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

TL;DR

This paper investigates the maximum order complexity of the sum of digits function in Zeckendorf base and its polynomial subsequences, showing they maintain high complexity, making them promising for cryptographic applications.

Contribution

It establishes a lower bound for the maximum order complexity of the Zeckendorf sum of digits function and demonstrates that polynomial subsequences preserve high complexity.

Findings

Sum of digits function in Zeckendorf base has high maximum order complexity.

Polynomial subsequences of this sequence also retain large complexity.

Results suggest potential cryptographic usefulness of these sequences.

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.