The article explores the mathematical and cryptographic properties of the secp256k1 elliptic curve and its relationship with another curve, secq256k1, forming a unique 2-cycle structure. Below is a summarized explanation:
Key Points
- Secp256k1 Curve: Defined by the equation $$y^2 = x^3 + 7$$ over the finite field $$\mathbb{F}_p$$, where $$p$$ is a large prime. It has an order $$q$$, another prime. Secp256k1 is widely used in Bitcoin and other cryptographic applications due to its efficiency and predictable design[1][3][4].
- Secq256k1 Curve: This curve uses the same equation but is defined over $$\mathbb{F}_q$$ (the scalar field of secp256k1). Its order is $$p$$, forming a 2-cycle with secp256k1. This cycle is useful for cryptographic implementations like proof-carrying data under discrete-log assumptions[2].
- Surprising Isomorphism: The two curves, although defined over different fields, share the same group order. This is unusual because random elliptic curves typically have orders within a large Hasse interval, making such coincidences rare[2].
- Origins of Secp256k1: Published by Certicom, secp256k1 belongs to Koblitz curves, which allow efficient scalar multiplication using endomorphisms. Certicom likely searched through finite fields to find a prime-order curve with desired properties[2][3].
- CM Curves: Both secp256k1 and secq256k1 are examples of CM (Complex Multiplication) curves. These curves exhibit a simple relationship between field characteristics and group orders, enabling efficient searches for specific properties[2].
- Implications: While these isomorphic curves do not pose security risks to secp256k1, their shared properties could be exploited for performance improvements in cryptographic systems like Zcash’s Halo 2 proofs. However, practical use is limited by the inefficiency of computing the actual isomorphism[2].
Summary
Secp256k1 and secq256k1 form a unique 2-cycle due to their shared equation and group orders over distinct finite fields. This structure arises from their classification as CM curves and has potential applications in cryptography, particularly in proof systems like Halo 2. Despite their theoretical significance, practical exploitation remains challenging due to computational limitations in deriving efficient isomorphisms[2][3].
Citations:
[1] https://en.bitcoin.it/wiki/Secp256k1
[2] https://hackmd.io/@dJO3Nbl4RTirkR2uDM6eOA/Bk0NvC8Vo
[3] https://www.nervos.org/knowledge-base/secp256k1_a_key%20algorithm_(explainCKBot)
[4] https://river.com/learn/terms/s/secp256k1/
[5] https://www.cyfrin.io/blog/elliptic-curve-digital-signature-algorithm-and-signatures
[6] https://bitcoin.stackexchange.com/questions/21907/what-does-the-curve-used-in-bitcoin-secp256k1-look-like
[7] https://www.johndcook.com/blog/2018/08/21/a-tale-of-two-elliptic-curves/
[8] https://learnmeabitcoin.com/technical/cryptography/elliptic-curve/