# The exact autocorrelation distribution and 2-adic complexity of a class of binary sequences with almost optimal autocorrelation

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## Abstract

Pseudo-random sequences with good statistical properties, such as low autocorrelation, high linear complexity and large 2-adic complexity, have been used in designing reliable stream ciphers. In this paper, we obtain the exact autocorrelation distribution of a class of binary sequences with three-level autocorrelation and analyze the 2-adic complexity of this class of sequences. Our results show that the 2-adic complexity of such a binary sequence with period *N* is at least (*N* + 1) − log_{2} (*N* + 1). We further show that it is maximal for infinitely many cases. This indicates that the 2-adic complexity of this class of sequences is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).

### Keywords

Stream ciphers Pseudo-random sequences Autocorrelation 2-adic complexity### Mathematics Subject Classification (2010)

94A55 94A60 65C10## Notes

### Acknowledgments

The authors wishes to thank the editor and the reviewers for the valuable comments, which make our work greatly improved.

Parts of this work were written during a very pleasant visit of the first author to Carleton University in Ottawa, Canada. She wishes to thank the hosts for their hospitality.

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