Abstract
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.
Highlights
Feedback shift register (FSR) sequences are used for many cryptographic applications such as pseudorandom number generators for stream cipher cryptosystems, see [10]
In order to determine this shortness for an infinite sequence S, Jansen [12, 13] introduced the notion of N th maximum order complexity of S, denoted as M(S, N ), which is the length of the shortest FSR that generates the first N elements of S
If the mapping F is a linear transformation, the FSR is called a linear feedback shift register (LFSR). This leads to the notion of the linear complexity profile of S denoted as L(S, N )
Summary
Feedback shift register (FSR) sequences are used for many cryptographic applications such as pseudorandom number generators for stream cipher cryptosystems, see [10]. In order to determine this shortness for an infinite sequence S, Jansen [12, 13] introduced the notion of N th maximum order complexity of S, denoted as M(S, N ), which is the length of the shortest FSR that generates the first N elements of S. If the mapping F is a linear transformation, the FSR is called a linear feedback shift register (LFSR) This leads to the notion of the linear complexity profile of S denoted as L(S, N ). Diem [7] observed that these sequences and sequences based on function expansion into expansion series can be efficiently computed from relatively short sequences This leads to the notion of expansion complexity, denoted as E(S, N ), see Definition 2 below for more details. We indicate our main results of this paper (Section 1.3)
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