Cube Theory and Stable k -Error Linear Complexity for Periodic Sequences

The linear complexity of a sequence has been used as an important measure of keystream strength, hence designing a sequence with high linear complexity and \(k\)-error linear complexity is a popular research topic in cryptography. In this paper, the concept of stable \(k\)-error linear complexity is proposed to study sequences with stable and large \(k\)-error linear complexity. In order to study linear complexity of binary sequences with period \(2^n\), a new tool called cube theory is developed. By using the cube theory, one can easily construct sequences with the maximum stable \(k\)-error linear complexity. For such purpose, we first prove that a binary sequence with period \(2^n\) can be decomposed into some disjoint cubes. Second, it is proved that the maximum \(k\)-error linear complexity is \(2^n-(2^l-1)\) over all \(2^n\)-periodic binary sequences, where \(2^{l-1}\le k<2^{l}\). Finally, continuing the work of Kurosawa et al., a characterization is presented about the minimum number \(k\) for which the second decrease occurs in the \(k\)-error linear complexity of a \(2^n\)-periodic binary sequence \(s\).