On the maximum order complexity of Thue-Morse and Rudin-Shapiro sequences along polynomial values

TL;DR

This paper investigates the maximum order complexity of Thue-Morse and Rudin-Shapiro sequences along polynomial subsequences, showing they retain high complexity and addressing an open problem in sequence complexity analysis.

Contribution

It generalizes previous results to any polynomial subsequence, extending understanding of sequence complexity and providing new insights into their cryptographic potential.

Findings

Thue-Morse sequence along polynomial subsequences maintains high maximum order complexity.

Rudin-Shapiro sequence exhibits similar complexity properties along polynomial subsequences.

Addresses and resolves an open problem posed by Sun and Winterhof.

Abstract

Both the Thue-Morse and Rudin-Shapiro sequences are not suitable sequences for cryptography since their expansion complexity is small and their correlation measure of order 2 is large. These facts imply that these sequences are highly predictable despite the fact that they have a large maximum order complexity. Sun and Winterhof (2019) showed that the Thue-Morse sequence along squares keeps a large maximum order complexity. Since, by Christol's theorem, the expansion complexity of this rarefied sequence is no longer bounded, this provides a potentially better candidate for cryptographic applications. Similar results are known for the Rudin-Shapiro sequence and more general pattern sequences. In this paper we generalize these results to any polynomial subsequence (instead of squares) and thereby answer an open problem of Sun and Winterhof. We conclude this paper by some open problems.