Research On Several Topics Of Cyclotomic Sequences

Pseudo random sequences are widely used in ranging systems, spread spectrum communication systems, code-division multiple access communication systems, glob-al positioning systems, software testing, and stream ciphers. For a periodic sequence, the length of the shortest linear feedback shift register (LFSR) which generates the sequence is called the linear complexity of sequences. In cryptography and the re-lated applications, pseudo-random sequences should have high linear complexity. In addition, binary sequences also can be generated by memory feedback shift register (FCSR), in order to resist the Berlekamp-Massey rational approximation algorithm. the sequences should possess high 2-adic complexity. A sequence is called perfect if the out-of-phase values of its periodic autocorrelation function are equal to zero. In communication systems and their related applications, low correlation is also an important indicator. In the thesis, we will study the linear complexity of a class of generalized cyclotomic sequences of period pq on general finite field, the 2-adic complexity of a class of generalized cyclotomic sequences of period pq, and the construction of perfect Guassian integer sequences based on generalized cyclotomic sequences of period p~2. The main results are listed as follows:1. Binary sequence can be viewed as a sequence over finite field GF(r),where r is an odd prime. We study the linear complexity of a class of generalized cyclotomic sequences with period pq over GF(r), The results show that the sequences over finite fields has high linear complexity.2. We study the 2-adic complexity of Ding-Helleseth generalized cyclotomic se-quence, and the results show that the sequence keeps having a high 2-adic complexity3. Based on the generalized cyclotomy of order 2 over Zp~2, p an odd prime, we construct a class of Gaussian integer sequence of period p~2 and determine its autocorrelation function distributions by cvclotomic number. Furthermore, we give the new constructions of perfect Gaussian integer sequences of period p~2.