What Is the Mathematical Relationship between a Private Key and a Public Key?

The public key is derived from the private key using a one-way function based on Elliptic Curve Digital Signature Algorithm (ECDSA). Specifically, the private key is multiplied by a defined generator point on the elliptic curve to produce the public key.

This operation is easy to compute in one direction (private to public), but computationally infeasible to reverse (public to private). This asymmetry is the core of Bitcoin's cryptography.