To solve the problem of constructing feedback functions of non-linear maximal length shift registers (NMLSR)

the method to construct the feedback functions for NMLSR was proposed based on root function.First of all

we expatiated the classification method of the root functions modulo 3

proveded the characteristic state sets of the root functions and extracted the eigenfunctions of the root functions.Secondly

we synthesized the feedback function for the nonlinear m subsequence shift register according to the filter of the eigenfunctions on the root functions.Finally

we conducted a great number of calculations and analyses for new type sequences generated from the nonlinear m subsequence shift registers.The results show unanimously that the nonlinear m subsequences dose not only possess better period property

balance property and run-path property

but also the sharp autocorrelation property and the ideal linear complexity.