Question

In the secp256k1 elliptic curve, what is the exact mathematical operation performed on a private key to derive the corresponding public key?

Accepted Answer

In the secp256k1 elliptic curve, the private key is a randomly selected large integer, referred to as the scalar. The public key is derived by performing elliptic curve point multiplication, which is the process of adding the curve&#x27;s base point to itself a specific number of times, determined by the private key. The base point, often called the generator point, is a fixed coordinate on the curve defined by the secp256k1 standard. Mathematically, the public key P is calculated as P = d * G, where d is the private key scalar and G is the base point. Point addition is not standard arithmetic addition but a specific geometric operation where two points on the curve are added to produce a third point on the same curve. To perform this, you draw a line through the two points, find the third point where the line intersects the curve, and reflect that point across the x-axis. Because the private key is a very large number, this operation is performed using an algorithm called double-and-add, which allows the calculation to be completed efficiently in logarithmic time. This mathematical relationship is a one-way function, meaning that while it is computationally trivial to calculate the public key from the private key, it is computationally infeasible to derive the private key from the public key due to the elliptic curve discrete logarithm problem.

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In the secp256k1 elliptic curve, what is the exact mathematical operation performed on a private key to derive the corresponding public key?