On the Maximum Order Complexity of the Thue-Morse and Rudin-Shapiro Sequence

• Published in: Uniform distribution theory Vol. 14; no. 2; pp. 33 - 42
• Main Authors: Sun, Zhimin, Winterhof, Arne
• Format: Journal Article
• Language: English
• Published: Sciendo 01.12.2019
• Subjects: 11B85; 11K45; automatic sequences; maximum order complexity; measures of pseudorandomness; Rudin-Shapiro sequence; Thue-Morse sequence
• ISSN: 2309-5377; 2309-5377
• DOI: 10.2478/udt-2019-0012

Expansion complexity and maximum order complexity are both finer measures of pseudorandomness than the linear complexity which is the most prominent quality measure for cryptographic sequences. The expected value of the th maximum order complexity is of order of magnitude log whereas it is easy to find families of sequences with th expansion complexity exponential in log . This might lead to the conjecture that the maximum order complexity is a finer measure than the expansion complexity. However, in this paper we provide two examples, the Thue-Morse sequence and the Rudin-Shapiro sequence with very small expansion complexity but very large maximum order complexity. More precisely, we prove explicit formulas for their th maximum order complexity which are both of the largest possible order of magnitude . We present the result on the Rudin-Shapiro sequence in a more general form as a formula for the maximum order complexity of certain pattern sequences.