About computation of the linear complexity of generalized cyclotomic sequences with period p n+1

Abstract

We propose a computation method for linear complexity of series of generalized cyclotomic sequences with period p n+1 . This method is based on using the polynomial of the classic cyclotomic sequences of period p. We found the linear complexity of generalized cyclotomic sequences corresponding to the classes of biquadratic residues and Hall sequences.

On the linear complexity of Hall’s sextic residue sequences over $${ GF}(q)$$ G F ( q )

“…With similar arguments as in [8] we obtain the following results for F(x). (6), and p ≡ −1 (mod q) or q / ∈ H (6).…”

“…The linear complexity over F 2 of s was investigated in [6,8]. Here we regard it over the finite field GF(q).…”

“…Let β be a primitive p n th root of unity in the extension of GF(q) such that β p n−1 = α. The summae j∈C k β j were investigated in [8] …”

“…The linear complexity of other binary cyclotomic sequences of order 6 over the finite field of order 2 was investigated in [3,7,15] (also, see references therein). Further, the linear complexity of series of generalized cyclotomic sextic sequences over the finite field of order 2 was studied in [6,8]. Authors [6] spoke of generalization of the Hall's sextic residue sequences in this case.…”

Autocorrelation Values of Generalized Cyclotomic Sequences with Period pn+1

“…In 2011 Edemskiy [20] proposed a method for computing the linear complexity of p n+1 -periodic generalized cyclotomic binary sequences. For details, suppose g is a primitive root of p n+1 , where p = dR + 1 is an odd prime, d, R are two non-negative integers, and n > 0 is a positive integer.…”