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The linear complexity and k-error linear complexity of a sequence have beenused as important measures of keystream strength, hence designing a sequencewith high linear complexity and $k$-error linear complexity is a popularresearch topic in cryptography. In this paper, the concept of stable $k$-errorlinear complexity is proposed to study sequences with stable and large$k$-error linear complexity. In order to study k-error linear complexity ofbinary 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 maximumstable $k$-error linear complexity. For such purpose, we first prove that abinary sequence with period $2^n$ can be decomposed into some disjoint cubesand further give a general decomposition approach. Second, it is proved thatthe 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, acharacterization is presented about the $t$th ($t>1$) decrease in the $k$-errorlinear complexity for a $2^n$-periodic binary sequence $s$ and this is acontinuation of Kurosawa et al. recent work for the first decrease of k-errorlinear complexity. Finally, A counting formula for $m$-cubes with the samelinear complexity is derived, which is equivalent to the counting formula for$k$-error vectors. The counting formula of $2^n$-periodic binary sequenceswhich can be decomposed into more than one cube is also investigated, whichextends an important result by Etzion et al..

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