Study On The Sequences Generated By Trace Function Over Finite Field

Applied widely in the data encryption,pseudo-random sequences can be generated not only by hardware(shift register)but also by trace function.And it is found that the sequences generated by trace function are much better than those generated by hardware,especially in the aspects of correlation properties.The trace function,as a linear transform from an extension victor space to its basic victor space,has been comprehensively employed to study the properties of algebra construction in finite field.In recent years,trace function has become a strong method of studying stream cipher. For example,the generation,period,correlation function,linear span and so on.It has been proved to be a powerful as well as simple tool to study sequence cipher by numerous documents.In design of sequences for CDMA system,the most important properties of the sequences are low periodic correlation between all pairs of distinct sequences and large family size.By using the theories and properties of trace function over finite field,two families(unbalanced and balanced)of nonbinary sequences with large size and optimal periodic correlation property and large linear span are constructed based on Helleseth-Gong sequences in this paper.These sequences have very high application value in CDMA scheme and privacy communication system.The linear span of periodic sequences is an important index of unpredictability and randomness of sequences.We analyze the linear span of p-ary GMW sequences,and give the linear span of p-ary extended sequences of period p^em—1 which are constructed from the p-ary sequences of period p^m—1.The conclusion indicated that this kind of -ary extended sequences has ideal autocorrelation and large linear span.By using trace representation and binomial coefficient over finite field,it is proved that the linear span of the p-ary periodic sequences equals to the rank of its left repetition's base matrix in this thesis.The trinomial properties of sequences have close relationship with the least distance of antithesis code whose sequences code space as well as the linear span of sequences.In this thesis,we show that necessary and sufficient conditions for the trinomial properties of p-ary sequences of period pⁿ—1 generated by trace function over finite field.We also show that the trinomial pairs of p-ary sequences generated by trace function over finite field can be divided into regular and non-regular.Besides,it is proved that the p-ary GMW and cascaded p-ary GMW sequences have regular trinomial property, and give the corresponding regular trinomial pairs in this thesis.