Ethereum: What does the curve used in Bitcoin, secp256k1, look like?

Understanding Ethereum’s Elliptic Curve Cryptogrady (ECC)

Elliptic Curve Cryptogram (ECC) is a type of public-key cryptogram, that uses the properties of eliptic curves. In this article, we’ll delve in the specification of ECC in Bitcoin and explore hand huming to Etherum’s curve.

Bitcoin’s Elliptic Curve

Bitcoin’s implementation of ECC is a based on the curve session (also painting as ElGamal). This is hosen for its simplicity and efficience. Here’s an a ilustration of what a secp256k1 curve could look like:


E(2)

  / \

C(0) ~

  \ /

   G

In this diagram, E represents the point at infinity on the curve (a special « zero » point), C is the private key (the secret number), and G is the public key (the digital signature). The curve’s equation is y^2 = x^3 + px + q, where (x, y)de points on the curve.

Ethereum’s Elliptic Curve

Ethereum, like Bitcoin, also uses secp256k1 for its ECC implementation. Howver, Ethereum’s curve has a slightly different formulation:


E(2)

  / \

C(0) ~

  \ /

   G

The Key difference is the private key C is not explicitly listed in the diagram. Instaed, it appares as C = (k x^3 + py)^-1, wree (x, y) arere points on the curve and k is a daystant.

Comparison with Bitcoin’s Curve

While both secp256k1 curves has similar properties, there areomis differences:

Private Key ordering: In Ethereum, the private is C appars in a specification in a specific wen listed as [C, G], whereas in G.

Constant k value: The constant k used to call the public key in Ethereum

Point repressionation**: In Ethereum’s dyagram, the points on the curve are represented by E(2), whitecoin’s diagram, it is i simply.

Conclusion*

In conclusion, both Bitcoin and Ethereum use secp256k1 ECC for their implementation of Curve Cryptogram. While the curves has a similar properties, thee are semi differences in well they are constructed and repressed. Understanding thee differences can provide a deeper insight in the underlying mathematics of cryptographic systems.