Maximum-Order Complexity and Correlation Measures

Işık, Leyla; Winterhof, Arne

doi:10.3390/cryptography1010007

Cited by 13 publications

(7 citation statements)

References 16 publications

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“…This paper presents generalizations of the results appearing in the articles [4] and [11]. Thanks to these results, it is possible to use these results mount correlation attacks in systems using standard families of binary sequences like Gold codes and Kasami families (see Table 1).…”

Section: Conclusion and Acknowledgmentsmentioning

confidence: 76%

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Correlation, Linear Complexity, Maximum order Complexity on Families of binary Sequences

Chen¹,

Gómez²,

Gómez-Pérez

et al. 2021

Preprint

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Correlation measure of order k is an important measure of randomness in binary sequences. This measure tries to look for dependence between several shifted version of a sequence. We study the relation between the correlation measure of order k and another two pseudorandom measures: the N th linear complexity and the N th 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.

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Section: Conclusion and Acknowledgmentsmentioning

confidence: 76%

“…Recently, Işık and Winterhof [11] have derived an analogous result concerning the Nth maximum-order complexity:…”

Section: Introductionmentioning

confidence: 97%

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Correlation, Linear Complexity, Maximum order Complexity on Families of binary Sequences

Chen¹,

Gómez²,

Gómez-Pérez

et al. 2021

Preprint

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show abstract

“…The correlation measure of order k introduced by Mauduit and Sárközy [12] is another figure of merit which is finer than the linear complexity, see [1]. A cryptographic sequence must have small correlation measure of all orders k up to a sufficiently large k. In [6], the maximum order complexity of a binary sequence was estimated in terms of its correlation measures. Roughly speaking, it was shown that any sequence with small correlation measure up to a sufficiently large order k cannot have very small maximum order complexity.…”

Section: Final Remarksmentioning

confidence: 99%

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

Sun

Winterhof

2019

Uniform Distribution Theory

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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.

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“…The maximum order complexity has been studied by several authors, for general results see [11,12,13,20,31] 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.…”

Section: Introductionmentioning

confidence: 99%

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

Jamet¹,

Popoli²,

Stoll³

2021

Preprint

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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.

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Maximum-Order Complexity and Correlation Measures

Abstract: Abstract:We estimate the maximum-order complexity of a binary sequence in terms of its correlation measures. Roughly speaking, we show that any sequence with small correlation measure up to a sufficiently large order k cannot have very small maximum-order complexity.

Cited by 13 publications

References 16 publications

Correlation, Linear Complexity, Maximum order Complexity on Families of binary Sequences

Correlation, Linear Complexity, Maximum order Complexity on Families of binary Sequences

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

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

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