Understand the cryptography of the Ethereum elliptical curve (ECC)
The cryptography of the elliptical curve (ECC) is a kind of cryptography of the public key that uses the properties of elliptical curves to ensure online transactions. In this article, we will deepen ECC details in Bitcoin and examine how it compares to the Ethereum curve.
Bitcoin elliptical curve
The Bitcoin ECC implementation is based on the SECP256K1 curve (also known as Elgamal). This curve is chosen by its simplicity and performance. Here is an illustration of how the SECP256K1 curve can be:
`''
E (2)
/
C (0) ~
\ /
GRAM
''
In this diagram and represents the infinity point in the curve (special point "point"), c is a private key (secret number), and G is a public key (digital signature). The curve equation isy^2 = x^3 + px + q, where (x, y)'are points in the curve.
Ethereum elliptical curve
Ethereum, as Bitcoin, also uses SECP256K1 to implement ECC. However, the Ethereum curve has a slightly different preparation:
''
E (2)
/
C (0) ~
\ /
GRAM
''
The key difference is that the private code C is not clearly listed in the diagram. Instead, it appears asC = (k x^3 + p y)^-1, where (x, y)are points in the yk curve is a fixed one.
comparison with the Bitcoin curve
While both curves SECP256K1 have similar properties, there are some differences:
* Private Key Order : In etherum, private code C appears in a certain order, when it is mentioned as [c, g], while in Bitcoins only G.
* Fixed value
: permanent K used to calculate the public key in Ethereum has a different value than the one used in Bitcoin.
* The point representation : In the Ethereum scheme, the points in the curve are represented by “E (2), while the bitcoin scheme is simplyG`.
Application
To sum up, both Bitcoin and Ethereum use SECP256K1 ECC to implement the cryptography of the elliptical curve. While curves have similar properties, there are some differences in the way they are built and represented. Understanding these differences can ensure a deeper vision of the mathematics of cryptographic systems underlying mathematics.
Additional resources
To read more about ECC and its applications:
- The Ethereum Warmia (2014) contains an review of the implementation of ECC Ethereum.
- The role of Bitcoin Green (2009) contains a description of the ECC Bitcoin algorithm.
- Outstanding research work of programmers, such as Vitalik Buterin and Gavin Wood, offer analysis and information about mathematics for ECC.