On the Stability of m-Sequences

We study the stability of m-sequences in the sense of determining the number of errors needed for decreasing the period of the sequences, as well as giving lower bounds on the k -error linear complexity of the sequences. For prime periods the results are straightforward so we concentrate on composite periods. We give exact results for the case when the period is reduced by a factor which is a Mersenne number and for the case when it is reduced by a prime p such that the order of 2 modulo p equals p −1. The general case is believed to be difficult due to its similarity to a well studied problem in coding theory. We also provide results about the relative frequencies of the different cases. We formulate a conjecture regarding the minimum number of errors needed for reducing the period at all. Finally we apply our results to the LFSR components of several well known stream ciphers.

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