American Institute of Mathematical Sciences

November  2019, 2(4): 279-297. doi: 10.3934/mfc.2019018

On the $k$-error linear complexity for $p^n$-periodic binary sequences via hypercube theory

 1 School of Computer Science, Anhui Univ. of Technology, Ma'anshan, 243002, China 2 Department of Computing, Curtin University, Perth, WA 6102, Australia 3 School of Computer Science, Anhui Univ. of Technology, Ma'anshan, 243002, China 4 Dept of Mathematics & Statistics, Curtin University, Perth, WA 6102, Australia

Published  December 2019

The linear complexity and the $k$-error linear complexity of a binary sequence are important security measures for the security of the key stream. By studying binary sequences with the minimum Hamming weight, a new tool, named as the hypercube theory, is developed for $p^n$-periodic binary sequences. In fact, the hypercube theory is based on a typical sequence decomposition and it is a very important tool for investigating the critical error linear complexity spectrum proposed by Etzion et al. To demonstrate the importance of hypercube theory, we first give a standard hypercube decomposition based on a well-known algorithm for computing linear complexity and show that the linear complexity of the first hypercube in the decomposition is equal to the linear complexity of the original sequence. Second, based on such decomposition, we give a complete characterization for the first decrease of the linear complexity for a $p^n$-periodic binary sequence. This significantly improves the current existing results in literature. As to the importance of the hypercube, we finally derive a counting formula for the $m$-hypercubes with the same linear complexity.

Citation: Jianqin Zhou, Wanquan Liu, Xifeng Wang, Guanglu Zhou. On the $k$-error linear complexity for $p^n$-periodic binary sequences via hypercube theory. Mathematical Foundations of Computing, 2019, 2 (4) : 279-297. doi: 10.3934/mfc.2019018
References:
 [1] C. S. Ding, G. Z. Xiao and W. J. Shan, The Stability Theory of Stream Ciphers[M], Lecture Notes in Computer Science, 561. Springer-Verlag, Berlin, 1991. doi: 10.1007/3-540-54973-0.  Google Scholar [2] T. Etzion, N. Kalouptsidis, N. Kolokotronis, K. Limniotis and K. G. Paterson, Properties of the error linear complexity spectrum, IEEE Transactions on Information Theory, 55 (2009), 4681-4686.  doi: 10.1109/TIT.2009.2027495.  Google Scholar [3] R. A. Games and A. H. Chan, A fast algorithm for determining the complexity of a binary sequence with period $2^n$, IEEE Trans on Information Theory, 29 (1983), 144-146.  doi: 10.1109/TIT.1983.1056619.  Google Scholar [4] F. Fu, H. Niederreiter and M. Su, The characterization of $2^n$-periodic binary sequences with fixed 1-error linear complexity, In: Gong G., Helleseth T., Song H.-Y., Yang K. (eds.) SETA 2006, LNCS, 4086 (2006), 88–103. doi: 10.1007/11863854_8.  Google Scholar [5] Y. K. Han, J. H. Chung and K. Yang, On the $k$-error linear complexity of $p^m$-periodic binary sequences, IEEE Transactions on Information Theory, 53 (2007), 2297-2304.  doi: 10.1109/TIT.2007.896863.  Google Scholar [6] K. Kurosawa, F. Sato, T. Sakata and W. Kishimoto, A relationship between linear complexity and $k$-error linear complexity, IEEE Transactions on Information Theory, 46 (2000), 694-698.  doi: 10.1109/18.825845.  Google Scholar [7] A. Lauder and K. Paterson, Computing the error linear complexity spectrum of a binary sequence of period $2^n$, IEEE Transactions on Information Theory, 49 (2003), 273-280.  doi: 10.1109/TIT.2002.806136.  Google Scholar [8] W. Meidl and H. Niederreiter, Linear complexity k-error linear complexity, and the discrete Fourier transform, J. Complexity, 18 (2002), 87-103.  doi: 10.1006/jcom.2001.0621.  Google Scholar [9] W. Meidl, How many bits have to be changed to decrease the linear complexity?, Des. Codes Cryptogr., 33 (2004), 109-122.  doi: 10.1023/B:DESI.0000035466.28660.e9.  Google Scholar [10] W. Meidl, On the stablity of $2^n$-periodic binary sequences, IEEE Transactions on Information Theory, 51 (2005), 1151-1155.  doi: 10.1109/TIT.2004.842709.  Google Scholar [11] R. A. Rueppel, Analysis and Design of Stream Ciphers, Berlin: Springer-Verlag, 1986. doi: 10.1007/978-3-642-82865-2.  Google Scholar [12] M. Stamp and C. F. Martin, An algorithm for the $k$-error linear complexity of binary sequences with period $2^{n}$, IEEE Trans. Inform. Theory, 39 (1993), 1398-1401.  doi: 10.1109/18.243455.  Google Scholar [13] S. M. Wei, G. Z. Xiao and Z. Chen, A fast algorithm for determining the minimal polynomial of a sequence with period $2p^n$ over $GF(q)$, IEEE Trans on Information Theory, 48 (2002), 2754-2758.  doi: 10.1109/TIT.2002.802609.  Google Scholar [14] 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 on Information Theory, 46 (2000), 2203-2206.  doi: 10.1109/18.868492.  Google Scholar [15] J. Q. Zhou, On the $k$-error linear complexity of sequences with period 2$p^n$ over GF(q), Des. Codes Cryptogr., 58 (2011), 279-296.  doi: 10.1007/s10623-010-9379-7.  Google Scholar [16] J. Q. Zhou, A counterexample concerning the 3-error linear complexity of $2^n$-periodic binary sequences, Des. Codes Cryptogr., 64 (2012), 285-286.  doi: 10.1007/s10623-011-9576-z.  Google Scholar [17] J. Q. Zhou and W. Q. Liu, The $k$-error linear complexity distribution for $2^n$-periodic binary sequences, Des. Codes Cryptogr., 73 (2014), 55-75.  doi: 10.1007/s10623-013-9805-8.  Google Scholar [18] J. Q. Zhou, W. Q. Liu and X. F. Wang, Complete characterization of the first descent point distribution for the $k$-error linear complexity of $2^n$-periodic binary sequences, Adv. in Math. of Comm., 11 (2017), 429-444.  doi: 10.3934/amc.2017036.  Google Scholar [19] J. Q. Zhou, W. Q. Liu and G. L. Zhou, Cube theory and stable $k$-error linear complexity for periodic sequences, Information Security and Cryptology, 70–85, Lecture Notes in Comput. Sci., 8567, Springer, Heidelberg, 2014. doi: 10.1007/978-3-319-12087-4_5.  Google Scholar [20] F. X. Zhu and W. F. Qi, The 2-error linear complexity of $2^n$-periodic binary sequences with linear complexity $2^n$-1, Journal of Electronics (China), 24 (2007), 390-395.   Google Scholar

show all references

References:
 [1] C. S. Ding, G. Z. Xiao and W. J. Shan, The Stability Theory of Stream Ciphers[M], Lecture Notes in Computer Science, 561. Springer-Verlag, Berlin, 1991. doi: 10.1007/3-540-54973-0.  Google Scholar [2] T. Etzion, N. Kalouptsidis, N. Kolokotronis, K. Limniotis and K. G. Paterson, Properties of the error linear complexity spectrum, IEEE Transactions on Information Theory, 55 (2009), 4681-4686.  doi: 10.1109/TIT.2009.2027495.  Google Scholar [3] R. A. Games and A. H. Chan, A fast algorithm for determining the complexity of a binary sequence with period $2^n$, IEEE Trans on Information Theory, 29 (1983), 144-146.  doi: 10.1109/TIT.1983.1056619.  Google Scholar [4] F. Fu, H. Niederreiter and M. Su, The characterization of $2^n$-periodic binary sequences with fixed 1-error linear complexity, In: Gong G., Helleseth T., Song H.-Y., Yang K. (eds.) SETA 2006, LNCS, 4086 (2006), 88–103. doi: 10.1007/11863854_8.  Google Scholar [5] Y. K. Han, J. H. Chung and K. Yang, On the $k$-error linear complexity of $p^m$-periodic binary sequences, IEEE Transactions on Information Theory, 53 (2007), 2297-2304.  doi: 10.1109/TIT.2007.896863.  Google Scholar [6] K. Kurosawa, F. Sato, T. Sakata and W. Kishimoto, A relationship between linear complexity and $k$-error linear complexity, IEEE Transactions on Information Theory, 46 (2000), 694-698.  doi: 10.1109/18.825845.  Google Scholar [7] A. Lauder and K. Paterson, Computing the error linear complexity spectrum of a binary sequence of period $2^n$, IEEE Transactions on Information Theory, 49 (2003), 273-280.  doi: 10.1109/TIT.2002.806136.  Google Scholar [8] W. Meidl and H. Niederreiter, Linear complexity k-error linear complexity, and the discrete Fourier transform, J. Complexity, 18 (2002), 87-103.  doi: 10.1006/jcom.2001.0621.  Google Scholar [9] W. Meidl, How many bits have to be changed to decrease the linear complexity?, Des. Codes Cryptogr., 33 (2004), 109-122.  doi: 10.1023/B:DESI.0000035466.28660.e9.  Google Scholar [10] W. Meidl, On the stablity of $2^n$-periodic binary sequences, IEEE Transactions on Information Theory, 51 (2005), 1151-1155.  doi: 10.1109/TIT.2004.842709.  Google Scholar [11] R. A. Rueppel, Analysis and Design of Stream Ciphers, Berlin: Springer-Verlag, 1986. doi: 10.1007/978-3-642-82865-2.  Google Scholar [12] M. Stamp and C. F. Martin, An algorithm for the $k$-error linear complexity of binary sequences with period $2^{n}$, IEEE Trans. Inform. Theory, 39 (1993), 1398-1401.  doi: 10.1109/18.243455.  Google Scholar [13] S. M. Wei, G. Z. Xiao and Z. Chen, A fast algorithm for determining the minimal polynomial of a sequence with period $2p^n$ over $GF(q)$, IEEE Trans on Information Theory, 48 (2002), 2754-2758.  doi: 10.1109/TIT.2002.802609.  Google Scholar [14] 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 on Information Theory, 46 (2000), 2203-2206.  doi: 10.1109/18.868492.  Google Scholar [15] J. Q. Zhou, On the $k$-error linear complexity of sequences with period 2$p^n$ over GF(q), Des. Codes Cryptogr., 58 (2011), 279-296.  doi: 10.1007/s10623-010-9379-7.  Google Scholar [16] J. Q. Zhou, A counterexample concerning the 3-error linear complexity of $2^n$-periodic binary sequences, Des. Codes Cryptogr., 64 (2012), 285-286.  doi: 10.1007/s10623-011-9576-z.  Google Scholar [17] J. Q. Zhou and W. Q. Liu, The $k$-error linear complexity distribution for $2^n$-periodic binary sequences, Des. Codes Cryptogr., 73 (2014), 55-75.  doi: 10.1007/s10623-013-9805-8.  Google Scholar [18] J. Q. Zhou, W. Q. Liu and X. F. Wang, Complete characterization of the first descent point distribution for the $k$-error linear complexity of $2^n$-periodic binary sequences, Adv. in Math. of Comm., 11 (2017), 429-444.  doi: 10.3934/amc.2017036.  Google Scholar [19] J. Q. Zhou, W. Q. Liu and G. L. Zhou, Cube theory and stable $k$-error linear complexity for periodic sequences, Information Security and Cryptology, 70–85, Lecture Notes in Comput. Sci., 8567, Springer, Heidelberg, 2014. doi: 10.1007/978-3-319-12087-4_5.  Google Scholar [20] F. X. Zhu and W. F. Qi, The 2-error linear complexity of $2^n$-periodic binary sequences with linear complexity $2^n$-1, Journal of Electronics (China), 24 (2007), 390-395.   Google Scholar
 [1] Jianqin Zhou, Wanquan Liu, Xifeng Wang. Structure analysis on the k-error linear complexity for 2n-periodic binary sequences. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1743-1757. doi: 10.3934/jimo.2017016 [2] 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 [3] Zhixiong Chen, Vladimir Edemskiy, Pinhui Ke, Chenhuang Wu. On $k$-error linear complexity of pseudorandom binary sequences derived from Euler quotients. Advances in Mathematics of Communications, 2018, 12 (4) : 805-816. doi: 10.3934/amc.2018047 [4] Alina Ostafe, Igor E. Shparlinski, Arne Winterhof. On the generalized joint linear complexity profile of a class of nonlinear pseudorandom multisequences. Advances in Mathematics of Communications, 2010, 4 (3) : 369-379. doi: 10.3934/amc.2010.4.369 [5] 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 [6] Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37 [7] Stefano Galatolo. Orbit complexity and data compression. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477 [8] Valentin Afraimovich, Lev Glebsky, Rosendo Vazquez. Measures related to metric complexity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1299-1309. doi: 10.3934/dcds.2010.28.1299 [9] Steffen Klassert, Daniel Lenz, Peter Stollmann. Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1553-1571. doi: 10.3934/dcds.2011.29.1553 [10] Valentin Afraimovich, Maurice Courbage, Lev Glebsky. Directional complexity and entropy for lift mappings. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3385-3401. doi: 10.3934/dcdsb.2015.20.3385 [11] Tsonka Baicheva, Iliya Bouyukliev. On the least covering radius of binary linear codes of dimension 6. Advances in Mathematics of Communications, 2010, 4 (3) : 399-404. doi: 10.3934/amc.2010.4.399 [12] Hua Liang, Jinquan Luo, Yuansheng Tang. On cross-correlation of a binary $m$-sequence of period $2^{2k}-1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$. Advances in Mathematics of Communications, 2017, 11 (4) : 693-703. doi: 10.3934/amc.2017050 [13] Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353 [14] Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1 [15] Easton Li Xu, Weiping Shang, Guangyue Han. Network encoding complexity: Exact values, bounds, and inequalities. Advances in Mathematics of Communications, 2017, 11 (3) : 567-594. doi: 10.3934/amc.2017044 [16] Valentin Afraimovich, Lev Glebsky. Measures related to $(\epsilon,n)$-complexity functions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 23-34. doi: 10.3934/dcds.2008.22.23 [17] Jiarong Peng, Xiangyong Zeng, Zhimin Sun. Finite length sequences with large nonlinear complexity. Advances in Mathematics of Communications, 2018, 12 (1) : 215-230. doi: 10.3934/amc.2018015 [18] Enrico Capobianco. Born to be big: Data, graphs, and their entangled complexity. Big Data & Information Analytics, 2016, 1 (2&3) : 163-169. doi: 10.3934/bdia.2016002 [19] Stefano Galatolo. Global and local complexity in weakly chaotic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1607-1624. doi: 10.3934/dcds.2003.9.1607 [20] Andrew Klapper. The asymptotic behavior of N-adic complexity. Advances in Mathematics of Communications, 2007, 1 (3) : 307-319. doi: 10.3934/amc.2007.1.307

Impact Factor: