The Distribution of \(2^n\)-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. Fu et al. studied the distribution of \(2^n\)-periodic binary sequences with 1-error linear complexity in their SETA 2006 paper. Recently, people 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. In this paper, we propose a new sieve method to solve this problem. We first define an equivalence relationship on error sequences and build a relation between the number of sequences with given k-error linear complexity and the number of pairwise non-equivalent error sequences. We introduce the concept of cube fragment and build specific equivalence relation based on the concept of the cube classes to figure out the number of pairwise non-equivalent error sequences. By establishing counting functions for several base cases and building recurrence relations for different cases of k and L, it is easy to manually get the complete counting function when k is not too large. And an efficient algorithm can be derived from this method to solve the problem using a computer when k is large.
KeywordsSequence Linear complexity k-Error linear complexity Counting function Cube theory
Many thanks go to the anonymous reviewers for their detailed comments and suggestions. This work was supported by the National Key R&D Program of China with No. 2016YFB0800100, CAS Strategic Priority Research Program with No. XDA06010701, National Key Basic Research Project of China with No. 2011CB302400 and National Natural Science Foundation of China with No. 61671448, No. 61379139.
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