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doi: 10.3934/amc.2020100
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Binary sequences derived from differences of consecutive quadratic residues

Arne Winterhof 1,, and  Zibi Xiao 2,

1. 

Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstr. 69, 4040 Linz, Austria
2. 

College of Science, Wuhan University of Science and Technology, Wuhan 430081, Hubei, China

* Corresponding author: Arne Winterhof

Received  March 2020 Revised  May 2020 Early access July 2020

Fund Project: The first author is partially supported by the Austrian Science Fund FWF Project P 30405-N32. The second author is supported by the Chinese Scholarship Council

For a prime $ p\ge 5 $ let $ q_0,q_1,\ldots,q_{(p-3)/2} $ be the quadratic residues modulo $ p $ in increasing order. We study two $ (p-3)/2 $-periodic binary sequences $ (d_n) $ and $ (t_n) $ defined by $ d_n = q_n+q_{n+1}\bmod 2 $ and $ t_n = 1 $ if $ q_{n+1} = q_n+1 $ and $ t_n = 0 $ otherwise, $ n = 0,1,\ldots,(p-5)/2 $. For both sequences we find some sufficient conditions for attaining the maximal linear complexity $ (p-3)/2 $.

Studying the linear complexity of $ (d_n) $ was motivated by heuristics of Caragiu et al. However, $ (d_n) $ is not balanced and we show that a period of $ (d_n) $ contains about $ 1/3 $ zeros and $ 2/3 $ ones if $ p $ is sufficiently large. In contrast, $ (t_n) $ is not only essentially balanced but also all longer patterns of length $ s $ appear essentially equally often in the vector sequence $ (t_n,t_{n+1},\ldots,t_{n+s-1}) $, $ n = 0,1,\ldots,(p-5)/2 $, for any fixed $ s $ and sufficiently large $ p $.

Keywords: Quadratic residues, sequences, linear complexity, pattern distribution, maximum order complexity.
Mathematics Subject Classification: Primary:94A55;Secondary:11T71.
Citation: Arne Winterhof, Zibi Xiao. Binary sequences derived from differences of consecutive quadratic residues. Advances in Mathematics of Communications, doi: 10.3934/amc.2020100
References:
[1]

N. Alon, Y Kohayakawa, C. Mauduit, C. G. Moreira and V. Rödl, Measures of pseudorandomness for finite sequences: typical values, Proc. Lond. Math. Soc., 95 (2007), 778-812.  doi: 10.1112/plms/pdm027.  Google Scholar

[2]

M. Caragiu, S. Tefft, A. Kemats and T. Maenle, A linear complexity analysis of quadratic residues and primitive roots spacings, Far East J. Math. Ed., 19 (2019), 27-37.  doi: 10.17654/ME019010027.  Google Scholar

[3]

T. W. Cusick, C. Ding and A. Renvall, Stream Ciphers and Number Theory, Elsevier Science B. V., Amsterdam, 2004.  Google Scholar

[4]

C. Ding, Pattern distributions of Legendre sequences, IEEE Trans. Inform. Theory, 44 (1998), 1693-1698.  doi: 10.1109/18.681353.  Google Scholar

[5]

O. Geil, F. Özbudak 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

[6]

L. Işık and A. Winterhof, Maximum-order complexity and correlation measures, Cryptography, 1 (2017), 1-7.   Google Scholar

[7]

C. J. A. Jansen, Investigations on Nonlinear Streamcipher Systems: Construction and Evaluation Methods, Ph.D thesis, Delft University of Technology, the Netherlands, 1989.  Google Scholar

[8]

Y. Luo, C. Xing and L. You, Construction of sequences with high nonlinear complexity from function fields, IEEE Trans. Inform. Theory, 63 (2017), 7646-7650.  doi: 10.1109/TIT.2017.2736545.  Google Scholar

[9]

C. Mauduit and A. Sárközy, On finite pseudorandom sequences of $k$ symbols, Indag. Math., 13 (2002), 89-101.  doi: 10.1016/S0019-3577(02)90008-X.  Google Scholar

[10] W. Meidl and A. Winterhof, Linear complexity of sequences and multisequences, In Handbook of Finite Fields, CRC Press, 2013.   Google Scholar
[11]

H. Niederreiter, Linear complexity and related complexity measures for sequences, In Progress in Cryptology-INDOCRYPT 2003, Lecture Notes in Comput. Sci., volume 2904, Springer, Berlin, 2003, 1-17. doi: 10.1007/978-3-540-24582-7_1.  Google Scholar

[12]

J. Peng, X. Zeng and Z. Sun, Finite length sequences with large nonlinear complexity, Adv. Math. Commun., 12 (2018), 215-230.  doi: 10.3934/amc.2018015.  Google Scholar

[13]

Z. Sun and A. Winterhof, On the maximum order complexity of the Thue-Morse and Rudin-Shapiro sequence, Unif. Distr. Th., 14 (2019), 33-42.   Google Scholar

[14]

Z. Sun and A. Winterhof, On the maximum order complexity of subsequences of the Thue-Morse and Rudin-Shapiro sequence along squares, Int. J. Comput. Math. Comput. Syst. Theory, 4 (2019), 30-36.  doi: 10.1080/23799927.2019.1566275.  Google Scholar

[15]

Z. Sun, X. Zeng, C. Li and T. Helleseth, Investigations on periodic sequences with maximum nonlinear complexity, IEEE Trans. Inform. Theory, 63 (2017), 6188-6198.  doi: 10.1109/TIT.2017.2714681.  Google Scholar

[16]

A. Topuzoǧlu glu and A. Winterhof, Pseudorandom sequences, in Topics in Geometry, Coding Theory and Cryptography, Algebr. Appl., volume 6, Springer, Dordrecht, (2007), 135-166. doi: 10.1007/1-4020-5334-4_4.  Google Scholar

[17]

A. Winterhof, Linear complexity and related complexity measures, in Selected Topics in Information and Coding Theory, Ser. Coding Theory Cryptol., volume 7, World Sci. Publ., Hackensack, NJ, (2010), 3-40. doi: 10.1142/9789812837172_0001.  Google Scholar

[18]

Z. Xiao, X. Zeng, C. Li and Y. Jiang, Binary sequences with period $N$ and nonlinear complexity $N-2$, Cryptogr. Commun., 11 (2019), 735-757.  doi: 10.1007/s12095-018-0324-3.  Google Scholar

show all references

References:
[1]

N. Alon, Y Kohayakawa, C. Mauduit, C. G. Moreira and V. Rödl, Measures of pseudorandomness for finite sequences: typical values, Proc. Lond. Math. Soc., 95 (2007), 778-812.  doi: 10.1112/plms/pdm027.  Google Scholar

[2]

M. Caragiu, S. Tefft, A. Kemats and T. Maenle, A linear complexity analysis of quadratic residues and primitive roots spacings, Far East J. Math. Ed., 19 (2019), 27-37.  doi: 10.17654/ME019010027.  Google Scholar

[3]

T. W. Cusick, C. Ding and A. Renvall, Stream Ciphers and Number Theory, Elsevier Science B. V., Amsterdam, 2004.  Google Scholar

[4]

C. Ding, Pattern distributions of Legendre sequences, IEEE Trans. Inform. Theory, 44 (1998), 1693-1698.  doi: 10.1109/18.681353.  Google Scholar

[5]

O. Geil, F. Özbudak 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

[6]

L. Işık and A. Winterhof, Maximum-order complexity and correlation measures, Cryptography, 1 (2017), 1-7.   Google Scholar

[7]

C. J. A. Jansen, Investigations on Nonlinear Streamcipher Systems: Construction and Evaluation Methods, Ph.D thesis, Delft University of Technology, the Netherlands, 1989.  Google Scholar

[8]

Y. Luo, C. Xing and L. You, Construction of sequences with high nonlinear complexity from function fields, IEEE Trans. Inform. Theory, 63 (2017), 7646-7650.  doi: 10.1109/TIT.2017.2736545.  Google Scholar

[9]

C. Mauduit and A. Sárközy, On finite pseudorandom sequences of $k$ symbols, Indag. Math., 13 (2002), 89-101.  doi: 10.1016/S0019-3577(02)90008-X.  Google Scholar

[10] W. Meidl and A. Winterhof, Linear complexity of sequences and multisequences, In Handbook of Finite Fields, CRC Press, 2013.   Google Scholar
[11]

H. Niederreiter, Linear complexity and related complexity measures for sequences, In Progress in Cryptology-INDOCRYPT 2003, Lecture Notes in Comput. Sci., volume 2904, Springer, Berlin, 2003, 1-17. doi: 10.1007/978-3-540-24582-7_1.  Google Scholar

[12]

J. Peng, X. Zeng and Z. Sun, Finite length sequences with large nonlinear complexity, Adv. Math. Commun., 12 (2018), 215-230.  doi: 10.3934/amc.2018015.  Google Scholar

[13]

Z. Sun and A. Winterhof, On the maximum order complexity of the Thue-Morse and Rudin-Shapiro sequence, Unif. Distr. Th., 14 (2019), 33-42.   Google Scholar

[14]

Z. Sun and A. Winterhof, On the maximum order complexity of subsequences of the Thue-Morse and Rudin-Shapiro sequence along squares, Int. J. Comput. Math. Comput. Syst. Theory, 4 (2019), 30-36.  doi: 10.1080/23799927.2019.1566275.  Google Scholar

[15]

Z. Sun, X. Zeng, C. Li and T. Helleseth, Investigations on periodic sequences with maximum nonlinear complexity, IEEE Trans. Inform. Theory, 63 (2017), 6188-6198.  doi: 10.1109/TIT.2017.2714681.  Google Scholar

[16]

A. Topuzoǧlu glu and A. Winterhof, Pseudorandom sequences, in Topics in Geometry, Coding Theory and Cryptography, Algebr. Appl., volume 6, Springer, Dordrecht, (2007), 135-166. doi: 10.1007/1-4020-5334-4_4.  Google Scholar

[17]

A. Winterhof, Linear complexity and related complexity measures, in Selected Topics in Information and Coding Theory, Ser. Coding Theory Cryptol., volume 7, World Sci. Publ., Hackensack, NJ, (2010), 3-40. doi: 10.1142/9789812837172_0001.  Google Scholar

[18]

Z. Xiao, X. Zeng, C. Li and Y. Jiang, Binary sequences with period $N$ and nonlinear complexity $N-2$, Cryptogr. Commun., 11 (2019), 735-757.  doi: 10.1007/s12095-018-0324-3.  Google Scholar

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