Linear complexity of periodically repeated random sequences

Abstract

On the linear complexity Ʌ(z) of a periodically repeated random bit sequence z, R. Rueppel proved that, for two extreme cases of the period T, the expected linear complexity E[Ʌ(z)] is almost equal to T, and suggested that E[Ʌ(z)] would be close to T in general [6, pp. 3352] [7, 8]. In this note we obtain bounds of E[Ʌ(z)], as well as bounds of the variance Var[Ʌ(z)], both for the general case of T, and we estimate the probability distribution of Λ(z). Our results on E[Λ(z)] quantify the closeness of E[Λ(z)] and T, in particular, formally confirm R. Rueppel’s suggestion.