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February  2018, 12(1): 215-230. doi: 10.3934/amc.2018015

Finite length sequences with large nonlinear complexity

Jiarong Peng , Xiangyong Zeng and  Zhimin Sun ,

The Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan, Hubei 430062, China

* Corresponding author: Zhimin Sun

Received  May 2017 Revised  November 2017 Published  March 2018

Fund Project: The authors were supported by the National Natural Science Foundation of China Grant (No. 61472120). X. Zeng and Z. Sun were also granted by National Natural Science Foundation of Hubei Province of China (No. 2017CFB143) and China Scholarship Council, respectively

Finite length sequences with large nonlinear complexity over $\mathbb{Z}_{p}\, (p≥ 2)$ are investigated in this paper. We characterize all $p$-ary sequences of length $n$ having nonlinear complexity $n-j$ for $j=2, 3$, where $n$ is an integer satisfying $n≥ 2j$. For $n≥ 8$, all binary sequences of length $n$ with nonlinear complexity $n-4$ are obtained. Furthermore, the numbers and $k$-error nonlinear complexity of these sequences are completely determined, respectively.

Keywords: Binary sequences, p-ary sequences, nonlinear complexity, k-error nonlinear complexity, finite length sequences.
Mathematics Subject Classification: Primary: 94A55; Secondary: 94A60.
Citation: Jiarong Peng, Xiangyong Zeng, Zhimin Sun. Finite length sequences with large nonlinear complexity. Advances in Mathematics of Communications, 2018, 12 (1) : 215-230. doi: 10.3934/amc.2018015
References:
[1]

C. J. A. Jansen, Investigations on Nonlinear Streamcipher Systems: Construction and Evaluation Methods, Ph. D thesis, Technical Univ. Delft, 1989. Google Scholar

[2]

C. J. A. Jansen and D. E. Boekee, The shortest feedback shift register that can generate a given sequence, in Adv. Crypt. -CRYPTO'89, 1990, 90–99. Google Scholar

[3]

N. Li and X. Tang, On the linear complexity of binary sequences of period $4N$ with optimal autocorrelation value/magnitude, IEEE Trans. Inf. Theory, 57 (2011), 7597-7604. Google Scholar

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F. T. Macwilliams and N. J. A. Sloane, Psedo-Random sequences and arrays, Proc. IEEE, 64 (1976), 1715-1729. Google Scholar

[5]

W. Meidl and A. Winterhof, On the linear complexity profile of some new explicit inverse pseudorandom numbers, J. Complexity, 20 (2004), 350-355. Google Scholar

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H. Niederreite, Random Number Generation and Quasi-Monte Carlo Methods, Soc. Ind. Appl. Math., Philadelphia, 1992. Google Scholar

[7]

H. Niederreiter, Linear complexity and related complexity measures for sequences, in Progr. Crypt. -INDOCRYPT 2003, 2003, 1–17. Google Scholar

[8]

P. Rizomiliotis, Constructing periodic binary sequences of maximum nonlinear span, IEEE Trans. Inf. Theory, 52 (2006), 4257-4261. Google Scholar

[9]

P. Rizomiliotis and N. Kalouptsidis, Results on the nonlinear span of binary sequences, IEEE Trans. Inf. Theory, 51 (2005), 1555-1563. Google Scholar

[10]

Z. Sun, X. Zeng, C. Li and T. Helleseth, Investigations on periodic sequences with maximum nonlinear complexity, IEEE Trans. Inf. Theory, 63 (2017), 6188-6198. Google Scholar

show all references

References:
[1]

C. J. A. Jansen, Investigations on Nonlinear Streamcipher Systems: Construction and Evaluation Methods, Ph. D thesis, Technical Univ. Delft, 1989. Google Scholar

[2]

C. J. A. Jansen and D. E. Boekee, The shortest feedback shift register that can generate a given sequence, in Adv. Crypt. -CRYPTO'89, 1990, 90–99. Google Scholar

[3]

N. Li and X. Tang, On the linear complexity of binary sequences of period $4N$ with optimal autocorrelation value/magnitude, IEEE Trans. Inf. Theory, 57 (2011), 7597-7604. Google Scholar

[4]

F. T. Macwilliams and N. J. A. Sloane, Psedo-Random sequences and arrays, Proc. IEEE, 64 (1976), 1715-1729. Google Scholar

[5]

W. Meidl and A. Winterhof, On the linear complexity profile of some new explicit inverse pseudorandom numbers, J. Complexity, 20 (2004), 350-355. Google Scholar

[6]

H. Niederreite, Random Number Generation and Quasi-Monte Carlo Methods, Soc. Ind. Appl. Math., Philadelphia, 1992. Google Scholar

[7]

H. Niederreiter, Linear complexity and related complexity measures for sequences, in Progr. Crypt. -INDOCRYPT 2003, 2003, 1–17. Google Scholar

[8]

P. Rizomiliotis, Constructing periodic binary sequences of maximum nonlinear span, IEEE Trans. Inf. Theory, 52 (2006), 4257-4261. Google Scholar

[9]

P. Rizomiliotis and N. Kalouptsidis, Results on the nonlinear span of binary sequences, IEEE Trans. Inf. Theory, 51 (2005), 1555-1563. Google Scholar

[10]

Z. Sun, X. Zeng, C. Li and T. Helleseth, Investigations on periodic sequences with maximum nonlinear complexity, IEEE Trans. Inf. Theory, 63 (2017), 6188-6198. Google Scholar

Table 1.  Binary sequences of length 11 with nonlinear complexity 7 and their distribution
$Set$Sequences # Seq.
$K_1$00000001000, 11111110111, 00000001001, 11111110011, 00000001101
11111110110, 00000001010, 11111110101, 00000001011, 00000001100
11111110100, 11111110010, 00000001110, 11111110001, 00000001111
11111110000, 01010101100, 10101010011, 01010101101, 10101010010
01010101110, 10101010001, 01010101111, 10101010000, 01111111000
10000000111, 01111111001, 10000000110, 01111111010, 10000000101
01111111011, 10000000100, 00100100110, 11011011001, 00100100111
11011011000, 00101010110, 11010101001, 00101010111, 11010101000
00111111100, 11000000011, 00111111101, 11000000010, 01001001010
10110110101, 01001001011, 10110110100, 01000000010, 10111111101
01000000011, 10111111100, 01101101110, 10010010001, 01101101111
10010010000		56
$K_2$00110011000, 11001100111, 00110110111, 11001001000, 01100110010
10011001101, 01100000001, 10011111110		8
$K_3$00010001001, 11101110110, 00010010011, 11101101100, 00010101011
11101010100, 00011111110, 11100000001, 00100000001, 11011111110
00100010000, 11011101111, 01000100011, 10111011100, 01011011010
10100100101, 01011111110, 10100000001, 01101010100, 10010101011
01110111010, 10001000101		22
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$Set$Sequences # Seq.
$K_1$00000001000, 11111110111, 00000001001, 11111110011, 00000001101
11111110110, 00000001010, 11111110101, 00000001011, 00000001100
11111110100, 11111110010, 00000001110, 11111110001, 00000001111
11111110000, 01010101100, 10101010011, 01010101101, 10101010010
01010101110, 10101010001, 01010101111, 10101010000, 01111111000
10000000111, 01111111001, 10000000110, 01111111010, 10000000101
01111111011, 10000000100, 00100100110, 11011011001, 00100100111
11011011000, 00101010110, 11010101001, 00101010111, 11010101000
00111111100, 11000000011, 00111111101, 11000000010, 01001001010
10110110101, 01001001011, 10110110100, 01000000010, 10111111101
01000000011, 10111111100, 01101101110, 10010010001, 01101101111
10010010000		56
$K_2$00110011000, 11001100111, 00110110111, 11001001000, 01100110010
10011001101, 01100000001, 10011111110		8
$K_3$00010001001, 11101110110, 00010010011, 11101101100, 00010101011
11101010100, 00011111110, 11100000001, 00100000001, 11011111110
00100010000, 11011101111, 01000100011, 10111011100, 01011011010
10100100101, 01011111110, 10100000001, 01101010100, 10010101011
01110111010, 10001000101		22
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