scalar multiplication
Repeated point addition on an elliptic curve: `kP = P + P + ⋯ + P` (k times). Computing `kP` from `(k, P)` is fast — double-and-add finishes in `O(log k)` group operations. Recovering `k` from `(P, kP)` is the discrete logarithm problem and is believed to take exponentially many group operations. This asymmetry — easy to multiply, hard to undo — is the entire cryptographic content of every elliptic-curve scheme: keys, signatures, key exchange.
1844 (Grassmann), 1853 (Hamilton) · Hermann Grassmann · Stettin (then Prussia)
Grassmann's *Ausdehnungslehre* introduced what we now call vectors and linear combinations. Almost no one read it for 40 years. Hamilton's quaternions (1843) and later Gibbs/Heaviside (1880s) finally pushed vector notation into physics and engineering.