Maximum-Order Complexity and Correlation Measures

Designs Codes and Cryptography, 1999

We obtain the upper bound O(2 14n/15 n −1/5) on the number of distinct values of all possible correlation functions between M-sequences of order n.

Uniform distribution theory

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.

Theoretical Computer Science, 2002

Our goal is to study the complexity of inÿnite binary recursive sequences. We introduce several measures of the quantity of information they contain. Some measures are based on size of programs that generate the sequence, the others are based on the Kolmogorov complexity of its ÿnite preÿxes. The relations between these complexity measures are established. The most surprising among them are obtained using a speciÿc two-players game 2 .

Indagationes Mathematicae, 2009

We estimate the linear complexity profile of m-ary sequences in terms of their correlation measure, which was introduced by Mauduit and Sárközy. For prime m this is a direct extension of a result of Brandstätter and the second author. For composite m, we define a new correlation measure for m-ary sequences, relate it to the linear complexity profile and estimate it in terms of the original correlation measure. We apply our results to sequences of discrete logarithms modulo m and to quaternary sequences derived from two Legendre sequences. s n+L = c L−1 s n+L−1 + · · · + c 0 s n , 0 n N − L − 1, MSC: 11K36, 94A55, 94A60

2004

In this paper we study the average behavior of the number of distinct substrings in a text of size n over an alphabet of cardinality k. This quantity is called the complexity index and it captures the “richness of the language” used in a sequence. For example, sequences with low complexity index contain a large number of repeated substrings and they eventually become periodic (eg, tandem repeats in a DNA sequence).

Science China Information Sciences, 2011

We improve previous results on the asymptotic behavior and the expected value of the joint linear complexity of random multisequences over finite fields. These results are of interest for word-based stream ciphers in cryptology.

The Ramanujan Journal, 2013

The linear complexity is an important and frequently used measure of unpredictability and pseudorandomness of binary sequences. In this paper our goal is to extend this notion to two dimensions. We will define and study the linear complexity of binary lattices. The linear complexity of a truly random binary lattice will be estimated. Finally, we will analyze the connection between the linear complexity and the correlation measures, and we will utilize the inequalities obtained in this way for estimating the linear complexity of an important special binary lattice. Finally, we will study the connection between the linear complexity of binary lattices and of the associated binary sequences.

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete

Journal of physics, 2019

In this paper, we study the linear complexity of series of binary sequences with optimal autocorrelation magnitude of length 4𝑁. These sequences are obtained from almost-perfect binary sequences and binary sequences with optimal autocorrelation of length 𝑁. The construction of these sequences were presented E.I. Krengel and P.V. Ivanov. We show that considered sequences have the high linear complexity. Also we derive the 1-error linear complexity of these sequences.

Information and Computation, 2002

Predictive complexity is a generalization of Kolmogorov complexity. It corresponds to an "optimal" prediction strategy and gives a natural lower bound to ability of any algorithm to predict elements of a sequence of outcomes. A natural question is studied: how complex can easy-to-predict sequences be? The standard measure of complexity, used in the paper, is Kolmogorov complexity K (which is close to predictive complexity for logarithmic loss function). The difficulty of prediction is measured by the notion of predictive complexity KG for bounded loss function (of nonlogarithmic type). We present an asymptotic relation sup x:l(x)=n K (x | n) KG(x) ∼ 1 a log n, when n → ∞, where a is a constant and l(x) is the length of a sequence x. An analogous asymptotic relation holds for relative complexities K (x | n)/n and KG(x)/n, where n = l(x). To obtain these results we present lower and upper bounds of the cardinality of all sequences of given predictive complexity.