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Character values of the Sidelnikov-Lempel-Cohn-Eastman sequences

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Abstract

Binary sequences with good autocorrelation properties and large linear complexity are useful in stream cipher cryptography. The Sidelnikov-Lempel-Cohn-Eastman (SLCE) sequences have nearly optimal autocorrelation. However, the problem of determining the linear complexity of the SLCE sequences is still open. It is well known that one can gain insight into the linear complexity of a sequence if one can say something about the divisors of the gcd of a certain pair of polynomials associated with the sequence. Helleseth and Yang (IEEE Trans. Inf. Theory 49(6), 1548–1552 2002), Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) and Meidl and Winterhof (Des. Codes Crypt. 8, 159–178 2006) were able to obtain some results of this type for the SLCE sequences. Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) mention that it would be nice to obtain more such results. We derive new divisibility results for the SLCE sequences in this paper. Our approach is to exploit the fact that character values associated with the SLCE sequences can be expressed in terms of a certain type of Jacobi sum. By making use of known evaluations of Gauss and Jacobi sums in the “pure” and “small index” cases, we are able to obtain new insight into the linear complexity of the SLCE sequences.

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Acknowledgments

The authors wish to thank the anonymous referees for their valuable comments. The research of Şaban Alaca is supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (RGPIN-2015-05208). Goldwyn Millar’s studies are supported by an Ontario Graduate Scholarship.

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  1. School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada

    Şaban Alaca & Goldwyn Millar

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  1. Şaban Alaca
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  2. Goldwyn Millar
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Correspondence to Şaban Alaca.

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Alaca, Ş., Millar, G. Character values of the Sidelnikov-Lempel-Cohn-Eastman sequences. Cryptogr. Commun. 9, 665–682 (2017). https://doi.org/10.1007/s12095-016-0208-3

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