# The exact autocorrelation distribution and 2-adic complexity of a class of binary sequences with almost optimal autocorrelation

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## Abstract

Pseudo-random sequences with good statistical properties, such as low autocorrelation, high linear complexity and large 2-adic complexity, have been used in designing reliable stream ciphers. In this paper, we obtain the exact autocorrelation distribution of a class of binary sequences with three-level autocorrelation and analyze the 2-adic complexity of this class of sequences. Our results show that the 2-adic complexity of such a binary sequence with period *N* is at least (*N* + 1) − log_{2} (*N* + 1). We further show that it is maximal for infinitely many cases. This indicates that the 2-adic complexity of this class of sequences is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).

## Keywords

Stream ciphers Pseudo-random sequences Autocorrelation 2-adic complexity## Mathematics Subject Classification (2010)

94A55 94A60 65C10## Notes

### Acknowledgments

The authors wishes to thank the editor and the reviewers for the valuable comments, which make our work greatly improved.

Parts of this work were written during a very pleasant visit of the first author to Carleton University in Ottawa, Canada. She wishes to thank the hosts for their hospitality.

## References

- 1.Cai, Y., Ding, C.: Binary sequences with optimal autocorrelation. Theor. Comput. Sci.
**410**, 2316–2322 (2009)MathSciNetCrossRefMATHGoogle Scholar - 2.Ding, C., Helleseth, T., Lam, K.Y.: Several classes of sequences with three-level autocorrelation. IEEE Trans. Inform. Theory
**45**, 2606–2612 (1999)MathSciNetCrossRefMATHGoogle Scholar - 3.Ding, C., Helleseth, T., Shan, W.: On the linear complexity of Legendre sequences. IEEE Trans. Inform. Theory
**45**, 693–698 (1998)MathSciNetMATHGoogle Scholar - 4.Edemskiy, V., Palvinskiy, A.: The linear complexity of binary sequences of length 2
*p*with optimal three-level autocorrelation. Inf. Process. Lett.**116**, 153–156 (2016)MathSciNetCrossRefMATHGoogle Scholar - 5.Etzion, T.: Linear complexity of de Bruijn sequences-old and new results. IEEE Trans. Inform. Theory
**45**, 693–698 (1999)MathSciNetCrossRefMATHGoogle Scholar - 6.Helleseth, T., Maas, M., Mathiassen, E., Segers, T.: Linear complexity over 𝔽,
*p*of Sidel’nikov sequences. IEEE Trans. Inform. Theory**50**, 2468–2472 (2004)MathSciNetCrossRefMATHGoogle Scholar - 7.Hu, H.: Comments on a new method to compute the 2-adic complexity of binary sequences. IEEE Trans. Inform. Theory
**60**, 5803–5804 (2014)MathSciNetCrossRefMATHGoogle Scholar - 8.Hu, L., Yue, Q., Wang, M.: The linear complexity of whiteman’s generalized cyclotomic sequences of period
*p*^{m+1},*q*^{n+1}. IEEE Trans. Inform. Theory**58**, 5534–5543 (2012)MathSciNetCrossRefMATHGoogle Scholar - 9.Kim, Y., Jang, J.W., Kim, S.H., No, J.S.: Linear complexity of quaternary sequences constructed from binary Legendre sequences. Int. Symp. Inf. Theory Appl., 611–614 (2012)Google Scholar
- 10.Klapper, A.: D-form sequences: Families of sequences with low correlation values and large linear spans. IEEE Trans. Inform. Theory
**41**, 423–431 (1995)MathSciNetCrossRefMATHGoogle Scholar - 11.Klapper, A., Chan, A.H., Goresky, M.: Cascaded GMW sequences. IEEE Trans. Inform. Theory
**39**, 177–183 (1995)CrossRefMATHGoogle Scholar - 12.Klapper, A., Goresky, M.: Feedback shift registers, 2-adic span, and combiners with memory. J. Cryptol.
**10**, 111–147 (1997)MathSciNetCrossRefMATHGoogle Scholar - 13.Kumanduri, R., Romero, C.: Number Theory with Computer Applications. Prentice Hall, New Jersey (1998)MATHGoogle Scholar
- 14.Li, N., Tang, X.: On the linear complexity of binary sequences of period 4
*N*with optimal autocorrelation/magnitude. IEEE Trans. Inform. Theory**57**, 7597–7604 (2011)MathSciNetCrossRefMATHGoogle Scholar - 15.Massey, J.L.: Shift-register synthesis and BCH decoding. IEEE Trans. Inform. Theory
**15**, 122–127 (1969)MathSciNetCrossRefMATHGoogle Scholar - 16.No, J.S.: New cyclic difference sets with singer parameters constructed from
*d*-Homogeneous functions. Des. Codes Cryptogr.**33**, 199–213 (2004)MathSciNetCrossRefMATHGoogle Scholar - 17.No, J.S.:
*p*-ary unified sequences:*p*-ary extended*d*-form sequences with the ideal autocorrelation property. IEEE Trans. Inform. Theory**48**, 2540–2546 (2002)MathSciNetCrossRefMATHGoogle Scholar - 18.No, J.S., Chung, H., Yun, M.S.: Binary pseudorandom sequences of period 2
^{m}− 1 with ideal autocorrelation generated by the polynomial*z*^{d}+ (*z*+ 1)^{d}. IEEE Trans. Inform. Theory**44**, 1278–1282 (1998)MathSciNetCrossRefMATHGoogle Scholar - 19.Scholtz, R.A., Welch, L.R.: GMW Sequences. IEEE Trans. Inform. Theory
**30**, 548–553 (1984)MathSciNetCrossRefMATHGoogle Scholar - 20.Sun, Y., Yan, T., Li, H.: The linear complexity of a class of binary sequences with three-level autocorrelation. IEICE Trans. Fundam.
**E96-A**, 1586–1592 (2013)CrossRefGoogle Scholar - 21.Tang, X., Ding, C.: New classes of balanced quaternary and almost balanced binary sequences with optimal autocorrelation Value. IEEE Trans. Inform. Theory
**56**, 6398–6405 (2010)MathSciNetCrossRefMATHGoogle Scholar - 22.Tang, X., Fan, P., Matsufuji, S.: Lower bounds on the maximum correlation of sequences with low or zero correlation zone. Electron. Lett.
**36**, 551–552 (2000)CrossRefGoogle Scholar - 23.Tian, T., Qi, W.: 2-Adic complexity of binary
*m*-sequences. IEEE Trans. Inform. Theory**56**, 450–454 (2010)MathSciNetCrossRefMATHGoogle Scholar - 24.Wang, Q.: The linear complexity of some binary sequences with three-level autocorrelation. IEEE Trans. Inform. Theory
**56**, 4046–4052 (2010)MathSciNetCrossRefMATHGoogle Scholar - 25.Wang, Q., Du, X.: The linear complexity of binary sequences with optimal autocorrelation. IEEE Trans. Inform. Theory
**56**, 6388–6397 (2010)MathSciNetCrossRefMATHGoogle Scholar - 26.Wang, Q., Jiang, Y., Lin, D.: Linear complexity of binary generalized cyclotomic sequences over GF(
*q*). J. Complex.**31**, 731–740 (2015)MathSciNetCrossRefMATHGoogle Scholar - 27.Xiong, H., Qu, L., Li, C., Fu, S.: Linear complexity of binary sequences with interleaved structure. IET Commun.
**7**, 1688–1696 (2013)CrossRefGoogle Scholar - 28.Xiong, H., Qu, L., Li, C.: A new method to compute the 2-adic complexity of binary sequences. IEEE Trans. Inform. Theory
**60**, 2399–2406 (2014)MathSciNetCrossRefMATHGoogle Scholar - 29.Xiong, H., Qu, L., Li, C.: 2-Adic complexity of binary sequences with interleaved structure. Finite Fields Appl.
**33**, 14–28 (2015)MathSciNetCrossRefMATHGoogle Scholar - 30.Yan, T., Du, X., Sun, Y., Xiao, G.: Construction of
*d*-form sequences with ideal autocorrelation. IEICE Trans. Fundam.**94**, 1696–1700 (2011)CrossRefGoogle Scholar - 31.Yan, T., Xiao, G.: Divisible difference sets, relative difference sets and sequences with ideal autocorrelation. Inform. Sci.
**249**, 143–147 (2013)MathSciNetCrossRefMATHGoogle Scholar - 32.Zhou, Z., Tang, X., Gong, G.: A new classes of sequences with zero or low correlation zone based on interleaving technique. IEEE Trans. Inform. Theory
**54**, 4267–4273 (2008)MathSciNetCrossRefMATHGoogle Scholar