Asymptotic analysis on the normalized k-error linear complexity of binary sequences

Abstract

Linear complexity and k-error linear complexity are the important measures for sequences in stream ciphers. This paper discusses the asymptotic behavior of the normalized k-error linear complexity $${L_{n,k}(\underline{s})/n}$$ of random binary sequences $${\underline{s}}$$ , which is based on one of Niederreiter's open problems. For k = n ?, where 0 ? ? ? 1/2 is a fixed ratio, the lower and upper bounds on accumulation points of $${L_{n,k}(\underline{s})/n}$$ are derived, which holds with probability 1. On the other hand, for any fixed k it is shown that $${\lim_{n\rightarrow\infty} L_{n,k}(\underline{s})/n = 1/2}$$ holds with probability 1. The asymptotic bounds on the expected value of normalized k-error linear complexity of binary sequences are also presented.