Based On The Theory Of Hyperbolic Arithmetic Determination Of Safe Primes

With the improvement of network information, the information security in this complex world play a very important role. However, in order to guarantee the absolute security of information, technology is necessary. If there has no advanced technology, we cannot understand the frontier or strategic information in time, then the development or research will be in a passive state. Therefore, with the development of technology, attacks by breakers and breakthroughs by scholars that indicate cryptosystem will face many various threatens. More and more researchers pay attention to security problems.On the basis of hyperbolic curve arithmetic theory, which was proposed by Professor Wang Rui, we can study the rules of hyperbolic curve whose points on the hyperbolic curve can consist of a hyperbolic curve multiplication group. According to the operation of hyperbolic curve group, It’s easy to solve the solutions of equation in finite field. And it cannot be restricted by too many parameters. To satisfy the basic requirement of encryption system, it can encrypt and decrypt information by hyperbolic curve arithmetic theory. Firstly, in the cryptography system the plaintext can be converted into hyperbolic points. Secondly, if we can randomly select the parameters, then the plaintexts can be encrypted and decrypted. However, Encrypting and decrypting information lie to select the large prime, this situation has some influences to break information. Therefore, whether a large number is prime or not, especially safe primes, the method to distinguish it is quite important.In this paper, I mainly calculate and prove the order of hyperbolic curve in the finite field. The result is that the different modulus will determine various orders of hyperbolic curve group. That is to say the simple polynomial can express the order of hyperbolic curve. In the process of determining primality, according to some theorems, judging modulus’s primality by inequality will transform into judge the primality of point’s order. It will make judgment blindness. In order to eliminate this disadvantage, according to Lagrange theory and multiplication group, I propose a new method for determining a prime number, the primality of larger number can be obtained from the order of hyperbolic point. Finally, an example prove the feasibility of the method.