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 $2^n$, 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 and further give a general decomposition approach. 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 $2^{l-1}\le k<2^{l}$. Thirdly, a characterization is presented about the $t$th ($t>1$) decrease in the $k$-error linear complexity for a $2^n$-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 $2^n$-periodic binary sequences which can be decomposed into more than one cube is also investigated, which extends an important result by Etzion et al..