# About the elliptic cryptographic algorithm

Issuing time:2018-09-20 00:00

The digital currency based theblock chain technique is called digital crypto-currency.  The economic module based on the block chaintechnique is called encryption economic. It indicates the importance ofcryptographic algorithm in block chain technology.

Let us look at theapplications of the cryptographic algorithm in the block chain.

1）、thecalculation of block hash value

2）、Merkel hashtree calculation

3）、Keycryptographic signature transaction

4）、Theownership of verification transactions

And so on

The first and second ones arethe applications of hash algorithm. The purpose of hash algorithm is to computea fixed-length summary of the input character string, similar as the ID numberof data.  This can be used to verify the integrity  of data. But hash algorithm is not our focusof discussion in this article. Let’s look at the third, fourth and fifthitem.  The realization of these threefunctions is closely related to a cryptographic algorithm. This cryptographicalgorithm is dissymmetry key cryptographicalgorithm .

What is dissymmetry keycryptographic algorithm?  Let’s look atthe traditional cryptographic algorithm, such as AES, DES, Triple DES, RC2,RC4, RC5 and

Blowfish etc, whichcan be used to encrypt and decrypt.  Although the algorithm process is different,but there is one thing in common that the secret key is symmetry.  It means that the key used for encryption isused for decryption at the same time. Take an example, the key we use to lock the door can open the door also.  This cryptographic algorithm, whoseencryption key and decryption key is the same one is called symmetry key cryptographic algorithm.  Obviously the key of this symmetry keycryptographic algorithm absolutely can’t be leaked out. In the opposite, thereis another kind called dissymmetry key cryptographic algorithm, whoseencryption and decryption key are different and separated.  One of the keys is public key, which can be opento the public.  Another secret key isprivate key, which has to be kept well. Because of the key open to the public, the dissymmetry key cryptographicis also called as public key algorithm.  What’s the feature of this public keyalgorithm.

One of the functionis used for ID verification.  Forexample,      a person A encrypted a piece of data (also known as a signature) with hisprivate key then he send to the internet. After receiving the data, the recipient need to verify if the data weresent by A (The data might be tampered by someone during the internet transmission) . The recipient decrypts the data through the public key exposed by A. It isproved that the data were not tampered if decrypted successfully.  It indicates that the data are not theoriginal ones sent by A, if the recipient found the decryption error.

The application of publicalgorithm has mainly two types RSA and ECC. The principle of RSA is to use the calculationdifficulty of decomposition of large prime numbers as well as thearithmetic difficulty of discrete logarithms.  EEC is the ellipticcurve algorithm introduced by us. In the block chain, from bit coin to ethereum and the others, they all useelliptic curve algorithm basically.

Let’s check the definition of the elliptic curve.

This is the standard equationof the elliptic curve. There are different curvilinear figures according todifferent parameters setup, as the following diagram probably:

Figure 1-1 elliptic curvefigure

As we can see, this figure hasno relation with the ellipse.  Why thisis called as an ellipse because its equation is very similar to the ellipsecircumference formula, and that’s it. Elliptic line is divided into real number fieldand finite field. The so-called realnumber filed means the point coordinates on the ellipse are in the range of thereal number.  But the elliptical linesused inencryption algorithm are based onfinite fields, and we always use the following equations.

As shown in the formula, weprocessed modulo calculation to the curve inthe real number filed, which limits the range of the points on the curve.  Through the modulo P calculation, the rangeof the points is 0~p~1.  Hereby let’slearn about the modulo calculation.  Infact, the modulo calculation is the complementarycalculus.  Let’s see a simpleexample.  Taking modulo 5 as theexamples, the percentage sign is used as the modulo operator according to thegrammar of the general programming language.

0 % 5 = 0

1 % 5 = 1

2 % 5 = 2

3 % 5 = 3

4 % 5 = 4

5 % 5 = 0

6 % 5 = 1

7 % 5 = 2

8 % 5 = 3

9 % 5 = 4

10 % 5 = 0

......

As shown above, the modulocalculation is a periodic cycle.  Formodulo 5 calculation, the result are always repeated between 0 to 4.  For another, to cryptographic algorithm, theprocess of encryption and decryption must be precise calculation, so theinteger point is chose as the usual point. Because of the feature of modulo calculation, once we choose P, Itsresult range is fixed, in which we can choose one point as the base point.  After selecting the base point, we canoperate scalar multiplication to thispoint.  It generates a periodic resultalso when we process scalar multiplication to a chosen base point on theelliptic curve.  Let’s check a sample:

Taking y2 = x3 + 2x + 3 forexample, moreover p=5, select one point (3,4) on the finite field.

Let’s look at theresult of the multiplication

n = 1 (3,4)

n = 2 (3,1)

n = 3 (infinity)

n = 4 (3,4)

n = 5 (3,1)

n = 6 (infinity)

...

At this moment, n=3 isthe “stage” of this finite field.

The multiplication of numberslike this is in the form of G=nP

In the elliptic curvecryptographic algorithm this “stage” is just like a secret key.  If course, this is only a simple example thatthe digit is not so large and the range is small.  We will design an elliptic curve equation meticulously,and adopt the digit with a large range.  Ifwe only know G and P, it’s very difficult to figure out n.  This difficult problem is called discrete logarithm problem.

Bit coin and ethereumincluding other kinds of block chain systems, adopt elliptic curve signaturealgorithm, the only difference is that the parameter of the curve equation usedmay be different.  For instance, bit coinadopts Secp256k1, while wisdom chain use ed25519, or another secret key shorterthan ed25519 but with more cryptographic strength and higher cryptographic computing performance.

As long with thedevelopment of quantum computing, the cryptographic algorithm will be updatedcontinuously.  In the future techniqueplanning, the wisdom chain will upgrade series of anti-quantum computation, soas to ensure the security of the future network.

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