Mathematics elliptic curve in blockchain

blocksight 2021-04-23 02:54:30 阅读数:532

mathematics elliptic curve blockchain

Write it at the front

As mentioned earlier, cryptography is the cornerstone of blockchain , No cryptography , Blockchain is a castle in the air , It's hard to exist . So what is the cornerstone of cryptography ? The answer is math . This section mainly talks about the background and basic properties of elliptic curve .

Projective plane

First of all, we have to talk about the projection plane , Because the standard square equation of elliptic curve is defined on the projective plane . What is a projective plane ? In a nutshell , It can be understood as an ordinary Euclidean plane plus an infinite point and an infinite line composed of infinite points . Why do this ? In doing so, there is a unity : Any two lines must intersect and there is only one intersection .

Let's think about what we learned in elementary geometry , Two parallel lines never intersect , But parallel lines never intersect. It's just an assumption , There's no way to strictly prove . This assumption is limited in explaining some complex problems . So we have infinity and projective plane geometry . The figure below assumes that parallel lines are at infinity P∞


A brief summary is as follows : Infinity : A straight line has only one infinite point , The two ends of a line intersect at infinity ( You can think of a straight line as a closed curve ), Two parallel lines can be regarded as intersecting at infinity , All parallel lines intersect at the same infinity . Infinite straight line : Also called ideal straight line . An imaginary straight line in the Euclidean plane , Is the set of infinity points on all lines in the projective plane . After introducing an infinite line in the plane , Every two planes in space have intersecting lines. A group of parallel planes intersect an infinite straight line belonging to the parallel planes .

Projective plane :2 Dimensional projective space . It can be regarded as an infinite line composed of an additional infinite point on the ordinary plane , It's algebraic geometry 、 The most basic object in projective geometry .

Projective plane coordinate system

The equation of a straight line in the ordinary Euclidean plane can be expressed as :ax+by+c=0 The above equation is modified , Make x=X/Z ,y=Y/Z(Z≠0), Then the new equation can be expressed as :aX+bY+cZ=0; This equation is the linear equation in the new photographic plane coordinate system . Illustrate with examples : spot (2,3) What are the coordinates in the new coordinate system ? introduce Z, Make X/Z=2 ,Y/Z=3(Z≠0) obtain X=2Z,Y=3Z So the new coordinates are (2Z:3Z:Z),Z≠0. namely (2:3:1)(4:6:2) Isomorphic (2Z:3Z:Z) The point of , You can see that the point of a single Euclidean plane expands into a projective line with the same slope . How do the new coordinates represent infinity ? Because infinity is defined as the intersection of parallel lines , One of the linear equations is :aX+bY+c1Z =0 The parallel equation of a line is aX+bY+c2Z =0( Because parallel lines have the same slope , so a,b identical ). Find the intersection of two lines , Combine the two equations , Available c2Z= c1Z= -(aX+bY) because c1!=c2, therefore Z=0, Infinity is expressed as (X,Y,0,), Ordinary point coordinates Z!=0, The infinite line equation Z=0. So the new coordinate system can represent all the points on the projective plane .

Elliptic curve equation

The standard elliptic curve equation is a three element secondary equation defined on the projective plane :


It's called the Weierstrass equation (Weierstrass), Satisfy :1 The elliptic curve equation is a homogeneous equation

2 Every point on the curve must be nonsingular ( smooth ), Partial derivative FX(X,Y,Z)、FY(X,Y,Z)、FZ(X,Y,Z) Different for 0

3 The shape of a circular curve , It's not elliptical . It's just because of the description equation of the elliptic curve , It is similar to the equation for calculating the circumference of an ellipse, so it is named

The next two are not elliptic curves ( It's not smooth at zero )


Elliptic curve ordinary equation :


In the above standard Weierstrass equation Z=1 Available , This is a two-dimensional curve equation of ordinary Euclidean plane . I said before. , A curve on a projective plane can be represented by a curve on an ordinary plane plus an infinite point , The infinity of the ordinary elliptic curve equation is (0, Y, 0). It can be seen that the projective plane coordinate system is downward compatible with the ordinary Euclidean plane coordinate system , let me put it another way , Euclidean plane coordinate is a special case of projective plane coordinate system (Z=1).

All right, that's it , Next time, let's talk about the operation of points on elliptic curves , Welcome to your attention !! If you have any questions, please leave a message