11 research outputs found
Maximum-order Complexity and Correlation Measures
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 cannot have very small
maximum-order complexity
Multisequences with high joint nonlinear complexity
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 -adic complexity
The -adic complexity has been well-analyzed in the periodic case. However,
we are not aware of any theoretical results on the th -adic complexity of
any promising candidate for a pseudorandom sequence of finite length or
results on a part of the period of length of a periodic sequence,
respectively. Here we introduce the first method for this aperiodic case. More
precisely, we study the relation between th maximum-order complexity and
th -adic complexity of binary sequences and prove a lower bound on the
th -adic complexity in terms of the th maximum-order complexity. Then
any known lower bound on the th maximum-order complexity implies a lower
bound on the th -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 -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 -adic complexity
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.Comment: 23 pages, 5 figures, 4 table