text: Domain parameters To use ECC, all parties must agree on all the elements defining the elliptic curve, that is, the domain parameters of the scheme. The size of the field used is typically either prime (and denoted as p) or is a power of two (); the latter case is called the binary case, and this case necessitates the choice of an auxiliary curve denoted by f.  Thus the field is defined by p in the prime case and the pair of m and f in the binary case. The elliptic curve is defined by the constants a and b used in its defining equation. Finally, the cyclic subgroup is defined by its generator (a.k.a. base point) G. For cryptographic application, the order of G, that is the smallest positive number n such that  (the point at infinity of the curve, and the identity element), is normally prime. Since n is the size of a subgroup of  it follows from Lagrange's theorem that the number  is an integer. In cryptographic applications, this number h, called the cofactor, must be small () and, preferably, . To summarize: in the prime case, the domain parameters are ; in the binary case, they are .
  Unless there is an assurance that domain parameters were generated by a party trusted with respect to their use, the domain parameters must be validated before use.
  The generation of domain parameters is not usually done by each participant because this involves computing the number of points on a curve which is time-consuming and troublesome to implement. As a result, several standard bodies published domain parameters of elliptic curves for several common field sizes. Such domain parameters are commonly known as "standard curves" or "named curves"; a named curve can be referenced either by name or by the unique object identifier defined in the standard documents:
  NIST, Recommended Elliptic Curves for Government Use
  SECG, SEC 2: Recommended Elliptic Curve Domain Parameters
  ECC Brainpool (), ECC Brainpool Standard Curves and Curve Generation
  SECG test vectors are also available. NIST has approved many SECG curves, so there is a significant overlap between the specifications published by NIST and SECG. EC domain parameters may be specified either by value or by name.