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

  • Authors:

  • Lin Tan;Wen-Feng Qi;Hong Xu

  • Affiliations:

  • Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou, People's Republic of China 450002;Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou, People's Republic of China 450002;Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou, People's Republic of China 450002

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2012

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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 驴, where 0 驴 驴 驴 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.