On the Maximum Order Complexity of Thue–Morse and Rudin–Shapiro Sequences along Polynomial Values

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.

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

“…Indeed, for an integer m and k large enough (of order of length of m), the expansion of mL k is the expansion of m centered around F k . Notice that this result is different from the q-base since the digits here appear on both sides of the expansion of m. We will see the impact in our main result Theorem 1.2 with the occurence of N 1/(2d) in the place of N 1/d that we got in the case of q-base expansion, see [21].…”

“…A proof of this conjecture would imply that the lower bound proved by Popoli [21] is optimal. The maximum order complexity of S ϕ is algorithmically more difficult to handle.…”

“…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.…”

“…Drmota, Mauduit et Rivat [8] showed that the Thue-Morse sequence along squares is normal and Moshe [19] showed that the polynomial subsequences of degree d ≥ 2 of the Thue-Morse sequence have an exponential subword complexity. Sun and Winterhof [27] and Popoli [21] showed that the maximum order complexity of the Thue-Morse sequence along polynomial subsequences remains comparatively large. A large maximum order complexity is desired, however, it should be noted that it should be not too large since the correlation measure of order 2 gets large in that case (see [12,Proposition 3.1]).…”

“…Indeed, as we will state later, the transition from k to k +1 adds exactly one block of 10. We have already used a similar method in the usual 2-base, see [21,proof of Lemma 6].…”