Characterization of 2 n -periodic binary sequences with fixed 2-error or 3-error linear complexity

Abstract

The linear complexity of sequences is an important measure of the cryptographic strength of key streams used in stream ciphers. The instability of linear complexity caused by changing a few symbols of sequences can be measured using k-error linear complexity. In their SETA 2006 paper, Fu et al. (SETA, pp. 88---103, 2006) studied the linear complexity and the 1-error linear complexity of 2 n -periodic binary sequences to characterize such sequences with fixed 1-error linear complexity. In this paper we study the linear complexity and the k-error linear complexity of 2 n -periodic binary sequences in a more general setting using a combination of algebraic, combinatorial, and algorithmic methods. This approach allows us to characterize 2 n -periodic binary sequences with fixed 2- or 3-error linear complexity. Using this characterization we obtain the counting function for the number of 2 n -periodic binary sequences with fixed k-error linear complexity for k = 2 and 3.