    February  2022, 16(1): 83-93. doi: 10.3934/amc.2020100

## Binary sequences derived from differences of consecutive quadratic residues

 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 Published  February 2022 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 $p$.

Citation: Arne Winterhof, Zibi Xiao. Binary sequences derived from differences of consecutive quadratic residues. Advances in Mathematics of Communications, 2022, 16 (1) : 83-93. doi: 10.3934/amc.2020100
##### References:
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##### References:
  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.   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.  T. W. Cusick, C. Ding and A. Renvall, Stream Ciphers and Number Theory, Elsevier Science B. V., Amsterdam, 2004.  C. Ding, Pattern distributions of Legendre sequences, IEEE Trans. Inform. Theory, 44 (1998), 1693-1698.  doi: 10.1109/18.681353.   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.   L. Işık and A. Winterhof, Maximum-order complexity and correlation measures, Cryptography, 1 (2017), 1-7. C. J. A. Jansen, Investigations on Nonlinear Streamcipher Systems: Construction and Evaluation Methods, Ph.D thesis, Delft University of Technology, the Netherlands, 1989.  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.   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.   W. Meidl and A. Winterhof, Linear complexity of sequences and multisequences, In Handbook of Finite Fields, CRC Press, 2013. 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.   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.   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. 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.   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.   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.   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.   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.   Liqin Hu, Qin Yue, Fengmei Liu. Linear complexity of cyclotomic sequences of order six and BCH codes over GF(3). Advances in Mathematics of Communications, 2014, 8 (3) : 297-312. doi: 10.3934/amc.2014.8.297  Jianqin Zhou, Wanquan Liu, Xifeng Wang. Complete characterization of the first descent point distribution for the k-error linear complexity of 2n-periodic binary sequences. Advances in Mathematics of Communications, 2017, 11 (3) : 429-444. doi: 10.3934/amc.2017036  Fuqing Sun, Qin Yue, Xia Li. on the 2-adic complexity of cyclotomic binary sequences of order three. 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