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

Automatic sequences such as the Thue-Morse sequence and the Rudin-Shapiro sequence are highly predictable and thus not suitable in cryptography. In particular, they have small expansion complexity. However, they still have a large maximum order complexity. Certain subsequences of automatic sequences are not automatic anymore and may be attractive candidates for applications in cryptography. In this paper we show that subsequences along the squares of certain pattern sequences including the Thue-Morse sequence and the Rudin-Shapiro sequence have also large maximum order complexity but do not suffer a small expansion complexity anymore.