 45 (1999), 1237-1243. doi: 10.1109/18.761276. Google Scholar

[8]

X.-D. Hou, On the norm and covering radius of first-order Reed-Muller codes, IEEE Transactions on Information Theory, 43 (1997), 1025-1027. doi: 10.1109/18.568715. Google Scholar

[9]

S. Kavut, Results on rotation-symmetric S-boxes, Information Sciences, 201 (2012), 93-113. doi: 10.1016/j.ins.2012.02.030. Google Scholar

[10]

S. Kavut and Sevdenur Baloǧlu, Results on symmetric S-boxes constructed by concatenation of RSSBs, Cryptography and Communications, 11 (2019), 641–660, http://dx.doi.org/10.1007/s12095-018-0318-1. doi: 10.1007/s12095-018-0318-1. Google Scholar

[11]

S. KavutS. MaitraS. Sarkar and M. D. Yücel, Enumeration of 9-variable rotation symmetric Boolean functions having nonlinearity $>240$, Lecture Notes in Computer Science, 4329 (2006), 266-279. doi: 10.1007/11941378_19. Google Scholar

[12]

S. KavutS. Maitra and M. D. Yücel, Search for Boolean functions with excellent profiles in the rotation symmetric class, IEEE Transactions on Information Theory, 53 (2007), 1743-1751. doi: 10.1109/TIT.2007.894696. Google Scholar

[13]

S. Kavut and M. D. Yücel, 9-variable Boolean functions with nonlinearity 242 in the generalized rotation symmetric class, Information and Computation, 208 (2010), 341-350. doi: 10.1016/j.ic.2009.12.002. Google Scholar

[14]

S. Maitra, Balanced Boolean function on 13-variables having nonlinearity strictly greater than the bent concatenation bound, Boolean Functions in Cryptology and Information Security, 173–182, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 18, IOS, Amsterdam, 2008. Available from: https://eprint.iacr.org/2007/309.pdf.Google Scholar

[15]

S. Maitra, S. Kavut and M. D. Yücel, Balanced Boolean function on 13-variables having nonlinearity greater than the bent concatenation bound, Proceedings of Boolean Functions: Cryptography and Applications, (2008), 109–118.Google Scholar

[16]

M. Matsui, Linear cryptanalysis method for DES cipher, Lecture Notes in Computer Science, 765 (1993), 386-397. doi: 10.1007/3-540-48285-7_33. Google Scholar

[17]

K. Nyberg, Differentially uniform mappings for cryptography, Lecture Notes in Computer Science, 765 (1994), 55-64. doi: 10.1007/3-540-48285-7_6. Google Scholar

[18]

N. J. Patterson and D. H. Wiedemann, The covering radius of the $(2^15, 16)$ Reed-Muller code is at least 16276, IEEE Transactions on Information Theory, 29 (1983), 354-356. doi: 10.1109/TIT.1983.1056679. Google Scholar

[19]

V. RijmenP. S. L. M. Barreto and D. L. Gazzoni Filho, Rotation symmetry in algebraically generated cryptographic substitution tables, Information Processing Letters, 106 (2008), 246-250. doi: 10.1016/j.ipl.2007.09.012. Google Scholar

[20]

S. Sarkar and S. Maitra, Idempotents in the neighbourhood of Patterson-Wiedemann functions having Walsh spectra zeros, Designs, Codes and Cryptography, 49 (2008), 95-103. doi: 10.1007/s10623-008-9181-y. Google Scholar

[21]

P. Stǎnicǎ and S. Maitra, Rotation symmetric Boolean functions - count and cryptographic properties, Discrete Applied Mathematics, 156 (2008), 1567-1580. doi: 10.1016/j.dam.2007.04.029. Google Scholar

[22]

P. StǎnicǎS. Maitra and J. Clark, Results on rotation symmetric bent and correlation immune Boolean functions, Lecture Notes in Computer Science, 3017 (2004), 161-177. Google Scholar

Table 1.  A summary of the highest nonlinearities for odd $ n\ge 9 $
Number of variables ($ n $) 9 11 13 15
Bounds
Bent concatenation bound 240 992 4032 16256
($ 2^{n-1}-2^\frac{n-1}{2} $)
Upper bound 244 1000 4050 16292
($ 2\left\lfloor 2^{n-2}-2^{\frac{n}{2}-2}\right\rfloor $)
Unbalanced nonlinearities
[18] $ - $ $ - $ $ - $ 16276
[13] 242 996 4040 $ - $
Balanced nonlinearities
[15] $ - $ $ - $ 4036 $ - $
[20] $ - $ $ - $ $ - $ 16272
Number of variables ($ n $) 9 11 13 15
Bounds
Bent concatenation bound 240 992 4032 16256
($ 2^{n-1}-2^\frac{n-1}{2} $)
Upper bound 244 1000 4050 16292
($ 2\left\lfloor 2^{n-2}-2^{\frac{n}{2}-2}\right\rfloor $)
Unbalanced nonlinearities
[18] $ - $ $ - $ $ - $ 16276
[13] 242 996 4040 $ - $
Balanced nonlinearities
[15] $ - $ $ - $ 4036 $ - $
[20] $ - $ $ - $ $ - $ 16272
Table 2.  Best achieved cryptographic properties [nonlinearity, differential uniformity, algebraic degree]
$ \# $ Representative
permutation
Space
size
Best result
(for involution S-boxes)
Best result
1 $ (7,6,2,1,8,5,4,3) $ $ 2^{147.93} $ $ [84,44,7] $ $ [84,44,7] $
2 $ (2,3,1,7,4,5,6,8) $ $ 2^{191.48} $ $ [84,52,7] $ $ [84,52,7] $
3 $ (6,7,5,8,4,3,1,2)^a $ $ 2^{208.29} $ $ \bf{[106,6,7]} $ $ \bf{[106,6,7]}, \bf{[108,8,6]} $
4 $ (4,3,2,5,8,1,7,6) $ $ 2^{227.35} $ $ [0, -, -] $ $ [0, -, -] $
5 $ (4,5,3,2,8,1,6,7) $ $ 2^{243.74} $ $ \bf {[106,6,7]} $ $ \bf {[106,6,7]} $
6 $ (8,3,4,6,7,1,5,2) $ $ 2^{277.78} $ $ [104,6,7] $ $ [104,6,7], {\bf{[106,8,7]}} $
7 $ (8,6,3,5,2,1,7,4) $ $ 2^{283.02} $ $ [104,10,7] $ $ \it {[104,8,7]} $
8 $ (4,6,7,5,1,2,3,8) $ $ 2^{357.97} $ $ [84,44,7] $ $ [84,44,7] $
9 $ (2,6,3,4,5,8,1,7) $ $ 2^{358.65} $ $ [100,10,7] $ $ [100,10,7],\it{[104,20,7]} $
10 $ (7,3,6,1,8,2,4,5) $ $ 2^{359.22} $ $ [0, -, -] $ $ [0, -, -] $
11 $ (7,6,1,2,3,8,5,4)^b $ $ 2^{412.21} $ $ [104,6,7] $ $ [104,6,7], {\bf{[106,8,7]}} $
12 $ (2,7,4,3,5,6,1,8) $ $ 2^{431.91} $ $ [0, -, -] $ $ [0, -, -] $
13 $ (6,4,8,2,1,7,5,3) $ $ 2^{440.19} $ $ [84,22,7] $ $ [84,22,7] $
14 $ (1,3,6,7,2,5,4,8) $ $ 2^{446.24} $ $ [84,22,7] $ $ [84,22,7] $
15 $ (1,5,6,4,3,2,7,8) $ $ 2^{476.86} $ $ [84,52,7] $ $ [84,52,7] $
16 $ (4,3,8,5,1,6,7,2) $ $ 2^{565.87} $ $ [104,6,7] $ $ [104,6,7] $
17 $ (1,6,3,4,2,5,7,8) $ $ 2^{693.43} $ $ [84,44,7] $ $ [84,44,7] $
18 $ (7,6,5,8,3,2,1,4)^c $ $ 2^{824.73} $ $ [104,6,7] $ $ [104,6,7] $
19 $ (1,5,8,4,2,7,6,3) $ $ 2^{835.24} $ $ [104,8,7] $ $ [104,8,7] $
20 $ (1,2,7,4,5,8,3,6) $ $ 2^{890.27} $ $ [84,22,7] $ $ [84,22,7] $
21 $ (8,2,3,4,5,6,7,1) $ $ 2^{1076.16} $ $ [0, -, -] $ $ [0, -, -] $
22 $ (1,2,3,4,5,6,7,8)^d $ $ 2^{1684} $ $ [102,6,7] $ $ \it{[104,6,7]} $
$ ^a: $ Linear equivalet to RSSBs
$ ^b: $ Linear equivalent to 2-RSSBs
$ ^c: $ Linear equivalent to 4-RSSBs
$ ^d: $ The search space of all bijective S-boxes
$ \# $ Representative
permutation
Space
size
Best result
(for involution S-boxes)
Best result
1 $ (7,6,2,1,8,5,4,3) $ $ 2^{147.93} $ $ [84,44,7] $ $ [84,44,7] $
2 $ (2,3,1,7,4,5,6,8) $ $ 2^{191.48} $ $ [84,52,7] $ $ [84,52,7] $
3 $ (6,7,5,8,4,3,1,2)^a $ $ 2^{208.29} $ $ \bf{[106,6,7]} $ $ \bf{[106,6,7]}, \bf{[108,8,6]} $
4 $ (4,3,2,5,8,1,7,6) $ $ 2^{227.35} $ $ [0, -, -] $ $ [0, -, -] $
5 $ (4,5,3,2,8,1,6,7) $ $ 2^{243.74} $ $ \bf {[106,6,7]} $ $ \bf {[106,6,7]} $
6 $ (8,3,4,6,7,1,5,2) $ $ 2^{277.78} $ $ [104,6,7] $ $ [104,6,7], {\bf{[106,8,7]}} $
7 $ (8,6,3,5,2,1,7,4) $ $ 2^{283.02} $ $ [104,10,7] $ $ \it {[104,8,7]} $
8 $ (4,6,7,5,1,2,3,8) $ $ 2^{357.97} $ $ [84,44,7] $ $ [84,44,7] $
9 $ (2,6,3,4,5,8,1,7) $ $ 2^{358.65} $ $ [100,10,7] $ $ [100,10,7],\it{[104,20,7]} $
10 $ (7,3,6,1,8,2,4,5) $ $ 2^{359.22} $ $ [0, -, -] $ $ [0, -, -] $
11 $ (7,6,1,2,3,8,5,4)^b $ $ 2^{412.21} $ $ [104,6,7] $ $ [104,6,7], {\bf{[106,8,7]}} $
12 $ (2,7,4,3,5,6,1,8) $ $ 2^{431.91} $ $ [0, -, -] $ $ [0, -, -] $
13 $ (6,4,8,2,1,7,5,3) $ $ 2^{440.19} $ $ [84,22,7] $ $ [84,22,7] $
14 $ (1,3,6,7,2,5,4,8) $ $ 2^{446.24} $ $ [84,22,7] $ $ [84,22,7] $
15 $ (1,5,6,4,3,2,7,8) $ $ 2^{476.86} $ $ [84,52,7] $ $ [84,52,7] $
16 $ (4,3,8,5,1,6,7,2) $ $ 2^{565.87} $ $ [104,6,7] $ $ [104,6,7] $
17 $ (1,6,3,4,2,5,7,8) $ $ 2^{693.43} $ $ [84,44,7] $ $ [84,44,7] $
18 $ (7,6,5,8,3,2,1,4)^c $ $ 2^{824.73} $ $ [104,6,7] $ $ [104,6,7] $
19 $ (1,5,8,4,2,7,6,3) $ $ 2^{835.24} $ $ [104,8,7] $ $ [104,8,7] $
20 $ (1,2,7,4,5,8,3,6) $ $ 2^{890.27} $ $ [84,22,7] $ $ [84,22,7] $
21 $ (8,2,3,4,5,6,7,1) $ $ 2^{1076.16} $ $ [0, -, -] $ $ [0, -, -] $
22 $ (1,2,3,4,5,6,7,8)^d $ $ 2^{1684} $ $ [102,6,7] $ $ \it{[104,6,7]} $
$ ^a: $ Linear equivalet to RSSBs
$ ^b: $ Linear equivalent to 2-RSSBs
$ ^c: $ Linear equivalent to 4-RSSBs
$ ^d: $ The search space of all bijective S-boxes
[1]

Pascale Charpin, Jie Peng. Differential uniformity and the associated codes of cryptographic functions. Advances in Mathematics of Communications, 2019, 13 (4) : 579-600. doi: 10.3934/amc.2019036

[2]

Manish K. Gupta, Chinnappillai Durairajan. On the covering radius of some modular codes. Advances in Mathematics of Communications, 2014, 8 (2) : 129-137. doi: 10.3934/amc.2014.8.129

[3]

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

[4]

Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025

[5]

Constanza Riera, Pantelimon Stănică. Landscape Boolean functions. Advances in Mathematics of Communications, 2019, 13 (4) : 613-627. doi: 10.3934/amc.2019038

[6]

Axel Kohnert, Johannes Zwanzger. New linear codes with prescribed group of automorphisms found by heuristic search. Advances in Mathematics of Communications, 2009, 3 (2) : 157-166. doi: 10.3934/amc.2009.3.157

[7]

Claude Carlet, Serge Feukoua. Three basic questions on Boolean functions. Advances in Mathematics of Communications, 2017, 11 (4) : 837-855. doi: 10.3934/amc.2017061

[8]

Sihem Mesnager, Gérard Cohen. Fast algebraic immunity of Boolean functions. Advances in Mathematics of Communications, 2017, 11 (2) : 373-377. doi: 10.3934/amc.2017031

[9]

Tsonka Baicheva, Iliya Bouyukliev. On the least covering radius of binary linear codes of dimension 6. Advances in Mathematics of Communications, 2010, 4 (3) : 399-404. doi: 10.3934/amc.2010.4.399

[10]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83

[11]

Jian Liu, Sihem Mesnager, Lusheng Chen. Variation on correlation immune Boolean and vectorial functions. Advances in Mathematics of Communications, 2016, 10 (4) : 895-919. doi: 10.3934/amc.2016048

[12]

Yu Zhou. On the distribution of auto-correlation value of balanced Boolean functions. Advances in Mathematics of Communications, 2013, 7 (3) : 335-347. doi: 10.3934/amc.2013.7.335

[13]

Yang Yang, Xiaohu Tang, Guang Gong. Even periodic and odd periodic complementary sequence pairs from generalized Boolean functions. Advances in Mathematics of Communications, 2013, 7 (2) : 113-125. doi: 10.3934/amc.2013.7.113

[14]

Claude Carlet, Brahim Merabet. Asymptotic lower bound on the algebraic immunity of random balanced multi-output Boolean functions. Advances in Mathematics of Communications, 2013, 7 (2) : 197-217. doi: 10.3934/amc.2013.7.197

[15]

Daria Bugajewska, Mirosława Zima. On the spectral radius of linearly bounded operators and existence results for functional-differential equations. Conference Publications, 2003, 2003 (Special) : 147-155. doi: 10.3934/proc.2003.2003.147

[16]

Pierre Magal. Global stability for differential equations with homogeneous nonlinearity and application to population dynamics. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 541-560. doi: 10.3934/dcdsb.2002.2.541

[17]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116

[18]

Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181

[19]

Tanja Eisner, Pavel Zorin-Kranich. Uniformity in the Wiener-Wintner theorem for nilsequences. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3497-3516. doi: 10.3934/dcds.2013.33.3497

[20]

Yi Ming Zou. Dynamics of boolean networks. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1629-1640. doi: 10.3934/dcdss.2011.4.1629

2018 Impact Factor: 0.879

Article outline

Figures and Tables

[Back to Top]