Research On Several Nonlinear Feedback Shift Registers Based On Semi-tensor Product Of Matrix

As the steam cipher is widely used in many areas such as military,business and diplomacy and so on.Hence,the research of steam cipher has attracted many scholars.The nonlinear feedback shift registers(NLFSR)are main build component of construction of generating the steam cipher.Hence,the research of NLFSR has been always received much attention.In recent years,with the appearance of the new tool of matrix computation,the research of steam cipher has arisen many new hot spots.In this paper,the tool of semi-tensor product(STP)of matrix is the main computation tool to inves-tigate NLFSR.We use the method of STP to investigate the equivalence between two kinds of NLFSR,the nonsingularity of cascade of NLFSR and the properties and stability of(n,k)NLFSR.The main work of this dissertation are listed as follows:In Chapter 1,there is an introduction of STP method of matrix.In Chapter 2,some notations and knowledge about semi-tensor product of matrices are presented,includ-ing definition,lemmas,properties and so on.In Chapter 3,we investigate the equivalence between Galois NLFSR and Fibonacci NLFSR and in which condition these two kinds of NLFSR are equivalent.By using the method of STP,we can turn the complex expression of NLFSR into a linear system,which provides a convenient way to investigate the equivalence of Galois NLFSR and Fibonacci NLFSR.Based on the properties of Galois NLFSR and Fibonacci NLFSR,we propose two algorithms to implement the transformation between Galois NLFSR and Fibonacci NLFSR.In this chapter,we also analyze the complexities of two algorithms proposed in this chapter.At last,we give an example to illustrate the feasibility of the algorithms in this chapter.In Chapter 4,we investigate the nonsingularity of Grain cascade NLFSR.We use the method of STP to turn algebraic forms of Grain cascade NLFSR into a linear system.In this chapter,we propose a necessary and sufficient condition of nonsingularity of Grain cascade NLFSR.Then the Grain cascade NLFSR can be regarded as a Boolean control network(BCN)with one input.we can investigate the nonsingularity of BCN to investigate the general Grain cascade NLFSR.At last,we give an example to illustrate the feasibility of the theorems in this chapter.In Chapter 5,we investigate the(n,k)NLFSR.At first,by using the method of STP,we turn(n,k)NLFSR into linear system.After that,based on the linear system,we investigate the stability of(n,k)NLFSR.Then,the period of(n,k)NLFSR is studied in this chapter,and we propose an algorithm to investigate the period of(n,k)NLFSR.Then,the period of composed(n,k)NLFSR is investigated.At last,three examples are given to illustrate the feasibility of the proposed methods in this chapter.In Chapter 6,a brief conclusion is presented to end this work,and the prospect for the work is also made.