An Algorithm for Counting the Number of 2n-Periodic Binary Sequences with Fixed k-Error Linear Complexity

The linear complexity and k-error linear complexity of sequences are important measures of the strength of key-streams generated by stream ciphers. The counting function of a sequence complexity measure gives the number of sequences with given complexity measure value and it is useful to determine the expected value and variance of a given complexity measure of a family of sequences. Fu et al. studied the distribution of 2n-periodic binary sequences with 1-error linear complexity in their SETA 2006 paper and peoples have strenuously promoted the solving of this problem from k = 2 to k = 4 step by step. Unfortunately, it still remains difficult to obtain the solutions for larger k and the counting functions become extremely complex when k become large. In this paper, we define an equivalent relation on error sequences. We use a concept of cube fragment as basic modules to construct classes of error sequences with specific structures. Error sequences with the same specific structures can be represented by a single symbolic representation. We introduce concepts of trace, weight trace and orbit of sets to build quantitative relations between different classes. Based on these quantitative relations, we propose an algorithm to automatically generate symbolic representations of classes of error sequences, calculate coefficients from one class to another and compute multiplicity of classes defined based on specific equivalence on error sequences. This algorithm can efficiently get the number of sequences with given k-error linear complexity. The time complexity of this algorithm is O(2k logk) in the worst case which does not depend on the period 2n.