On the $k$-error linear complexity for $2^n$-periodic binary sequences via Cube Theory

The linear complexity and k-error linear complexity of a sequence have been used as important measures 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 k-error linear complexity of binary sequences with period 2, a new tool called cube theory is developed. By using the cube theory, one can easily construct sequences with the maximum stable kerror linear complexity. For such purpose, we first prove that a binary sequence with period 2 can be decomposed into some disjoint cubes and further give a general decomposition approach. Second, it is proved that the maximum k-error linear complexity is 2 − (2 − 1) over all 2-periodic binary sequences, where 2 ≤ k < 2. Thirdly, a characterization is presented about the tth (t > 1) decrease in the k-error linear complexity for a 2-periodic binary sequence s and this is a continuation of Kurosawa et al. recent work for the first decrease of k-error linear complexity. Finally, A counting formula for m-cubes with the same linear complexity is derived, which is equivalent to the counting formula for k-error vectors. The counting formula of 2-periodic binary sequences which can be decomposed into more than one cube is also investigated, which extends an important result by Etzion et al..