The Cube Theory for 2n-Periodic Binary Sequences

The linear complexity and k-error linear complexity of a sequence have been used as important measures for keystream strength, hence designing a sequence with high linear complexity and k-error linear complexity is a popular research topic in cryptography. In order to study k-error linear complexity of binary sequences with period 2ⁿ, a new tool called cube theory is developed. In this paper, we first give a general decomposition approach to decompose a binary sequence with period 2ⁿ into some disjoint cubes. Second, 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 2ⁿ-periodic binary sequences which can be decomposed into more than one cube is also investigated, which extends an important result by Etzion et al.