blocksight 2021-04-23 20:24:21 阅读数:287

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mathematics
blockchain

As mentioned earlier, cryptography is the cornerstone of blockchain , No cryptography , Blockchain is a castle in the air , It's hard to exist . The cornerstone of cryptography is mathematics . In the last section, we introduced the equation of elliptic curve , This section mainly deals with some knowledge of number theory and addition operation on elliptic curve .

First, let's talk about the basis of group theory . A group in an algebra is simply a set of elements and operations defined on them . For example, all integers form a group , Operations include addition and so on . The set here uses G（group） Express , The set is going to be a group , Generally, the following properties should be satisfied ：1. Sealing property ： If a and b All belong to G aggregate , that a+b Also belong to G;

2. Associative law ：(a+b)+c=a+(b+c)

3. There are unit elements （* In binary operations , A unit element is an element whose value is not changed by operation with any element , Take real numbers for example , The unit of multiplication is 1, The unit of addition is 0*）O bring a+O=O+a=a ;

4. Every element has an inverse element , That is to say, for any element a There must be b bring a+b=O(O It's unit yuan ) .

A set satisfying the above four properties is called a group , There are also some special groups , Such as Abelian group （Abelian Group） Besides satisfying the basic properties of groups , It also satisfies the law of exchange, that is ：a+b=b+a So Abelian group is also called Abelian group . According to these properties , We can know the set of integers Z It's an Abelian group , Set of natural numbers N But it's not a group , Because it doesn't satisfy the fourth property . Okay , The basic knowledge of group theory comes here first （ It seems simple , There are also many complex features that are temporarily omitted ）. It should be noted that , Set elements in a group , It can be numbers , It can also be other types of elements , For example, resolving the midpoint of a set （ Coordinate form ） etc. .

With the basic knowledge of groups , We can further define groups on elliptic curves in a similar way . Just now we said that the elements of a group can be of any type ,** A group element on an elliptic curve is a point on an elliptic curve ** The unit element is infinite, and the point is denoted as O（ For infinity, please refer to the previous article Projective plane and elliptic curve ）. Any point P The inverse of is this point about x Axisymmetric point . The addition of elliptic curve groups is also different from that of integers , Its additive definition can be described as ： Given three collinear nonzero points P ,Q,R , Their sum is P+Q+R=O. The geometric meaning is ： too 𝑃、𝑄 Make a straight line at two 𝐿, Intersect the elliptic curve at the third point , The point is about X The point of symmetry of the axis is what is sought 𝑅 spot . The addition of elliptic curve has a clear geometric meaning . As shown below ：

Here are some exceptions ：1 𝑂+𝑂=𝑂, For any 𝑃, Yes 𝑃+𝑂=𝑃;𝑂 See zero 2 𝑃=(𝑥,𝑦) The negative element of is about X The point of symmetry in −𝑃=(𝑥,−𝑦)（ Not about the origin symmetry ）,𝑃+(−𝑃)=𝑂, Can be seen as P And -P The line intersects the elliptic curve at infinity 3 Calculation 𝑃 spot （𝑃≠𝑂） Twice as much as , It's tangent to that point , Then take the intersection of tangent and elliptic curve 𝑆 About X The point of symmetry of the axis −𝑆, That is to say 2𝑃=𝑃+𝑃=−𝑆, obtain 2 The multiple value can be recursively reduced to several times .

It can be seen that , Point set of elliptic curve （ Including infinity O） And the addition operations defined above form an Abelian group ： The unit is 𝑂 spot ,𝑃(𝑥,𝑦) The inverse element of is 𝑃(𝑥,−𝑦), Sealing property , Associativity and exchangeability are also obviously satisfied .

Geometric interpretation is convenient to understand the meaning of elliptic curve point addition , Algebraic interpretation is easier to operate . Over the curve 𝑃(,) and 𝑄(,) At two o 'clock （𝑃 and 𝑄 It's not negative for each other ） Make a straight line , The problem of finding the third point of intersection with a curve can be easily described by algebraic methods .

That is to ask for ：

Where the slope 𝑘=

take （2） Plug in （1） It is easy to find the symmetry point of the third intersection by using the method of degree alignment , That is to say 𝑃,𝑄 The sum of the 𝑅(,) by ：

=−−=−+𝑘(−) If P=Q, The sum of the two is the multiplication , Multiple points can be added repeatedly to get . for example 𝑃+𝑃=2𝑃=𝑅, When ≠0 when , Algebra is described as ：

=−2

=(−)−

Come here , The operation of elliptic curve group elements in real number field is basically clear , But is it enough to implement encryption and decryption on elliptic curve ？ The answer is not enough , But it's close . In the next section, we will introduce the group domain and operation of elliptic curve algorithm in cryptography .

Welcome to your attention , If you have any questions, please leave a message ！

版权声明：本文为[blocksight]所创，转载请带上原文链接，感谢。 https://netfreeman.com/2021/04/20210423201642203o.html

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