# American Institute of Mathematical Sciences

doi: 10.3934/amc.2020091

## On finite length nonbinary sequences with large nonlinear complexity over the residue ring $\mathbb{Z}_{m}$

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

* Corresponding author: Xiangyong Zeng

Received  September 2019 Revised  March 2020 Published  July 2020

Fund Project: L. Yi and Z. Sun were supported by the Natural Science Foundation of Hubei province of China (2019CFB543). X. Zeng was supported by Major Technological Innovation Special Project of Hubei Province (No. 2019ACA144)

In this paper, we characterize all nonbinary sequences of length $n$ with nonlinear complexity $n-4$ for $n\geq9$ and establish a formula on the number of such sequences. More generally, we characterize other finite length nonbinary sequences with large nonlinear complexity over $\mathbb{Z}_{m}$.

Citation: Lin Yi, Xiangyong Zeng, Zhimin Sun. On finite length nonbinary sequences with large nonlinear complexity over the residue ring $\mathbb{Z}_{m}$. Advances in Mathematics of Communications, doi: 10.3934/amc.2020091
##### References:
 [1] C. Ding, Linear complexity of generalized cyclotomic binary sequences of order 2, Finite Fields Appl., 3 (1997), 159-174.  doi: 10.1006/ffta.1997.0181.  Google Scholar [2] O. Geil, O. Ferruh and D. Ruano, Constructing sequences with high nonlinear complexity using the Weierstrass semigroup of a pair of distinct points of a hermitian curve, Semigroup Forum, 98 (2019), 543-555.  doi: 10.1007/s00233-018-9976-8.  Google Scholar [3] C. J. A. Jansen, Investigations on Nonlinear Streamciper Systems: Construction and Evaluation Methods, Ph.D thesis, Dept. Elect. Eng., TU Delft, Delft, The Netherlands, 1989.  Google Scholar [4] C. J. A. Jansen and D. E. Boekee, The shortest feedback shift register that can generate a given sequence, in Advanced in Cryptography - CRYPTO'89, LNCS 435 (1990), 90–99. doi: 10.1007/0-387-34805-0_10.  Google Scholar [5] 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.  doi: 10.1109/TIT.2011.2159575.  Google Scholar [6] K. Limniotis, N. Kolokotronis and N. Kalouptsidis, On the nonlinear complexity and lempel-Ziv complexity of finite length sequences, IEEE Trans. Inf. Theory, 53 (2007), 4293-4302.  doi: 10.1109/TIT.2007.907442.  Google Scholar [7] Y. Luo, C. Xing and L. You, Construction of sequences with high nonlinear complexity from function fields, IEEE Trans. Inf. Theory, 63 (2017), 7646-7650.  doi: 10.1109/TIT.2017.2736545.  Google Scholar [8] J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Inf. Theory, 15 (1969), 122-127.  doi: 10.1109/tit.1969.1054260.  Google Scholar [9] W. Meidl and A. Winterhof, On the linear complexity profile of some new explicit inversive pseudorandom numbers, J. Complexity, 20 (2004), 350-355.  doi: 10.1016/j.jco.2003.08.017.  Google Scholar [10] H. Niderreiter, Linear complexity and related complexity measures for sequences, Progress in Cryptology - INDOCRYPT 2003, LNCS 2904 (2003), 1–17. doi: 10.1007/978-3-540-24582-7_1.  Google Scholar [11] H. Niderreiter and C. Xing, Sequences with high nonlinear complexity, IEEE Trans. Inf. Theory, 60 (2014), 6696-6701.  doi: 10.1109/TIT.2014.2343225.  Google Scholar [12] J. Peng, X. Zeng and Z. Sun, Finite length sequences with large nonlinear complexity, Advance in Mathematics of Communication, 12 (2018), 215-230.  doi: 10.3934/amc.2018015.  Google Scholar [13] P. Rizomiliotis, Constructing periodic binary sequences with maximum nonlinear span, IEEE Trans. Inf. Theory, 52 (2006), 4257-4261.  doi: 10.1109/TIT.2006.880054.  Google Scholar [14] P. Rizomiliotis and N. Kalouptsidis, Results on the nonlinear span of binary sequences, IEEE Trans. Inf. Theory, 51 (2005), 1555-1563.  doi: 10.1109/TIT.2005.844090.  Google Scholar [15] 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.  doi: 10.1109/TIT.2017.2714681.  Google Scholar [16] Z. Xiao, X. Zeng, C. Li and Y. Jiang, Binary sequences with period $N$ and nonlinear complexity $N-2$, Cryptography and Communications, 11 (2019), 735-757.  doi: 10.1007/s12095-018-0324-3.  Google Scholar

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##### References:
 [1] C. Ding, Linear complexity of generalized cyclotomic binary sequences of order 2, Finite Fields Appl., 3 (1997), 159-174.  doi: 10.1006/ffta.1997.0181.  Google Scholar [2] O. Geil, O. Ferruh and D. Ruano, Constructing sequences with high nonlinear complexity using the Weierstrass semigroup of a pair of distinct points of a hermitian curve, Semigroup Forum, 98 (2019), 543-555.  doi: 10.1007/s00233-018-9976-8.  Google Scholar [3] C. J. A. Jansen, Investigations on Nonlinear Streamciper Systems: Construction and Evaluation Methods, Ph.D thesis, Dept. Elect. Eng., TU Delft, Delft, The Netherlands, 1989.  Google Scholar [4] C. J. A. Jansen and D. E. Boekee, The shortest feedback shift register that can generate a given sequence, in Advanced in Cryptography - CRYPTO'89, LNCS 435 (1990), 90–99. doi: 10.1007/0-387-34805-0_10.  Google Scholar [5] 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.  doi: 10.1109/TIT.2011.2159575.  Google Scholar [6] K. Limniotis, N. Kolokotronis and N. Kalouptsidis, On the nonlinear complexity and lempel-Ziv complexity of finite length sequences, IEEE Trans. Inf. Theory, 53 (2007), 4293-4302.  doi: 10.1109/TIT.2007.907442.  Google Scholar [7] Y. Luo, C. Xing and L. You, Construction of sequences with high nonlinear complexity from function fields, IEEE Trans. Inf. Theory, 63 (2017), 7646-7650.  doi: 10.1109/TIT.2017.2736545.  Google Scholar [8] J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Inf. Theory, 15 (1969), 122-127.  doi: 10.1109/tit.1969.1054260.  Google Scholar [9] W. Meidl and A. Winterhof, On the linear complexity profile of some new explicit inversive pseudorandom numbers, J. Complexity, 20 (2004), 350-355.  doi: 10.1016/j.jco.2003.08.017.  Google Scholar [10] H. Niderreiter, Linear complexity and related complexity measures for sequences, Progress in Cryptology - INDOCRYPT 2003, LNCS 2904 (2003), 1–17. doi: 10.1007/978-3-540-24582-7_1.  Google Scholar [11] H. Niderreiter and C. Xing, Sequences with high nonlinear complexity, IEEE Trans. Inf. Theory, 60 (2014), 6696-6701.  doi: 10.1109/TIT.2014.2343225.  Google Scholar [12] J. Peng, X. Zeng and Z. Sun, Finite length sequences with large nonlinear complexity, Advance in Mathematics of Communication, 12 (2018), 215-230.  doi: 10.3934/amc.2018015.  Google Scholar [13] P. Rizomiliotis, Constructing periodic binary sequences with maximum nonlinear span, IEEE Trans. Inf. Theory, 52 (2006), 4257-4261.  doi: 10.1109/TIT.2006.880054.  Google Scholar [14] P. Rizomiliotis and N. Kalouptsidis, Results on the nonlinear span of binary sequences, IEEE Trans. Inf. Theory, 51 (2005), 1555-1563.  doi: 10.1109/TIT.2005.844090.  Google Scholar [15] 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.  doi: 10.1109/TIT.2017.2714681.  Google Scholar [16] Z. Xiao, X. Zeng, C. Li and Y. Jiang, Binary sequences with period $N$ and nonlinear complexity $N-2$, Cryptography and Communications, 11 (2019), 735-757.  doi: 10.1007/s12095-018-0324-3.  Google Scholar
Binary sequences of length 13 with nonlinear complexity 9 and their distribution
 set Sequences $\#$ Seq. $E_3(A_{10})$ 0000000001000, 0000000001100, 0000000001010, 0000000001110, 0000000001001 0000000001101, 0000000001011, 0000000001111, 1111111110000, 1111111110100 16 1111111110010, 1111111110110, 1111111110001, 1111111110101, 1111111110011 1111111110111 $E_2(A_{11})$ 0101010101100, 0101010101110, 0101010101101, 0101010101111, 0111111111000 0111111111010, 0111111111001, 0111111111011, 1010101010000, 1010101010010 16 1010101010001, 1010101010011, 1000000000100, 1000000000110, 1000000000101 1000000000111 $E_1(A_{12})$ 0010010010000, 0010010010001, 0010101010110, 0010101010111, 0011111111100 0011111111101, 1101101101110, 1101101101111, 1101010101000, 1101010101001 24 1100000000010, 1100000000011, 0100100100110, 0100100100111, 0100000000010 0100000000011, 1011011011000, 1011011011001, 1011111111100, 1011111111101 0110110110100, 0110110110101, 1001001001010, 1001001001011 $B_1$ 0011001100111, 0011011011010, 1100110011000, 1100100100101, 0110011001101 8 0110000000001, 1001100110010, 1001111111110 $B_2$ 0001000100011, 0001001001000, 0001010101011, 0001111111110, 1110111011100 1110110110111, 1110101010100, 1110000000001, 0010001000101, 0010000000001 1101110111010, 1101111111110, 0100010001001, 1011101110110, 0101101101100 22 0101111111110, 1010010010011, 1010000000001, 0110101010100, 1001010101011 0111011101111, 1000100010000
 set Sequences $\#$ Seq. $E_3(A_{10})$ 0000000001000, 0000000001100, 0000000001010, 0000000001110, 0000000001001 0000000001101, 0000000001011, 0000000001111, 1111111110000, 1111111110100 16 1111111110010, 1111111110110, 1111111110001, 1111111110101, 1111111110011 1111111110111 $E_2(A_{11})$ 0101010101100, 0101010101110, 0101010101101, 0101010101111, 0111111111000 0111111111010, 0111111111001, 0111111111011, 1010101010000, 1010101010010 16 1010101010001, 1010101010011, 1000000000100, 1000000000110, 1000000000101 1000000000111 $E_1(A_{12})$ 0010010010000, 0010010010001, 0010101010110, 0010101010111, 0011111111100 0011111111101, 1101101101110, 1101101101111, 1101010101000, 1101010101001 24 1100000000010, 1100000000011, 0100100100110, 0100100100111, 0100000000010 0100000000011, 1011011011000, 1011011011001, 1011111111100, 1011111111101 0110110110100, 0110110110101, 1001001001010, 1001001001011 $B_1$ 0011001100111, 0011011011010, 1100110011000, 1100100100101, 0110011001101 8 0110000000001, 1001100110010, 1001111111110 $B_2$ 0001000100011, 0001001001000, 0001010101011, 0001111111110, 1110111011100 1110110110111, 1110101010100, 1110000000001, 0010001000101, 0010000000001 1101110111010, 1101111111110, 0100010001001, 1011101110110, 0101101101100 22 0101111111110, 1010010010011, 1010000000001, 0110101010100, 1001010101011 0111011101111, 1000100010000
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