Elliptic curve cryptosystems are analogs of public-key cryptosystems such as RSA and ElGamal, in which modular multiplication is replaced by the elliptic curve addition operation.

The curves used in elliptic curve analogs of discrete logarithm cryptosystems are normally of the form

y2 = x3 + ax + b (mod p),

where p is prime. The problem tapped by the discrete logarithm analogs in elliptic curves is the elliptic curve logarithm problem, defined as follows: given a point G on an elliptic curve with order r (number of points on the curve) and another point Y on the curve, find a unique x (0 x r - 1) such that Y = xG, i.e., Y is the xth multiple of G.