The Implementation Of Classical Public Key Cryptography Algorithm In Hyperbolic Curve Arithmetic

In 1976,the concept of public key cryptography was proposed by Whitfield Diffie and Martin Hellman.Public key cryptography has been developing for more than 40 years.Classical public-key cryptography algorithms:Elliptic curve cryptography and RSA algorithms have weaknesses in the arithmetic theory,so this This thesis proposes a public-key cryptosystem based on hyperbolic arithmetic to make up for some deficiencies of classical public-key cryptography algorithms.The main work of this thesis is as follows:(1)This thesis introduces hyperbolic arithmetic:?Unit hyperbola on a finite field GF(p);? Hyperbolic curve bundle on a finite field GF(p);?Unit hyperbola on compound modulo.The entire solution of the unit hyperbola and hyperbolic bundle on the finite field GF(p)are cyclic groups.Hyperbolic arithmetic can apply to the public key cryptography algorithm,and the selection of the base point G has a theoretical basis.(2)In this thesis,the classical Diffie-Hellman algorithm,ElGamal encryption algorithm and RSA algorithm are implemented one by one in hyperbola arithmetic,respectively forming HC-Diffie-Hellman algorithm,HC-ElGamal encryption algorithm and HC-RSA algorithm.This thesis not only verify the correctness of hyperbolic arithmetic,but also prove that these popular public key cryptography algorithms can be easily realized in hyperbolic arithmetic.(3)This thesis analyzes the security and advantages of the new algorithm in detail.This thesis proves that the time complexity of decoding HC-Diffie-Hellman algorithm and HC-ElGamal encryption algorithm is higher than that of the original algorithm.The HC-RSA algorithm can adjust the parameters to choose the plaintext space,which makes up for the lack of RSA algorithm to determine the modulus and the plaintext space can not be changed.(4)The order of hyperbolic cipher has a definite formula to make up for the elliptic curve cryptogram of the order not clear.When(D/P)=-1,where(D/P)is a Legendre symbol,making passwords on hyperbolic clusters on finite field GF(p),there is no need to encode the encrypted message,which makes up for the lack of coding for the elliptic curve encryption.At the same time,hyperbolic curve bundle cipher space is the square of the elliptic curve cipher space size,which means that deciphering of the hyperbolic curve bundle cipher is a square level higher than that of the deciphering elliptic curve cryptography.