# American Institute of Mathematical Sciences

August  2017, 11(3): 429-444. doi: 10.3934/amc.2017036

## Complete characterization of the first descent point distribution for the k-error linear complexity of 2n-periodic binary sequences

 1 Department of Computing, Curtin University, Perth, WA 6102, Australia 2 School of Computer Science, Anhui Univ. of Technology, Maanshan, Anhui 243032, China

Received  September 2014 Revised  August 2015 Published  August 2017

In this paper, a new constructive approach of determining the first descent point distribution for the $k$-error linear complexity of $2^n$-periodic binary sequences is developed using the sieve method and Games-Chan algorithm. First, the linear complexity for the sum of two sequences with the same linear complexity and minimum Hamming weight is completely characterized and this paves the way for the investigation of the $k$-error linear complexity. Second we derive a full representation of the first descent point spectrum for the $k$-error linear complexity. Finally, we obtain the complete counting functions on the number of $2^n$-periodic binary sequences with given $2^m$-error linear complexity and linear complexity $2^n-(2^{i_1}+2^{i_2}+···+2^{i_m})$, where $0≤ i_1<i_2<···<i_m<n.$ In summary, we depict a full picture on the first descent point of the $k$-error linear complexity for $2^n$-periodic binary sequences and this will help us construct some sequences with requirements on linear complexity and $k$-error complexity.

Citation: 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
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##### References:
 [1] C. S. Ding, Lower bounds on the weight complexity of cascaded binary sequences, in Proc. Auscrypt'90 Adv. Crypt. , Springer-Verlag, 1990, 39–43. doi: 10.1007/BFb0030350. Google Scholar [2] C. S. Ding, G. Z. Xiao and W. J. Shan, The Stability Theory of Stream Ciphers, SpringerVerlag, Berlin, 1991, 85–88. doi: 10.1007/3-540-54973-0. Google Scholar [3] T. Etzion, N. Kalouptsidis, N. Kolokotronis, K. Limniotis and K. G. Paterson, Properties of the error linear complexity spectrum, IEEE Trans. Inf. Theory, 55 (2009), 4681-4686. doi: 10.1109/TIT.2009.2027495. Google Scholar [4] F. Fu, H. Niederreiter and M. Su, The characterization of 2n-periodic binary sequences with fixed 1-error linear complexity, in SETA 2006 (eds. G. Gong et al), Springer, 2006, 88–103. doi: 10.1007/11863854_8. Google Scholar [5] R. A. Games and A. H. Chan, A fast algorithm for determining the complexity of a binary sequence with period $2^n$, IEEE Trans. Inf. Theory, 29 (1983), 144-146. doi: 10.1109/TIT.1983.1056619. Google Scholar [6] Y. K. Han, J. H. Chung and K. Yang, On the $k$-error linear complexity of $p^m$-periodic binary sequences, IEEE Trans. Inf. Theory, 53 (2007), 2297-2304. doi: 10.1109/TIT.2007.896863. Google Scholar [7] T. Kaida, S. Uehara and K. Imamura, An algorithm for the $k$-error linear complexity of sequences over GF($p^m$) with period $p^n$, $p$ a prime, Inf. Comput., 151 (1999), 134-147. doi: 10.1006/inco.1998.2768. Google Scholar [8] R. Kavuluru, Characterization of $2^n$-periodic binary sequences with fixed 2-error or 3-error linear complexity, Des. Codes Crypt., 53 (2009), 75-97. doi: 10.1007/s10623-009-9295-x. Google Scholar [9] N. Kolokotronis, P. Rizomiliotis and N. Kalouptsidis, Minimum linear span approximation of binary sequences, IEEE Trans. Inf. Theory, 48 (2002), 2758-2764. doi: 10.1109/TIT.2002.802621. Google Scholar [10] K. Kurosawa, F. Sato, T. Sakata and W. Kishimoto, A relationship between linear complexity and $k$-error linear complexity, IEEE Trans. Inf. Theory, 46 (2000), 694-698. doi: 10.1109/18.825845. Google Scholar [11] A. Lauder and K. Paterson, Computing the error linear complexity spectrum of a binary sequence of period $2^n$, IEEE Trans. Inf. Theory, 49 (2003), 273-280. doi: 10.1109/TIT.2002.806136. Google Scholar [12] W. Meidl, How many bits have to be changed to decrease the linear complexity?, Des. Codes Crypt., 33 (2004), 109-122. doi: 10.1023/B:DESI.0000035466.28660.e9. Google Scholar [13] W. Meidl, On the stablity of $2^n$-periodic binary sequences, IEEE Trans. Inf. Theory, 51 (2005), 1151-1155. doi: 10.1109/TIT.2004.842709. Google Scholar [14] F. Pi and W. F. Qi, The 4-error linear complexity of $2^n$-periodic binary sequences with linear complexity $2^n-2^m-2^l$, ACTA Electr. Sin. (in Chinese), 39 (2011), 2914-2920. Google Scholar [15] R. A. Rueppel, Analysis and Design of Stream Ciphers Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-82865-2. Google Scholar [16] M. Stamp and C. F. Martin, An algorithm for the $k$-error linear complexity of binary sequences with period $2^n$, IEEE Trans. Inf. Theory, 39 (1993), 1398-1401. doi: 10.1109/18.243455. Google Scholar [17] G. Z. Xiao, S. M. Wei, K. Y. Lam and K. Imamura, A fast algorithm for determining the linear complexity of a sequence with period $p^n$ over $GF(q)$, IEEE Trans. Inf. Theory, 46 (2000), 2203-2206. doi: 10.1109/18.868492. Google Scholar [18] J. Q. Zhou, On the $k$-error linear complexity of sequences with period 2$p^n$ over GF(q), Des. Codes Crypt., 58 (2011), 279-296. doi: 10.1007/s10623-010-9379-7. Google Scholar [19] J. Q. Zhou and W. Q. Liu, The $k$-error linear complexity distribution for $2^n$-periodic binary sequences, Des. Codes Crypt., 73 (2014), 55-75. doi: 10.1007/s10623-013-9805-8. Google Scholar [20] J. Q. Zhou, J. Liu and W. Q. Liu, The 4-error linear complexity distribution for $2^n$-periodic binary sequences 2013, preprint, arXiv: 1310.0132 doi: 10.1007/s10623-013-9805-8. Google Scholar [21] J. Q. Zhou, W. Q. Liu and G. L. Zhou, Cube theory and stable k-error linear complexity for periodic sequences, in Inscrypt 2013, Springer, 70–85. doi: 10.1007/978-3-319-12087-4_5. Google Scholar [22] F. X. Zhu and W. F. Qi, The 2-error linear complexity of $2^n$-periodic binary sequences with linear complexity $2^n$-1, J. Electronics (China), 24 (2007), 390-395. doi: 10.1007/s11767-006-0005-9. Google Scholar
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