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 (s)/n of random binary sequences s, which is based on one of Niederreiter's open problems. For k = nθ , where 0 ≤ θ ≤ 1/2 is a fixed ratio, the lower and upper bounds on accumulation points of L n,k (s)/n are derived, which holds with probability 1. On the other hand, for any fixed k it is shown that lim n→∞ L n,k (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.

Research about the Characteristics of Chaotic Systems Based on Multi-Scale Entropy

“…A multivariate multi-scale distribution entropy (MMSDE) was also presented in the literature [ 11 ], which was used to assess the complexity of a complex dynamical system. In literature [ 12 , 13 ], randomness and quantifying complexity from information theory were described in detail, and the complexity analysis method was an important measure for the sequences in the stream ciphers. The authors of [ 14 ] defined the concept of uniformity through learning the theory of an exclusive sphere, and made a comparative analysis between the Lyapunov exponent and the uniformity of a logistic system, and concluded that the chaotic characteristic of a logistic system is more and more random with the parameter increasing.…”