Some Properties Of σ - Polynomials

Theσ-LFSR is a kind of linear feedback shift register based on word., the sequence generated from which has the quality of both security and realize the efficiency and resource consumption. In the password code, to get the pseudo random sequence with the largest circle is a basic research topic, while a primitive σ-LFSR sequence is the one that can achieves theoretically the greatest sequences, In order to get a primitive σ-LFSR sequence, the judgment of primitive of a polynomial ofσ-LFSR sequence becomes a key problem urgently need to solve.This paper studies properties ofs-polynomial, which is a continuation of the research work of Han Wenbao. Han Wenbao get the conclusion that "the primitive of a sequence of σ-LFSR is equivalent to that of its characteristic polynomial determinant", but a method of computing the determinant of the relatives-polynomial is still expected. In this paper, first we give a simple methods of computing the determinant of a polynomial over a finite field; Then, through translating a s- polynomial into a polynomial matrix under some base, we conclude that the smith standard form of the polynomial matrix of a primitive s-polynomial is unique. Moreover, for a special kind ofs- polynomial, we give a fast algorithm of translation from thes- polynomial to its corresponding polynomial matrix, and then we give a screening method for this special primitive polynomial.Finally, according to another work of Han Wenbao, by studying the relations of sequence and it’s coordinate sequence, we obtain that the determinant ofs-polynomial is just the minimal polynomial of it’s coordinate sequence.