Linear complexity of generalized cyclotomic sequences of order 4 over F_l

TL;DR

This paper investigates the linear complexity of generalized cyclotomic sequences of order 4 over finite fields, demonstrating that under certain conditions, these sequences possess high linear complexity, which is desirable for cryptographic applications.

Contribution

It extends the analysis of cyclotomic sequences from order 2 to order 4, providing explicit formulas and conditions for high linear complexity over finite fields.

Findings

Sequences of order 4 often have high linear complexity

Derived explicit formulas for linear complexity of these sequences

High linear complexity under certain conditions

Abstract

Generalized cyclotomic sequences of period pq have several desirable randomness properties if the two primes p and q are chosen properly. In particular,Ding deduced the exact formulas for the autocorrelation and the linear complexity of these sequences of order 2. In this paper, we consider the generalized sequences of order 4. Under certain conditions, the linear complexity of these sequences of order 4 is developed over a finite field F_l. Results show that in many cases they have high linear complexity.