# A further study of the linear complexity of new binary cyclotomic sequence of length \(p^r\)

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## Abstract

Recently, a conjecture on the linear complexity of a new class of generalized cyclotomic binary sequences of period \(p^r\) was proposed by Xiao et al. (Des Codes Cryptogr 86(7):1483–1497, 2018). Later, for the case *f* being the form \(2^a\) with \(a\ge 1\), Vladimir Edemskiy proved the conjecture (arXiv:1712.03947). In this paper, under the assumption of \(2^{p-1} \not \equiv 1 \bmod p^2\) and \(\gcd (\frac{p-1}{\mathrm{{ord}}_{p}(2)},f)=1\), the conjecture proposed by Xiao et al. is proved for a general *f* by using the Euler quotient. Actually, a generic construction of \(p^r\)-periodic binary sequences based on the generalized cyclotomy is introduced in this paper, which admits a flexible support set and contains Xiao’s construction as a special case, and then an efficient method to compute the linear complexity of the sequence by the generic construction is presented, based on which the conjecture proposed by Xiao et al. could be easily proved under the aforementioned assumption.

## Keywords

Linear complexity Generalized cyclotomy Binary sequence Euler quotient## Notes

### Acknowledgements

The authors would like to thank the reviewers and editors for their detailed and constructive comments, which substantially improved the presentation of the paper. This work was supported by the National Natural Science Foundation of China (No. 61772292, 61772476), Foundation of Fujian Educational Committee (No. JAT170627), Key Scientific Research Projects of Colleges and Universities in Henan Province (No. 18A110029) and Fujian Normal University Innovative Research Team (No. IRTL1207).

## References

- 1.Tomasevic, V., Bojanic, S., Niteo-Taladriz, O.: Finding an internal state of RC4 stream cipher. Inf. Sci.
**177**, 1715–1727 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Golomb, G., Gong, G.: Signal Designs with Good Correlations: For Wireless Communications, Cryptography and Radar Applications. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
- 3.Massey, J.L.: Shift register synthesis and BCH decoding. IEEE Trans. Inf. Theory
**15**(1), 122–127 (1969)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Massey, J.L., Schaub, T.: Linear complexity in coding theory. In: Cohen, G., Godlewski, P.H. (eds.) Coding Theory and Applications. Lecture Notes in Computer Science, vol. 311, pp. 19–32. Springer, Heidelberg (1988)CrossRefGoogle Scholar
- 5.Ding, C., Hellseth, T., Shan, W.: On the linear complexity of Legendre sequences. IEEE Trans. Inf. Theory
**44**(3), 1276–1278 (1998)MathSciNetCrossRefGoogle Scholar - 6.Zeng, X.Y., Cai, H., Tang, X.H., Yang, Y.: Optimal frequency hopping sequences of odd length. IEEE Trans. Inf. Theory
**59**(5), 3237–3248 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Xiao, Z.B., Zeng, X.Y., Li, C.L., Helleseth, T.: New generalized cyclotomic binary sequences of period \(p^2\). Des. Codes Cryptogr.
**86**(7), 1483–1497 (2018)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Edemskiy, V.: The linear complexity of new binary cyclotomic sequences of period \(p^n\). arXiv:1712.03947
- 9.Chen, Z.X., Niu, Z.H., Wu, C.H.: On the k-error linear complexity of binary sequences derived from polynomial quotients. Sci. China Inf. Sci.
**58**(9), 092107:1–092107:15 (2015)MathSciNetCrossRefGoogle Scholar - 10.Chen, Z.X., Du, X.N., Marzouk, R.: Trace representation of pseudorandom binary sequences derived from Euler quotients. Appl. Algebra Eng. Commun. Comput.
**26**(6), 555–570 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Ye, Z.F., Ke, P.H., Zhang, S.Y., Chang, Z.L.: Some notes on pseudorandom binary sequences derived from Fermat-Euler quotients. IEICE Trans. Fundam.
**98**(10), 2199–2202 (2015)CrossRefGoogle Scholar - 12.Ye, Z. F., Ke, P. H., Chen, Z. X.: Further results on pseudorandom binary sequences derived from Fermat-Euler quotients. In: 10th International Conference on Information, Communications and Signal Processing (ICICS), pp. 1–4 (2015)Google Scholar
- 13.Du, X.N., Wu, C.H., Wei, W.Y.: An extension of binary threshold sequence from Fermat quotient. Adv. Math. Commun.
**10**(4), 743–752 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Dorais, F. G., Klyve, D.: A Wieferich prime search up to \(6.7\times 10^{15}\). J. Integer Seq.
**14**(9) (2011). https://cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.html. Accessed 20 Feb 2018