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

Understanding Ethereum Elliptic Curve Cryptography (ECC)

Elliptic Curve Cryptography (ECC) is a type of public-key cryptography that uses the properties of elliptic curves to secure online transactions. In this article, we will delve into the specifics of ECC in Bitcoin and examine how it compares to Ethereum’s curve.

Bitcoin Elliptic Curve

Bitcoin’s ECC implementation is based on the secp256k1 curve (also known as ElGamal). This curve was chosen for its simplicity and efficiency. Here is an illustration of what a secp256k1 curve might 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 equation of the curve is “y^2 = x^3 + px + q”, where “(x, y)” are points on the curve.

Ethereum Elliptic Curve

Ethereum, like Bitcoin, also uses secp256k1 for its ECC implementation. However, the Ethereum curve has a slightly different formulation:

E(2)
/ \
C(0) ~
\ /
G

The key difference is that the private key C is not explicitly listed in the diagram. Instead, it appears as “C = (k x^3 + py)^-1″, where “(x, y)” are points on the curve and “k” is a constant.

Comparison with Bitcoin Curve

While both secp256k1 curves have similar properties, there are some differences:

• Private Key Ordering: In Ethereum, the private key C appears in a specific order when listed as “[C, G]”, while in Bitcoin it only shows G.

• Value of Constant k: The constant k used to calculate the public key in Ethereum has a different value than the one used in Bitcoin.

• Point Representation: In the Ethereum diagram, the points on the curve are represented by “E(2)”, while in the Bitcoin diagram it is simply “G”.

Conclusion

In conclusion, both Bitcoin and Ethereum use secp256k1 ECC for their implementation of elliptic curve cryptography. Although the curves have similar properties, there are some differences in the way they are constructed and represented. Understanding these differences can provide a deeper insight into the underlying mathematics of cryptographic systems.

Additional Resources

For further reading on ECC and its applications:

• The Ethereum White Paper (2014) provides an overview of Ethereum’s ECC implementation.

• The Bitcoin Green Paper (2009) includes a description of Bitcoin’s ECC algorithm.

• Research papers by prominent developers, such as Vitalik Buterin and Gavin Wood, offer in-depth analysis and insights into the mathematics behind ECC.

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