The cube theory for 2n-periodic binary sequences

Abstract

The linear complexity and k-error linear complexity of a sequencehave been used as important measures for keystream strength, hencedesigning a sequence with high linear complexity and k-errorlinear complexity is a popular research topic in cryptography. In order to study k-error linear complexity of binarysequences with period 2n, a new tool called cube theory isdeveloped. In this paper, we first give ageneral decomposition approach to decompose a binary sequence with period 2n into some disjoint cubes. Second, a countingformula for m-cubes with the same linear complexity is derived, which is equivalent to the counting formula for k-error vectors. The counting formulaof 2n-periodic binary sequences which can be decomposedinto more than one cube is also investigated, which extends an important result by Etzion et al.