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

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 Nth maximum order complexity is of order of magnitude log N whereas it is easy to find families of sequences with Nth 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 the 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.

On the pseudorandomness of automatic sequences

“…In Section 5 we will see that such a large maximum order complexity points to undesirable structure in a sequence. The N th maximum order complexity of the Rudin-Shapiro sequence and some generalizations is also of order of magnitude N , see [92,Theorem 2]. In particular we have…”

Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences

“…Indeed, such a sequence can be built with relatively short blocks of consecutive terms. However, a sequence with large maximum order complexity can still be predictable, as it is known for the Thue-Morse sequence, see [15,28]. Note also that the largest possible order of magnitude of M(S, N) is N and the expected value of M(S, N) is log N , see [12,20].…”

“…The maximum order complexity has been studied by several authors, for general results see [11-13, 20, 31] or [21,[27][28][29] for applications to some particular sequences such as the Thue-Morse sequence or the Rudin-Shapiro sequence. From a computational perspective, Jansen [12,Proposition 3.17] showed how Blumer's DAWG (Direct Acyclic Weighted Graph) algorithm [3] can be used to compute the maximum order complexity in linear time and memory.…”

“…The Thue-Morse sequence has a large maximum order complexity, see [28], but this sequence is far from being pseudorandom. Indeed, it is well-known that E(T , N) ≤ 5 for all N since h(x, y)…”

Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences

“…Indeed, such a sequence can be built with relatively short blocks of consecutive terms. However, a sequence with large maximum order complexity can still be predictable, as it is known for the Thue-Morse sequence, see [15,28]. Note also that the largest possible order of magnitude of M (S, N ) is N and the expected value of M (S, N ) is log N , see [12,20].…”