The Application Of Chebyshev Polynomial On Pulic Key Cryptography

With the rapid development of communication technology, The application of public key cryptography is becoming more and more popular and in-depth in the political,economic, military and other fields, followed by public key cryptosystem security issues have also aquired more and more concern and attention.In this paper,we study some cryptography systems based on Chebyshev polynomial.By using provable security ideas we anylyze securities of these cryptography systems in detail. We found that in some cases the security of Chebyshev public key cryptosystem is equivalent to discrete logarithm problem while in other is diffcult than solving discrete logarithm problem.Chebyshev polynomial can be seen as a linear shift register sequence,hence we discuss the period of Chebyshev polynomial through linear shift register sequence.and how the period influence the security of the cryptography system.Based on this we propose some suggestions on the selection of parameters.We conclude that in finite field Fp P-1and P+1should respectively have a large prime factor,so that small period won’t influence the seurity of Chebyshev public key cryptosystem.We study the advantage and disadvantage of the cryptography system established in finite field and ring and put forward some opinion to the rapid algorithm of these cryptography systems.This algorithm largely reduces bit operation and it can be calculated by parallel computing,hence it can significantly improve the running speed.RSA,Lucas and Chebyshev public key cryptography system are applications of Dickson polynomial,we study their periods by linear shift register sequence and analyze their securities through period and propose some suggestions on the selection of parameters.By the study of period we point out that cycling attacks take advantage of small period and put forward a more efficient attack against cryptography system similar to RSA.Cycling attacks require repeated power operation,this attack simply needs ciphertext multiplied by itself many times. To resist cycling attacks in RSA,p-1and q-1should have a big unreducible factor, a concise proof is proposed.