Journal of the Korea Institute of Information and Communication Engineering (한국정보통신학회논문지)

The Korea Institute of Information and Commucation Engineering (한국정보통신학회)
권호 표지
DOI QR코드

DOI QR Code

Analysis of Linear Span of Non-linear Binary Sequences with Decimation d=2m-2(2m+3)

데시메이션이 d=2m-2(2m+3)인 비선형 이진수열의 선형스팬 분석

  • Yim, Ji-Mi (Department of Applied Mathematics, Pukyong National University) ;
  • Cho, Sung-Jin (Department of Applied Mathematics, Pukyong National University) ;
  • Kim, Han-Doo (Department of Applied Mathematics, Inje University) ;
  • Kim, Seok-Tae (Department of Information and Communication Engineering, Pukyong National University)
  • Received : 2013.12.05
  • Accepted : 2014.01.10
  • Published : 2014.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

Large linear span makes difficult to predict, so this study is important to the security and code system. It has been studied about the non-linear binary sequences having low correlation values and large linear span. In this paper we analyze the linear span of $S^r_a(t)=Tr^m_1\{[Tr^n_m(a{\alpha}^t+{\alpha}^{dt})]^r\}$ ($a{\in}GF(2^m)$, $0{\leq}t{\leq}2^m-2$) where n=2m and $d=2^{m-2}(2^m+3)$.

선형스팬이 클수록 예측을 어렵게 하기 때문에 선형스팬을 크게 하는 것은 보안 및 암호 시스템에서 중요한 문제이다. 낮은 상관함숫값을 가지면서 큰 선형스팬을 가지는 비선형 이진수열에 대한 연구는 계속 이루어져 왔다. 본 논문에서는 n=2m이고 데시메이션이 $d=2^{m-2}(2^m+3)$인 비선형 이진수열 $S^r_a(t)=Tr^m_1\{[Tr^n_m(a{\alpha}^t+{\alpha}^{dt})]^r\}$ ($a{\in}GF(2^m)$, $0{\leq}t{\leq}2^m-2$)에 대한 선형스팬을 분석한다.

Keywords

References

  1. Y. Niho, "Multi-valued cross-correlation functions between two maximal linear recursive sequences", Ph.D thesis, University of Southern California,1972.
  2. T. Helleseth, "Some results about the cross-correlation function between two maximal linear sequences", Discrete Mathematics, vol. 16, No. 3, pp. 209-232, 1976. https://doi.org/10.1016/0012-365X(76)90100-X
  3. P. Rosendahl, "Niho type cross-correlation functions and related equations", Ph.D thesis, Turku centre for computer science, 2004.
  4. X.H. Tang, T. Helleseth, L. Hu, and W. Jiang, "Two new families of optimal binary sequences obtained from quaternary sequences", IEEE Trans. Inf. Theory, vol. 55, no. 4, pp. 433-436, 2009. https://doi.org/10.1109/TIT.2008.2008136
  5. F.X. Zeng and Z.Y. Zhang, "Several Families of Sequences with Low Correlation and Large Linear Span", IEEE Trans. Fundamentals. vol. E91-A, pp. 2263-2268, 2008. https://doi.org/10.1093/ietfec/e91-a.8.2263
  6. K.-U. Schmidt, "Z4-valued quadratic forms and quaternary sequence families", IEEE. Trans. Inf. Theory, vol. 55, no. 12, pp. 5803-5810, 2009. https://doi.org/10.1109/TIT.2009.2032818
  7. P. Udaya, X.H. Tang, "On the Construction of Binary Sequence Families With Low Correlation and Large Sizes", IEEE. Trans. Inf. Theory, vol. 59, no. 2, pp. 1082-1089, 2013. https://doi.org/10.1109/TIT.2012.2219691
  8. T. Helleseth and P.V. Kumar, "Sequences with low correlation", in Handbook of Coding Theory, ed. V. Pless and C. Huffman, Elsevier, Amsterdam, The Netherlands, 1998.
  9. M.K. Simon, J.K. Omura, R.A. Scholtz, and B.K. Levitt, "Spread spectrum communication", vol. I, Computer Science Press, Rockville, MD, 1985.
  10. K. Fazel and S. Kaiser, "Multi-carrier and spread spectrum system", John Wiley and Sons Ltd., 2003.
  11. D.V. Sarwarte and M.B. Pursley, "Crosscorrelation properties of pseudorandom and related sequences", Proc. IEEE, vol. 68, no. 5, pp. 593-620, May. 1980. https://doi.org/10.1109/PROC.1980.11697
  12. R.A. Scholtz, "The origins of spread-spectrum communications", IEEE Trans. Commun., vol. COM-30, pp.822-854, May. 1982.
  13. P.V. Kumar and R.A. Scholtz, "Bounds on the linear span of bent sequences", IEEE Trans. Inform. Theory, vol. IT-29, no. 6, pp. 854-862, Nov. 1983.
  14. J.D. Olsen, R.A. Scholtz, and L. R. Welch, "Bent-function sequences", IEEE Trans. Inform. Theory, vol. IT-28. no. 6, pp.858-864, Nov. 1982.
  15. R. Gold, "Maximal recursive sequences with 3-valued recursive crosscorrelation functions", IEEE Trans. Inform. Theory, vol. IT-14, no. 1, pp. 154-156, Jan. 1968.
  16. R. Gold, "Optimal binary sequences for spread spectrum multiplexing", IEEE Trans. Inform. Theory, vol. IT-13, no. 5, pp.619-621, Oct. 1967.
  17. T. Kasami, "Weight distribution formula for some class of cyclic codes", Coordinated Science Laboratory, University of Illinois, Urbana, Tech. Rep. R-285(AD632574), 1966.
  18. T. Kasami, "Weight distribution of Bose-Chaudhuri-Hocquenghem codes in Combinatorial Mathematics and its Applications", Chapel Hill, NC: University of North Carolina Press, 1969.
  19. J.S. No and P.V. Kumar, "A New Family of Binary Pseudorandom Sequences Having Optimal Periodic Correlation Properties and Large Linear Span", IEEE Trans. Inform. Theory, vol. 35, no. 2, pp. 371-379, Mar. 1989. https://doi.org/10.1109/18.32131
  20. E.L. Key, "An analysis of the structure and complexity of non-linear binary sequence generators", IEEE Trans. Inform. Theory, IT-22, pp. 732-736, 1976.
  21. R. Lidl and H. Niederreiter, "Finite fields", Cambridge University Press, 1997.
  22. S.J. Cho, J.M. Yim, "Analysis of binary sequences generated by GMW sequences and No sequences", J. Korea Inst. Inf. Commun. Eng., vol. 15, no. 10, pp. 2183-2187, 2011. https://doi.org/10.6109/jkiice.2011.15.10.2181