11 research outputs found

    Maximum-order Complexity and Correlation Measures

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    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 kk cannot have very small maximum-order complexity

    Multisequences with high joint nonlinear complexity

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    We introduce the new concept of joint nonlinear complexity for multisequences over finite fields and we analyze the joint nonlinear complexity of two families of explicit inversive multisequences. We also establish a probabilistic result on the behavior of the joint nonlinear complexity of random multisequences over a fixed finite field

    Maximum-order complexity and 22-adic complexity

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    The 22-adic complexity has been well-analyzed in the periodic case. However, we are not aware of any theoretical results on the NNth 22-adic complexity of any promising candidate for a pseudorandom sequence of finite length NN or results on a part of the period of length NN of a periodic sequence, respectively. Here we introduce the first method for this aperiodic case. More precisely, we study the relation between NNth maximum-order complexity and NNth 22-adic complexity of binary sequences and prove a lower bound on the NNth 22-adic complexity in terms of the NNth maximum-order complexity. Then any known lower bound on the NNth maximum-order complexity implies a lower bound on the NNth 22-adic complexity of the same order of magnitude. In the periodic case, one can prove a slightly better result. The latter bound is sharp which is illustrated by the maximum-order complexity of \ell-sequences. The idea of the proof helps us to characterize the maximum-order complexity of periodic sequences in terms of the unique rational number defined by the sequence. We also show that a periodic sequence of maximal maximum-order complexity must be also of maximal 22-adic complexity

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

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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.Comment: 23 pages, 5 figures, 4 table
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