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doi: 10.3934/amc.2020115

A New Construction of odd-variable Rotation symmetric Boolean functions with good cryptographic properties

Bingxin Wang and  Sihong Su ,

School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China

*Corresponding author: Sihong Su (E-mail: sush@henu.edu.cn)

Received  April 2020 Revised  August 2020 Published  October 2020

Fund Project: The second author is supported by the National Natural Science Foundation of China (Grant No. 61502147)

Rotation symmetric Boolean functions constitute a class of cryptographically significant Boolean functions. In this paper, based on the theory of ordered integer partitions, we present a new class of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity by modifying the support of the majority function. Compared with the existing rotation symmetric Boolean functions on odd variables, the newly constructed functions have the highest nonlinearity.

Keywords: Rotation symmetric Boolean functions, integer partition, Walsh spectrum, algebraic immunity, nonlinearity.
Mathematics Subject Classification: 06E30, 94D10.
Citation: Bingxin Wang, Sihong Su. A New Construction of odd-variable Rotation symmetric Boolean functions with good cryptographic properties. Advances in Mathematics of Communications, doi: 10.3934/amc.2020115
References:
[1]

A. Canteaut and M. Trabbia, Improved fast correlation attacks using parity-check equations of weight 4 and 5, in Advances in Cryptology-EUROCRYPT 2000 (eds. B. Preneel), Springer, Berlin, Heidelberg, 2000,573–588. doi: 10.1007/3-540-45539-6.  Google Scholar

[2]

C. Carlet, Boolean functions for cryptography and error correcting codes, to appear in Cambridge University Press. doi: 10.1017/CBO9780511780448.011.  Google Scholar

[3]

C. Carlet and K. Feng, An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity, in ASIACRYPT 2008 (eds. J. Pieprzyk), Lecture Notes in Computer Science, 5350, Springer, Heidelberg, 2008,425–440. doi: 10.1007/978-3-540-89255-7_26.  Google Scholar

[4]

C. Carlet, G. Gao and W. Liu, A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions, J. Combin. Theory Ser. A, 127 (2014), 161-175.  doi: 10.1016/j.jcta.2014.05.008.  Google Scholar

[5]

C. Carlet, X. Zeng, C. Li and L. Hu, Further properties of several classes of Boolean functions with optimum algebraic immunity, Des. Codes Cryptogr., 52 (2009), 303-338.  doi: 10.1007/s10623-009-9284-0.  Google Scholar

[6]

Y. Chen, F. Guo and J. Ruan, Constructing odd-variables RSBFs with optimal algebraic immunity, good nonlieanrity and good behavior against fast algebraic attarcks, Discrete Appl. Math., 262 (2019), 1-12.  doi: 10.1016/j.dam.2019.02.041.  Google Scholar

[7]

N. Courtois and W. Meier, Algebraic attacks on stream ciphers with linear feedback, in EUROCRYPT 2003, Lecture Notes in Computer Science, 2656, Springer-Verlag, Heidelberg, 2003,345–359. doi: 10.1007/3-540-39200-9_21.  Google Scholar

[8]

D. Dalai, S. Maitra and S. Sarkar, Basic theory in construction of Boolean functions with maximum possible annihilator immunity, Des. Codes Cryptogr., 40 (2006), 41-58.  doi: 10.1007/s10623-005-6300-x.  Google Scholar

[9]

C. Ding, G. Xiao and W. Shan, The stability theory of stream ciphers, in Lecture Notes in Computer Science, 561, Springer-Verlag, Berlin, 1991. doi: 10.1007/3-540-54973-0.  Google Scholar

[10]

J. Du, Q. Wen, J. Zhang and S. Pang, Constructions of resilient rotation symmetric Boolean functions on given number of variables, IET Inform. Secur., 8 (2014), 265-272.  doi: 10.1049/iet-ifs.2013.0090.  Google Scholar

[11]

S. Fu, J. Du, L. Qu and C. Li, Construction of odd-variable rotation symmetric boolean functions with maximum algebraic immunity, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 99 (2016), 853-855.  doi: 10.1016/j.dam.2016.06.005.  Google Scholar

[12]

G. Gao, Constructions of quadratic and cubic rotation symmetric bent functions, IEEE Trans. Inform. Theory, 58 (2012), 4908-4913.  doi: 10.1109/TIT.2012.2193377.  Google Scholar

[13]

W. Meier, E. Pasalic and C. Carlet, Algebraic attacks and decomposition of Boolean functions, in Advances in Cryptology-EUROCRYPT 2004, Lecture Notes in Computer Science, 3027, Springer Heidelberg, 2004,474–491. doi: 10.1007/978-3-540-24676-3_28.  Google Scholar

[14]

S. Sarkar and S. Maitra, Construction of rotation symmetric Boolean functions with maximum algebraic immunity on odd number of variables, in AAECC 2007, Lecture Notes in Computer Science, 4851, 2007,271–280. doi: 10.1007/978-3-540-77224-8_32.  Google Scholar

[15]

P. Stnic and S. Maitra, Rotation symmetric Boolean functions-count and cryptographic properties, Discrete Appl. Math., 156 (2002), 1567-1580.  doi: 10.1016/j.dam.2007.04.029.  Google Scholar

[16]

S. Su, A new construction of rotation symmetric bent functions with maximal algebraic degree, Adv. Math. Commun., 13 (2019), 253-265.  doi: 10.3934/amc.2019017.  Google Scholar

[17]

S. Su and X. Tang, Construction of rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity, Des. Codes Cryptogr., 71 (2014), 183-199.  doi: 10.1007/s10623-012-9727-x.  Google Scholar

[18]

S. Su and X. Tang, Systematic constructions of rotation symmetric bent functions, 2-rotation symmetric bent functions, and bent idempotent functions, IEEE Trans. Inf. Theory, 63 (2017), 4658-4667.  doi: 10.1109/TIT.2016.2621751.  Google Scholar

[19]

L. Sun, F. Fu and X. Guang, Two classes of 1-resilient prime-variable rotation symmetric Boolean functions, IEICE Trans Fund. Electron. Comm. Comput. Sci., E100-A (2017), 902-907.  doi: 10.1587/transfun.E100.A.902.  Google Scholar

[20]

H. Zhang and S. Su, A new construction of rotation symmetric Boolean functions with optimal algebraic immunity and higher nonlinearity, Discrete Appl. Math., 262 (2019), 13-28.  doi: 10.1016/j.dam.2019.02.030.  Google Scholar

[21]

Q. Zhao, G. Han, D. Zheng and X. Li, Constructing odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity, Chinese J. Electron., 28 (2019), 45-51.  doi: 10.1007/s12190-019-01245-2.  Google Scholar

show all references

References:
[1]

A. Canteaut and M. Trabbia, Improved fast correlation attacks using parity-check equations of weight 4 and 5, in Advances in Cryptology-EUROCRYPT 2000 (eds. B. Preneel), Springer, Berlin, Heidelberg, 2000,573–588. doi: 10.1007/3-540-45539-6.  Google Scholar

[2]

C. Carlet, Boolean functions for cryptography and error correcting codes, to appear in Cambridge University Press. doi: 10.1017/CBO9780511780448.011.  Google Scholar

[3]

C. Carlet and K. Feng, An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity, in ASIACRYPT 2008 (eds. J. Pieprzyk), Lecture Notes in Computer Science, 5350, Springer, Heidelberg, 2008,425–440. doi: 10.1007/978-3-540-89255-7_26.  Google Scholar

[4]

C. Carlet, G. Gao and W. Liu, A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions, J. Combin. Theory Ser. A, 127 (2014), 161-175.  doi: 10.1016/j.jcta.2014.05.008.  Google Scholar

[5]

C. Carlet, X. Zeng, C. Li and L. Hu, Further properties of several classes of Boolean functions with optimum algebraic immunity, Des. Codes Cryptogr., 52 (2009), 303-338.  doi: 10.1007/s10623-009-9284-0.  Google Scholar

[6]

Y. Chen, F. Guo and J. Ruan, Constructing odd-variables RSBFs with optimal algebraic immunity, good nonlieanrity and good behavior against fast algebraic attarcks, Discrete Appl. Math., 262 (2019), 1-12.  doi: 10.1016/j.dam.2019.02.041.  Google Scholar

[7]

N. Courtois and W. Meier, Algebraic attacks on stream ciphers with linear feedback, in EUROCRYPT 2003, Lecture Notes in Computer Science, 2656, Springer-Verlag, Heidelberg, 2003,345–359. doi: 10.1007/3-540-39200-9_21.  Google Scholar

[8]

D. Dalai, S. Maitra and S. Sarkar, Basic theory in construction of Boolean functions with maximum possible annihilator immunity, Des. Codes Cryptogr., 40 (2006), 41-58.  doi: 10.1007/s10623-005-6300-x.  Google Scholar

[9]

C. Ding, G. Xiao and W. Shan, The stability theory of stream ciphers, in Lecture Notes in Computer Science, 561, Springer-Verlag, Berlin, 1991. doi: 10.1007/3-540-54973-0.  Google Scholar

[10]

J. Du, Q. Wen, J. Zhang and S. Pang, Constructions of resilient rotation symmetric Boolean functions on given number of variables, IET Inform. Secur., 8 (2014), 265-272.  doi: 10.1049/iet-ifs.2013.0090.  Google Scholar

[11]

S. Fu, J. Du, L. Qu and C. Li, Construction of odd-variable rotation symmetric boolean functions with maximum algebraic immunity, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 99 (2016), 853-855.  doi: 10.1016/j.dam.2016.06.005.  Google Scholar

[12]

G. Gao, Constructions of quadratic and cubic rotation symmetric bent functions, IEEE Trans. Inform. Theory, 58 (2012), 4908-4913.  doi: 10.1109/TIT.2012.2193377.  Google Scholar

[13]

W. Meier, E. Pasalic and C. Carlet, Algebraic attacks and decomposition of Boolean functions, in Advances in Cryptology-EUROCRYPT 2004, Lecture Notes in Computer Science, 3027, Springer Heidelberg, 2004,474–491. doi: 10.1007/978-3-540-24676-3_28.  Google Scholar

[14]

S. Sarkar and S. Maitra, Construction of rotation symmetric Boolean functions with maximum algebraic immunity on odd number of variables, in AAECC 2007, Lecture Notes in Computer Science, 4851, 2007,271–280. doi: 10.1007/978-3-540-77224-8_32.  Google Scholar

[15]

P. Stnic and S. Maitra, Rotation symmetric Boolean functions-count and cryptographic properties, Discrete Appl. Math., 156 (2002), 1567-1580.  doi: 10.1016/j.dam.2007.04.029.  Google Scholar

[16]

S. Su, A new construction of rotation symmetric bent functions with maximal algebraic degree, Adv. Math. Commun., 13 (2019), 253-265.  doi: 10.3934/amc.2019017.  Google Scholar

[17]

S. Su and X. Tang, Construction of rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity, Des. Codes Cryptogr., 71 (2014), 183-199.  doi: 10.1007/s10623-012-9727-x.  Google Scholar

[18]

S. Su and X. Tang, Systematic constructions of rotation symmetric bent functions, 2-rotation symmetric bent functions, and bent idempotent functions, IEEE Trans. Inf. Theory, 63 (2017), 4658-4667.  doi: 10.1109/TIT.2016.2621751.  Google Scholar

[19]

L. Sun, F. Fu and X. Guang, Two classes of 1-resilient prime-variable rotation symmetric Boolean functions, IEICE Trans Fund. Electron. Comm. Comput. Sci., E100-A (2017), 902-907.  doi: 10.1587/transfun.E100.A.902.  Google Scholar

[20]

H. Zhang and S. Su, A new construction of rotation symmetric Boolean functions with optimal algebraic immunity and higher nonlinearity, Discrete Appl. Math., 262 (2019), 13-28.  doi: 10.1016/j.dam.2019.02.030.  Google Scholar

[21]

Q. Zhao, G. Han, D. Zheng and X. Li, Constructing odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity, Chinese J. Electron., 28 (2019), 45-51.  doi: 10.1007/s12190-019-01245-2.  Google Scholar

Table 1.  The nonlinearities of the rotation symmetric Boolean functions
function nonlinearity
[14] $ {2^{n-1}-{n-1\choose k}}+2 $
[17] $ 2^{n-1}-{n-1\choose k}+2^k-2 $
[11] $ 2^{n-1}-{n-1\choose k}+2^k+2^{k-2}-k $
[21] $ 2^{n-1}-{n-1\choose k}+2^k+2^{k-1}-2k $
[20] $ 2^{n-1}-{n-1\choose k}+(k-5)2^{k-1}+2k+2 $
[6] $ 2^{n-1}-{n-1\choose k}+\sum_{h=3}^k(n-2h)|T_h|+L_k $
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function nonlinearity
[14] $ {2^{n-1}-{n-1\choose k}}+2 $
[17] $ 2^{n-1}-{n-1\choose k}+2^k-2 $
[11] $ 2^{n-1}-{n-1\choose k}+2^k+2^{k-2}-k $
[21] $ 2^{n-1}-{n-1\choose k}+2^k+2^{k-1}-2k $
[20] $ 2^{n-1}-{n-1\choose k}+(k-5)2^{k-1}+2k+2 $
[6] $ 2^{n-1}-{n-1\choose k}+\sum_{h=3}^k(n-2h)|T_h|+L_k $
Table 2.  The entries of the vectors in $ T $ for $ n = 13 $
$ \alpha_1 $ $ \alpha_2 $ $ \alpha_3 $ $ \alpha_4 $ $ \alpha_5 $ $ \alpha_6 $ $ \alpha_7 $ $ \alpha_8 $ $ \alpha_9 $ $ \alpha_{10} $ $ \alpha_{11} $ $ \alpha_{12} $ $ \alpha_{13} $
$ 0 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$ 2 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$ 3 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $
$ 4 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $
$ 5 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 0 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $
$ 6 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $
$ 7 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $
$ 8 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $
$ 9 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$ 10 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $
$ 11 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $
$ 12 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $
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$ \alpha_1 $ $ \alpha_2 $ $ \alpha_3 $ $ \alpha_4 $ $ \alpha_5 $ $ \alpha_6 $ $ \alpha_7 $ $ \alpha_8 $ $ \alpha_9 $ $ \alpha_{10} $ $ \alpha_{11} $ $ \alpha_{12} $ $ \alpha_{13} $
$ 0 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$ 2 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$ 3 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $
$ 4 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $
$ 5 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 0 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $
$ 6 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $
$ 7 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $
$ 8 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $
$ 9 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$ 10 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $
$ 11 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $
$ 12 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $
Table 3.  Comparison of the nonlinearities
$ n $ $ 9 $ 11 13 15 17 19 21
$ F(x) $ 186 772 3172 12952 52666 213524 863820
[3] 232 980 3988 16212 65210 261428 1046552
[17] $ - $ 802 3234 13078 52920 214034 864842
[21] $ - $ 810 3256 13130 53034 214274 865336
[20] $ - $ 784 3218 13096 53068 214568 866402
[6] $ - $ 794 3230 13098 53044 214486 866294
$ f $ in (13) 188 782 3208 13064 52988 214406 866160
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$ n $ $ 9 $ 11 13 15 17 19 21
$ F(x) $ 186 772 3172 12952 52666 213524 863820
[3] 232 980 3988 16212 65210 261428 1046552
[17] $ - $ 802 3234 13078 52920 214034 864842
[21] $ - $ 810 3256 13130 53034 214274 865336
[20] $ - $ 784 3218 13096 53068 214568 866402
[6] $ - $ 794 3230 13098 53044 214486 866294
$ f $ in (13) 188 782 3208 13064 52988 214406 866160
Table 4.  Comparison of the nonlinearities
$ n $ 27 37 47 57
$ F(x) $ 56708264 59644341436 62135313450064 64408903437167496
[17] 56716454 59644603578 62135321838670 64408903705602950
[21] 56720526 59644734616 62135326032930 64408903839820624
[20] 56741060 59646045410 62135388947584 64408906524175298
[6] 56748298 59648002864 62135605652036 64408924613659456
$ f $ in (13) 56747394 59647951550 62135614817362 64408926590774154
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$ n $ 27 37 47 57
$ F(x) $ 56708264 59644341436 62135313450064 64408903437167496
[17] 56716454 59644603578 62135321838670 64408903705602950
[21] 56720526 59644734616 62135326032930 64408903839820624
[20] 56741060 59646045410 62135388947584 64408906524175298
[6] 56748298 59648002864 62135605652036 64408924613659456
$ f $ in (13) 56747394 59647951550 62135614817362 64408926590774154
Table 5.  Comparison of the nonlinearity biases
$ n $ 9 11 13 15 17 19 21 27 37 47 57
$ F(x) $ 0.273 0.246 0.226 0.209 0.1964 0.1855 0.1762 0.15498 0.132061 0.11700409 0.10614691
[3] 0.094 0.043 0.026 0.010 0.0050 0.0027 0.0019 $ - $ $ - $ $ - $ $ - $
[17] $ - $ 0.217 0.210 0.202 0.1925 0.1835 0.1752 0.15486 0.132057 0.11700397 0.10614690
[21] $ - $ 0.209 0.205 0.199 0.1908 0.1826 0.1748 0.15480 0.132055 0.11700391 0.10614690
[20] $ - $ 0.234 0.214 0.201 0.1902 0.1815 0.1737 0.15449 0.132036 0.11700302 0.10614686
[6] $ - $ 0.224 0.211 0.201 0.1906 0.1818 0.1738 0.15438 0.132007 0.11699994 0.10614661
$ f $ in (13) 0.266 0.236 0.217 0.203 0.1915 0.1821 0.1740 0.15440 0.132008 0.11699981 0.10614658
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$ n $ 9 11 13 15 17 19 21 27 37 47 57
$ F(x) $ 0.273 0.246 0.226 0.209 0.1964 0.1855 0.1762 0.15498 0.132061 0.11700409 0.10614691
[3] 0.094 0.043 0.026 0.010 0.0050 0.0027 0.0019 $ - $ $ - $ $ - $ $ - $
[17] $ - $ 0.217 0.210 0.202 0.1925 0.1835 0.1752 0.15486 0.132057 0.11700397 0.10614690
[21] $ - $ 0.209 0.205 0.199 0.1908 0.1826 0.1748 0.15480 0.132055 0.11700391 0.10614690
[20] $ - $ 0.234 0.214 0.201 0.1902 0.1815 0.1737 0.15449 0.132036 0.11700302 0.10614686
[6] $ - $ 0.224 0.211 0.201 0.1906 0.1818 0.1738 0.15438 0.132007 0.11699994 0.10614661
$ f $ in (13) 0.266 0.236 0.217 0.203 0.1915 0.1821 0.1740 0.15440 0.132008 0.11699981 0.10614658
Table 6.  Comparison of the fast algebraic immunities
$ n $ $ 9 $ 11 13 15
[3] 8 10 12 14
[21] $ - $ 10 12 14
[20] $ - $ 10 12 13
[6] $ - $ 10 12 14
$ f $ in (13) 6 8 10 10
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$ n $ $ 9 $ 11 13 15
[3] 8 10 12 14
[21] $ - $ 10 12 14
[20] $ - $ 10 12 13
[6] $ - $ 10 12 14
$ f $ in (13) 6 8 10 10
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