We study statistical properties of N-ary FCSR sequences with odd prime connection integer q and least period (q−1)/2, which are called half- ℓ-sequences. More precisely, for any N > 1 our first aim is to give upper bounds on the number of occurrences of two symbols with a fixed distance between them and the number of occurrences of three consecutive symbols using certain character sums. Our second aim is to give a bound on the autocorrelation of half- ℓ-sequences for N = 2. Finally, we give bounds on the number of occurrences of two symbols with a fixed distance between them in an ℓ-sequence and obtain conditions on the connection integer that guarantee the distribution is highly uniform.
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The authors would like to thank the anonymous reviewers for their valuable suggestions. We would also like to thank Dr. Zhixiong Chen for his help and advice in this work. This material is based upon work supported by the National Science Foundation under Grant No. CNS-1420227. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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Gu, T., Klapper, A. Statistical properties of half-ℓ-sequences. Cryptogr. Commun. 8, 383–400 (2016). https://doi.org/10.1007/s12095-015-0152-7
- Feedback with carry shift register (FCSR)
- Half- ℓ-sequences
Mathematics Subject Classification (2010)