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On the expected value of the linear complexity and the k-error linear complexity of periodic sequences

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2 Author(s)
Meidl, W. ; Inst. of Discrete Math., Austrian Acad. of Sci., Vienna, Austria ; Niederreiter, H.

Rueppel (1986) conjectured that periodic binary sequences have expected linear complexity close to the period length N. In this paper, we determine the expected value of the linear complexity of N-periodic sequences explicitly and confirm Rueppel's conjecture for arbitrary finite fields. Cryptographically strong sequences should not only have a large linear complexity, but also the change of a few terms should not cause a significant decrease of the linear complexity. This requirement leads to the concept of the k-error linear complexity of N-periodic sequences. We present a method to establish a lower bound on the expected k-error linear complexity of N-periodic sequences based on the knowledge of the counting function 𝒩N,0(c), i.e., the number of N-periodic sequences with given linear complexity c. For some cases, we give explicit formulas for that lower bound and we also determine 𝒩N,0(c)

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Information Theory, IEEE Transactions on  (Volume:48 ,  Issue: 11 )