The Application Of Hyperbolic Curve In Public Key Cryptography

Since Diffie and Hellman proposed the concept of public key cryptography in 1976, because of its great advantages both in key distribution and digital signature, scholars have put forward many kinds of public key cryptography algorithm. The most typical are:RSA, Rabin, ElGamal and Elliptic Curve cryptosystems. These public key cryptosystems are based on a difficult mathematical function, such as the RSA algorithm based on NP-hardness of the integer factorization, with science and technology progress as well as cryptanalysts to constantly optimize decipher algorithms, the classic cryptographic algorithms will also face a variety of threats. If we pay no attention to the innovation of cryptographic algorithms, the consequences can be very serious. In addition to improving the existing public key cryptographic algorithms, a new public key cryptography algorithm is urgently needed.Based on the above background, in order to ensure the safety of information, a new public key cryptosystem called the hyperbola cryptosystem is discussed here under the guidance of Professor Wang Rui. The hyperbola cryptosystem is based on the hyperbolic arithmetic theory, which is derived from the Pell equation. Over the integer ring, the points on the hyperbola form an infinite cyclic group. Hyperbola cryptosystem is discussed over the finite field GF (P), namely HGF(p) Over the finite field, the point multiplication on the hyperbola HGF(p) is a finite cyclic group. Moreover, the generated element of this cyclic group is not necessarily the basic solution of the Pell equation. The security of the hyperbolic cryptosystem is based on the discrete logarithm over the hyperbola. The encryption and decryption of the hyperbola cryptosystem is a polynomial time complexity, while the attack requires exponent time complexity. In this paper, the hyperbola key exchange, encryption and decryption and digital signature are discussed in this paper. In the end, the performance of the hyperbola, together with the advantages of the hyperbolic system are analyzed and discussed.In this paper, an important advantage of the hyperbola cryptosystem is that the order of the hyperbolic group is certain, and there is an exact formula. In addition, the hyperbola parameters are less, and the user can quickly select the appropriate hyperbola.Moreover, both the ELGamal cryptosystem and the hyperbola two-key cryptosystem, which are implemented with the hyperbola, do not need to embed the plaintext message into the hyperbola. This means that the encryption process is simplified. On the other hand, by changing the parameters of HGF(p), we can get different hyperbola groups. The hyperbola cryptosystem has rich group structures and muti-choice, which provide an alternative space for the design of the hyperbola cryptosystem system. Finally, according to the SChnorr signature algorithm and the DSA signature algorithm, we can establish a hyperbola digital signature system. In short, the hyperbola cryptosystem is a secure and feasible public key cryptosystem. It has broad application prospects.