Reduced RSA Public Key Cryptographic Algorithm Of Polynomial Form On Finite Fields

The public key cryptosystem was firstly presented by Diffie and Hellman in1976. Since then on, it has aroused too much attention and has been widely applied in real life. In1978, Rivest, Shamir and Adleman proposed the first practical RSA public-key cryptosystemand has been widely applied in practice. Qi Sun proposed a polynomial on finite fields of the RSA public key cryptosystem in1985. Until1999, Zhenfu Cao pointed out the insecurity of the algorithm proposed by Qi Sun, and then he improved the algorithm and proposed a RSA analog of polynomial over finite fields, however, it esisted ciphertext extended problem. Then Bin Zhang solved the ciphertext extended problem and presented a better analog of RSA subsequently. Since2002, the research of RSA analog on finite fields which was proposed by Qi Sun, Zhenfu Cao and Bin Zhang et al. didn’t get further development and have not cited instance to verify such algorithms in a lot of literatures which I read.Based on the previous work what present above, this paper apply kth power residue theory into the RSA analog algorithm of polynomial form on the finite fields, reduce it and then get a new algorithm called reduced RSA public key cryprographic algorithm of polynomial form on finite fields, and abbreviate PK-RSA. The algorithm uses its factor k to replace its mode(?)r(g(x)), makes the public and private key length shortened signigicantly, and then improve the speed of encryption and decryption. At the same time, the security of the algorithm is mainly based on the double difficulties of solving large integer factorization and discrete logarithm problems, thus it has a high level of security. And then we use a simple instance to verify the feasibility of the algorithm.Then, we analysis the functionality of the algoritm in detail, the two most significant functions are the speed improvement of encryption and decryption and hierarchical management. Due to we apply kth power residue theory, it makes the mode value become smaller significantly and public and private key length become shorter. Therefore, the speed of encryption and decryption are improved significantly. At the same time, we can encrypt and decrypt for different secutiry levels of departments or files depending on the size of k value, thus it disperses the permissions of the key computing and load and has a practical significance.Lastly, we summarize the characteristics of the algorithm and give lights to the future of the algorithm based on the existed problems. In the future, we can improve the speed of password system through parallel computing and multi-processor, and ensure the greater security of password system through the research of more general kth equation.