Distribution of the Second Descent Points for the $k$-error Linear Complexity of $2^n$-periodic Binary Sequences

In this paper, a structural approach for determining CELCS (critical error linear complexity spectrum) for the k-error linear complexity distribution of 2 n -periodic binary sequences is developed via the sieve method and Games-Chan algorithm. Accordingly, the second descent point (critical point) distribution of the k-error linear complexity for 2 n -periodic binary sequences is characterized. As a consequence, we obtain the complete counting functions on the k-error linear complexity of 2 n -periodic binary sequences as the second descent point for k = 3,4.

[1]  Kaoru Kurosawa,et al.  A relationship between linear complexity and kapa-error linear complexity , 2000, IEEE Trans. Inf. Theory.

[2]  Wenfeng Qi,et al.  The 2-error linear complexity of 2n-periodic binary sequences with linear complexity 2n − 1 , 2007 .

[3]  R. A. Rueppel.  Analysis and Design of Stream Ciphers , 2012 .

[4]  Jianqin Zhou.  A counterexample concerning the 3-error linear complexity of 2n-periodic binary sequences , 2012, Des. Codes Cryptogr..

[5]  Guozhen Xiao,et al.  A fast algorithm for determining the minimal polynomial where of a sequence with period 2pn over GF(q) , 2002, IEEE Trans. Inf. Theory.

[6]  Jianqin Zhou,et al.  On the k-error linear complexity of sequences with period 2pn over GF(q) , 2008, 2008 International Conference on Computational Intelligence and Security.

[7]  Wanquan Liu,et al.  The $$k$$-error linear complexity distribution for $$2^n$$-periodic binary sequences , 2014, Des. Codes Cryptogr..

[8]  Wilfried Meidl.  On the stability of 2/sup n/-periodic binary sequences , 2005, IEEE Transactions on Information Theory.

[9]  Guozhen Xiao,et al.  A Fast Algorithm for Determining the Linear Complexity of a Sequence with Period Over GF , 2000 .

[10]  Jin-Ho Chung,et al.  On the k-Error Linear Complexity of pm -Periodic Binary Sequences , 2007, IEEE Trans. Inf. Theory.

[11]  Kenneth G. Paterson,et al.  Computing the error linear complexity spectrum of a binary sequence of period 2n , 2003, IEEE Trans. Inf. Theory.

[12]  Ramakanth Kavuluru,et al.  Characterization of 2n-periodic binary sequences with fixed 2-error or 3-error linear complexity , 2009, Des. Codes Cryptogr..

[13]  Mark Stamp,et al.  An algorithm for the k-error linear complexity of binary sequences with period 2n , 1993, IEEE Trans. Inf. Theory.

[14]  Richard A. Games,et al.  A fast algorithm for determining the complexity of a binary sequence with period 2n , 1983, IEEE Trans. Inf. Theory.

[15]  Satoshi Uehara,et al.  An Algorithm for thek-Error Linear Complexity of Sequences over GF(pm) with Period pn, pa Prime , 1999, Inf. Comput..

[16]  James L. Massey,et al.  Shift-register synthesis and BCH decoding , 1969, IEEE Trans. Inf. Theory.

[17]  Nicholas Kolokotronis,et al.  Minimum linear span approximation of binary sequences , 2002, IEEE Trans. Inf. Theory.

[18]  Cunsheng Ding,et al.  The Stability Theory of Stream Ciphers , 1991, Lecture Notes in Computer Science.

[19]  Wilfried Meidl,et al.  How Many Bits have to be Changed to Decrease the Linear Complexity? , 2004, Des. Codes Cryptogr..

[20]  Harald Niederreiter,et al.  The Characterization of 2n-Periodic Binary Sequences with Fixed 1-Error Linear Complexity , 2006, SETA.

[21]  Kenneth G. Paterson,et al.  Properties of the Error Linear Complexity Spectrum , 2009, IEEE Transactions on Information Theory.