Abstract
Based on the parity of the number of occurrences of a pattern 10 as a scattered subsequence in the binary representation of integers, a Rudin-Shapiro-like sequence is defined by Lafrance, Rampersad and Yee. The N th maximum order complexity and the expansion complexity of this Rudin-Shapiro-like sequence are calculated in this paper.
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References
Allouche, J.P., sequences, J. Shallit.: Automatic Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)
Allouche, J.P., Liardet, P.: Generalized Rudin-Shapiro sequences. Acta Arithmet LX.1, 1–27 (1991)
Allouche, J.P.: On a Golay-Shapiro-like sequence. Unif. Distrib. Theory 11(2), 205–210 (2016)
Brillhart, J., Morton, P.: A case study in mathematical research: the Golay-Rudin-Shapiro sequence. Am. Math. Mon. 103(10), 854–869 (1996)
Chan, L., Grimm, U.: Spectrum of a Rudin-Shapiro-like sequence. Adv. Appl. Math. 87, 16–23 (2017)
Diem, C.: On the use of expansion series for stream ciphers. LMS J. Comput. Math. 15, 326–340 (2012)
Jansen, C.J.A.: Investigations on Nonlinear Streamcipher Systems: Construction and Evaluation Methods. Ph.D.dissertation, Technical University of Delft, Delft (1989)
Jansen, C.J.A.: The Maximum Order Complexity of Sequence Ensembles. In: Davies, D.W. (ed.) Advances in Cryptology - EUROCRYPT ’91, Lect. Notes Comput. Sci., vol. 547, pp 153–159. Springer, Berlin (1991)
Lafrance, P., Rampersad, N., Yee, R.: Some properties of a Rudin-Shapiro-like sequence. Adv. Appl. Math. 63, 19–40 (2015)
Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences II. The Champernowne, Rudin-Shapiro, and Thue-Morse sequences, a further construction. J. Number Theory 73(2), 256–276 (1998)
Mérai, L., Niederreiter, H., Winterhof, A.: Expansion complexity and linear complexity of sequences over finite fields. Cryptogr. Commun. 9, 501–509 (2017)
Mérai, L., Winterhof, A.: On the N th linear complexity of automatic sequences. J. Number Theory 187, 415–429 (2018)
Müllner, C.: The Rudin-Shapiro sequence and similar sequences are normal along squares. Can. J. Math. 70(5), 1096–1129 (2018)
Sun, Z., Winterhof, A.: On the maximum order complexity of the Thue-Morse sequence and the Rudin-Shapiro sequence. Preprint (2017)
Sun, Z., Winterhof, A.: On the maximum order complexity of subsequences of the Thue-Morse sequence and the Rudin-Shapiro sequence along squares. Int. J. Comput. Math. Comput. Syst. Theory 4(1), 30–36 (2019)
Xing, C.P., Lam, K.Y.: Sequence with almost perfect linear complexity profiles and curves over finite fields. IEEE Trans. Inf. Theory 45(4), 1267–1270 (1999)
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The authors are supported by the National Natural Science Foundation of China Grant 61472120. The first author is also supported by China Scholarship Council.
This article is part of the Topical Collection on Special Issue on Sequences and Their Applications
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Sun, Z., Zeng, X. & Lin, D. On the Nth maximum order complexity and the expansion complexity of a Rudin-Shapiro-like sequence. Cryptogr. Commun. 12, 415–426 (2020). https://doi.org/10.1007/s12095-019-00396-0
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DOI: https://doi.org/10.1007/s12095-019-00396-0