Definition
Elliptic curve point multiplication, also called scalar multiplication, is the core operation of every ECC system. Given a point P on the curve and an integer scalar k, it computes Q = kP by repeatedly adding the point to itself. In Bitcoin, the scalar is your private key and P is the fixed generator point G, so your public key is simply Q = kG.
Double-and-add
Naively adding G to itself k times would take astronomically many steps, since k can be up to roughly 2²⁵⁶. Instead, implementations use the double-and-add algorithm, which walks the binary representation of k, doubling the running point at each bit and adding G only where a bit is set. This reduces the work from linear to logarithmic, so a full 256-bit multiplication needs only a few hundred curve operations. Production libraries harden this further with constant-time and windowed variants so that timing never leaks the secret scalar.
The one-way property
Point multiplication is easy to compute forward but believed infeasible to reverse: recovering k from Q and P is the discrete logarithm problem. That asymmetry is exactly what lets you publish a public key without exposing the private key behind it. Each result also stays inside the curve's finite field by construction.
Every signature your wallet produces and every address it derives ultimately reduces to one or more of these scalar multiplications, making it the single most-performed cryptographic step in Bitcoin.
In Simple Terms
Elliptic curve point multiplication, also called scalar multiplication, is the core operation of every ECC system. Given a point P on the curve and an…