Constructions And Applications Of Generalized Cyclotomic Sequences Derived From Fermat Quotients

This thesis will construct several classes of pseudo-random sequence based on Fermat quotients and analyze their properties and applications.Specifically,fol-lowing topics will be addressed including the structure and properties of binary sequences with periodic pq derived from Fermat quotients,the structure and prop-erties of binary sequences with periodic p^m+1qⁿ⁺¹ derived from Fermat quotients,trace representation of quaternary sequences over Z₄ based on generalized cyclotom-ic classes modulo p² derived from Fermat quotients.Our main research works are summarized as follows.1.We consider Fermat quotients over Zpq,using Fermat quotients to construct a new cyclotomy of order p with respect to pq.Then,we construct new classes of binary sequences with periodic pq based on the new cyclotomy,which in-clude several classes of binary sequences derived from Fermat quotients.The advantage of the constructions is that they allow one to choose their support set freely.Furthermore,under certain conditions,with some constrains on the support set,the new sequences axe proved to possess large linear complexities.2.We consider Fermat quotients over Zp^m+1qⁿ⁺¹.A new cyclotomy of order p with respect to p^m+1qⁿ⁺¹ derived from Fermat quotients is proposed in this paper.Then,we construct new classes of binary sequences with periodic p^m+1qⁿ⁺¹ based on the new cyclotomy.Furthermore,under certain conditions,the linear complexity of the proposed sequences is also determined.3.Based on Fermat quotients,we define a family of quaternary sequences over the residue class ring modulo 4 of length p².Using the theory of trace function in the finite field and coset,we calculate the discrete Fourier transform?DFT?of the sequences.The DFT helps us to determine the trace representation of the sequences.