scholarly journals Maximum-Order Complexity and Correlation Measures

Cryptography ◽  
2017 ◽  
Vol 1 (1) ◽  
pp. 7 ◽  
Author(s):  
Leyla Işık ◽  
Arne Winterhof
Keyword(s):  
Maximum Order ◽  
Correlation Measures ◽  
Maximum Order Complexity

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

2022 ◽  
Vol 78 ◽  
pp. 101977
Author(s):  
Zhixiong Chen ◽  
Ana I. Gómez ◽  
Domingo Gómez-Pérez ◽  
Andrew Tirkel
Keyword(s):  
Linear Complexity ◽  
Binary Sequences ◽  
Correlation Measure ◽  
Maximum Order ◽  
Maximum Order Complexity

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

2019 ◽  
Vol 14 (2) ◽  
pp. 33-42
Author(s):  
Zhimin Sun ◽  
Arne Winterhof
Keyword(s):  
Linear Complexity ◽  
Expected Value ◽  
Quality Measure ◽  
Explicit Formulas ◽  
Maximum Order ◽  
Order Of Magnitude ◽  
Large Maximum ◽  
Maximum Order Complexity

AbstractExpansion 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.

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

2020 ◽  
Vol 15 (2) ◽  
pp. 9-22
Author(s):  
Pierre Popoli
Keyword(s):  
Open Problem ◽  
General Pattern ◽  
Correlation Measure ◽  
Open Problems ◽  
Maximum Order ◽  
Large Maximum ◽  
Maximum Order Complexity

AbstractBoth 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.

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

2021 ◽  
Author(s):  
Damien Jamet ◽  
Pierre Popoli ◽  
Thomas Stoll
Keyword(s):  
Natural Generalization ◽  
Correlation Measure ◽  
Pseudorandom Sequences ◽  
Maximum Order ◽  
Automatic Sequences ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
Large Maximum ◽  
Maximum Order Complexity ◽  
Subword Complexity

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

2019 ◽  
Vol 12 (3) ◽  
pp. 415-426
Author(s):  
Zhimin Sun ◽  
Xiangyong Zeng ◽  
Da Lin
Keyword(s):  
Maximum Order ◽  
Maximum Order Complexity

scholarly journals Residual dimension of nilpotent groups

2020 ◽  
Vol 23 (5) ◽  
pp. 801-829
Author(s):  
Mark Pengitore
Keyword(s):  
New York ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Nontrivial Element ◽  
Maximum Order ◽  
Asymptotic Bounds ◽  
Group Morphism ◽  
A Finite Group

AbstractThe function {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of {\mathrm{F}_{N}(n)} can be fully characterized.

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