Maximum Order Complexity Research Articles

Maximum-Order Complexity and 2-Adic Complexity

Maximum-Order Complexity and 2-Adic Complexity

Note on a Fibonacci parity sequence

Let ftm = 0111010010001⋯ be the analogue of the Thue-Morse sequence in Fibonacci representation. In this note we show how, using the Walnut theorem-prover, to obtain the exact value of its maximum order complexity, previously studied by Jamet, Popoli, and Stoll. We strengthen one of their theorems and disprove one of their conjectures.

Correlation measure, linear complexity and maximum order complexity for families of binary sequences

The correlation measure of order k is an important measure of pseudorandomness for binary sequences. This measure tries to look for dependence between several shifted versions of a sequence. We study the relation between the correlation measure of order k and two other pseudorandom measures: the Nth linear complexity and the Nth maximum order complexity. We simplify and improve several state-of-the-art lower bounds for these two measures using the Hamming bound as well as weaker bounds derived from it.

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

Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue–Morse sequence, are better candidates for pseudorandom sequences. A natural generalization of automatic sequences are morphic sequences, given by a fixed point of a prolongeable morphism that is not necessarily uniform. In this paper we prove a lower bound for the maximum order complexity of the sum of digits function in Zeckendorf base which is an example of a morphic sequence. We also prove that the polynomial subsequences of this sequence keep large maximum order complexity, such as the Thue–Morse sequence.

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.

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.

A Note on Hall’s Sextic Residue Sequence: Correlation Measure of Order $k$ and Related Measures of Pseudorandomness

It is known that Hall’s sextic residue sequence has some desirable features of pseudorandomness: an ideal two-level autocorrelation and linear complexity of the order of magnitude of its period $p$ . Here we study its correlation measure of order $k$ and show that it is, up to a constant depending on $k$ and some logarithmic factor, of order of magnitude $p^{1/2}$ , which is close to the expected value for a random sequence of length $p$ . Moreover, we derive from this bound a lower bound on the $N$ th maximum order complexity of order of magnitude $\log p$ , which is the expected order of magnitude for a random sequence of length $p$ .

On the Nth maximum order complexity and the expansion complexity of a Rudin-Shapiro-like sequence

Based on the parity of the number of occurrences of a pattern 10 as a scattered subsequence in the binary representation of integers, a Rudin-Shapiro-like sequence is defined by Lafrance, Rampersad and Yee. The N th maximum order complexity and the expansion complexity of this Rudin-Shapiro-like sequence are calculated in this paper.

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

ABSTRACTAutomatic 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.

An Approximate Distribution for the Maximum Order Complexity

In this paper we give an approximate probability distribution for the maximum order complexity of a random binary sequence. This enables the development of statistical tests based on maximum order complexity for the testing of a binary sequence generator. These tests are analogous to those based on linear complexity.