Bitcoin’s secp256k1 Curve: Mathematical Foundations and Structure

Ever wondered what exactly makes secp256k1, the curve behind Bitcoin, so special?
In this video, we explore the full mathematical and structural story of the secp256k1 specification — from how public keys are represented to how the mysterious y-coordinate is recovered from just the x-value.

You’ll learn:

📘 What defines the secp256k1 elliptic curve (parameters, equation, field)

🔢 Different public key formats (uncompressed, compressed, hybrid — as per BIP-137)

🧭 How the y-coordinate is derived from the x-coordinate using the curve equation

♾️ The role of quadratic residues and modular square roots in determining the correct y-value

⚙️ The fast modular exponentiation algorithm that makes this efficient

🧠 General idea of digital signature verification and security hardness of public key

By the end, you’ll see how pure math meets practical engineering to define the foundation of every Bitcoin transaction.