text: Projective coordinates A close examination of the addition rules shows that in order to add two points, one needs not only several additions and multiplications in  but also an inversion operation. The inversion (for given  find  such that ) is one to two orders of magnitude slower than multiplication. However, points on a curve can be represented in different coordinate systems which do not require an inversion operation to add two points. Several such systems were proposed: in the projective system each point is represented by three coordinates  using the following relation: , ; in the Jacobian system a point is also represented with three coordinates , but a different relation is used: , ; in the López–Dahab system the relation is , ; in the modified Jacobian system the same relations are used but four coordinates are stored and used for calculations ; and in the Chudnovsky Jacobian system five coordinates are used . Note that there may be different naming conventions, for example, IEEE P1363-2000 standard uses "projective coordinates" to refer to what is commonly called Jacobian coordinates. An additional speed-up is possible if mixed coordinates are used.
  Fast reduction (NIST curves)
  Reduction modulo p (which is needed for addition and multiplication) can be executed much faster if the prime p is a pseudo-Mersenne prime, that is ; for example,  or  Compared to Barrett reduction, there can be an order of magnitude speed-up. The speed-up here is a practical rather than theoretical one, and derives from the fact that the moduli of numbers against numbers near powers of two can be performed efficiently by computers operating on binary numbers with bitwise operations.
  The curves over  with pseudo-Mersenne p are recommended by NIST. Yet another advantage of the NIST curves is that they use a = −3, which improves addition in Jacobian coordinates.
  According to Bernstein and Lange, many of the efficiency-related decisions in NIST FIPS 186-2 are suboptimal. Other curves are more secure and run just as fast.
  Security