Method Of Primality Testing Based On Hyperbolic Curve Arithmetic And Its Numerical Results

Predication of big prime number is one of the most difficult problems what people face in all ages.Historically,a lot of excellent mathematicians like Fermat,Dirichlet,Legendre and Euler,and so on,have spent a great deal of time and energy studying the problem about primes.In modern times,the famous mathematicians Jing-run Chen,Zhe-xuan Tao and Yi-tang Zhang were also constantly trying to explore the primes.With the development of computer science and Internet technology,prime number has been applied importantly in aspects of coding and modern public key cryptosystem,etc.So,the study of large prime numbers is still a hot and meaningful thing today.After introducing the basic theory of hyperbolic curve,this article proposes a new arithmetic theory-hyperbolic curve arithmetic.Also,the article finds that the points on the hyperbolic curve make up an Abel group in the finite field.According to the classical result,it can be further judged that the points on the hyperbolic curve make up a cyclic group in the finite field.In addition,a formula for calculating the order of hyperbolic curve is given.Since the order of the hyperbolic curve has a concise expression,it will be more efficient and simple of calculating on the hyperbolic curve than elliptic curve.The unit hyperbolic curve can be extended to the hyperbolic curve bundle,which further expands the space of the group.It will get more key space if the hyperbolic curve bundle is used in the public key cryptosystem,and there are more options while the hyperbolic curve bundle is applied in primality testing.Based on the new hyperbolic curve arithmetic theory,this article proposes a certain method of primality testing.This method transforms the primality testing of module into the primality testing of the order of hyperbolic curve,and it can further reduce the number that we need to judge by using this method over and over again,so that the computation quantity is smaller and the time is more efficient.Besides,due to the variability of hyperbolic curve parameter,this method is more flexible and even can be applied to other special forms of integers like Mersenne numbers.In the course of study,this article also finds a Carmichael Phenomenon on the hyperbolic curve which,in fact,is can be regarded as a generalize of method of primality testing based on hyperbolic curve arithmetic.