Decomposing Approach for Error Vectors of k-Error Linear Complexity of Certain Periodic Sequences

Abstract

The k-error linear complexity of periodic sequences is an important security index of stream cipher systems. By using an interesting decomposing approach, we investigate the intrinsic structure for the set of 2ⁿ-periodic binary sequences with fixed complexity measures. For k ≤ 4, we construct the complete set of error vectors that give the k-error linear complexity. As auxiliary results we obtain the counting functions of the k-error linear complexity of 2ⁿ-periodic binary sequences for k ≤ 4, as well as the expectations of the k-error linear complexity of a random sequence for k ≤ 3. Moreover, we study the 2^t-error linear complexity of the set of 2ⁿ-periodic binary sequences with some fixed linear complexity L, where t < n-1 and the Hamming weight of the binary representation of 2ⁿ-L is t. Also, we extend some results to pⁿ-periodic sequences over F_p. Finally, we discuss some potential applications.