Polynomials Over Finite Fields And Their Applications In The Public Key Cryptosystem

Content:The public key cryptosystem was firstly presented by Diffie and Hellman in 1976.From then on,it has aroused too much attention and has been widely used in real life.Different kinds of analogs of the public key cryptosystem have been carried out in different areas. Based on the polynomial properties over finite fields,a new RSA public key cryptosystem and a new ElGamal public key cryptosystem are presented,the security of the new cryptosystems is higher.The paper firstly discusses the factoring of polynomials over finite fields,gives some theorems,which are about the number of the irreducible polynomials and the primitive polynomials,the necessary and sufficient condition of a primitive polynomial and so on.Principles of obtaining new irreducible polynomials with higher degree from the polynomials with lower degree are also described in the paper.Extensive and profound discussions about the properties of the irreducible polynomials and the primitive polynomials are provided.Secondly,based on the polynomial properties over finite fields,a new analog of the RSA is given,whose security is higher than the anologs of the RSA type public key cryptosystem presented in the papers by Qi Sun,Zhenfu Cao and Bin Zhang.And it is also shown that the ciphertext extension problem in Zhenfu Cao's system no longer exists by properly choosing some parameters.The security of the new cryptosystems mainly bases on the factorization of large integers.Finally,anew ElGamal public key cryptosystem based on a polynomial over finite fields is presented.