User:KimitoHorie/Sandbox

Hyperbolic curve cryptosystem
Most general cryptosystem in the World may be the RSA presented at 1970’s as a public cryptosystem. In U.S. the DES has been a standard of cryptosystem, which is based on the complexity of the combinations of the matrix. But a speed of the computers has grown year by year, then these cryptosystem have to increase its bit numbers more large, or its cycle times more frequent, to get the amount of calculations time to keep its security. In the half of 1980’s, the elliptic curve cryptosystem was born as a public cryptosystem, recently studied with a heat in Japan, in connection with the resolutions of the mathematically difficult problems. The elliptic curve cryptosystem is based on the discrete logarithm problem of the group operation, and has a merit of using less bits numbers as compared with RSA.

In the year of 2007, as a one of public cryptosystem the hyperbolic cryptosystem was proposed by myself, which is based the discrete logarithm problem as a same with the elliptic curve cryptosystem. The hyperbolic cryptosystem is the name of general technologies, like key generation method, cryptosystem, signature verification method, and so on, using a new quadratic hyperbolic curve group, which was discovered at that time. A quadratic hyperbolic curve group is a finite commutative group, a kind of cyclic group, defined on the finite ring (in this case the residual ring Z/pZ), have a group operation consist of the 3 roots of the crossing points made from the function $$y=\frac{dx+e}{ax^{2}+bx+c}$$and the line. Above function can be converted by a linear transformation of the coefficients to the form of

$$y=\frac{x-b}{x^{2}+cx-a} (a,b,c,x,y \in Z/pZ)$$

, and this form includes the history of the birth of a quadratic hyperbolic curve group, so I use it later. Said quadratic hyperbolic curve is a number theory function, include the denominator that means the inverse element of the residual ring Z/pZ, and do not have a relation with Geometry. So you have to consider it on the sight of Algebra with the relation of roots and coefficients. A quadratic hyperbolic curve group has a group condition such that (x²+cx-a) is a non-quadratic residue, which means $$(\frac{x^{2}+cx-a}{p})=-1$$ by using Jacobi’s symbol. This condition is unique on the group theory, included for satisfying the closure of the group operation and conserving the definition area.

Next, I try to take up the example of a simple quadratic hyperbolic curve group, which has a modulo p=11 on the residual ring Z/11Z with a curve parameters a=7,b=2,c=0 on the function of $$y=\frac{x-2}{x^{2}-7} (mod 11)$$.

This function has the points as like (x,y), that (5,2),(6,10),(9,5), (8,3), (3,6),(2,0) are all just on its function in this case. The operation is done by the two times crossing of the function with the line, and especially second time crossing is took between the fixed point O on the function and the line. The fixed point O becomes the zero element by a theory of the quadratic hyperbolic curve group, and this case already selected such O=(2,0) in accordance with the curve parameters. When the base point is selected, for example like P=(5,2), the group addictive operation (+) is used to make the calculation of mP=P+P+,,,+P. Because the quadratic hyperbolic curve group is a kind of a cyclic group, there exist positive number k, that satisfy kP=O, and this k is called the rank of the group. In reality, P=(5,2); 2P=(6,10); 3P=(9,5); 4P=(8,3); 5P=(3,6); 6P=(2,0) are resulted by the group addictive operation. Well, the residual ring Z/11Z has the 11 points in it, but as seen above only 6 points are appeared for the condition of (x²-7) is a non-quadratic residue, and the rest of 5 points have the condition of (x²-7) is a quadratic residue. If (x²-7) is a quadratic residue, such x can not treated with the group operation and also can not be executed. That is the reason why the quadratic hyperbolic curve group has deserted such elements.