On the maximum order complexity of the Thue-Morse and Rudin-Shapiro sequence

TL;DR

This paper demonstrates that the Thue-Morse and Rudin-Shapiro sequences have very small expansion complexity but maximal maximum order complexity, challenging the idea that expansion complexity is a finer measure of pseudorandomness.

Contribution

It provides explicit formulas for the maximum order complexity of these sequences, showing they can have large maximum order complexity despite small expansion complexity.

Findings

Thue-Morse sequence has maximum order complexity of order N.

Rudin-Shapiro sequence also has maximum order complexity of order N.

These sequences have small expansion complexity but large maximum order complexity.

Abstract

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 $N$th maximum order complexity is of order of magnitude $\log N$ whereas it is easy to find families of sequences with $N$th expansion complexity exponential in $\log N$. 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 $N$th maximum order complexity which are both of largest possible order of magnitude $N$. 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.