I will demonstrate techniques to derive the addition law on an arbitrary elliptic curve. The derived addition laws are applied to provide methods for efficiently adding points. The contributions immediately find applications in cryptology such as the efficiency improvements for elliptic curve scalar multiplication and cryptographic pairing computations. In particular, contributions are made to case of the following five forms of elliptic curves: (a) Short Weierstrass form, y^2 = x^3 + ax + b, (b) Extended Jacobi quartic form, y^2 = dx^4 + 2ax^2 + 1, (c) Twisted Hessian form, ax^3 + y^3 + 1 = dxy, (d) Twisted Edwards form, ax^2 + y^2 = 1 + dx^2y^2, (e) Twisted Jacobi intersection form, bs^2 + c^2 = 1, as^2 + d^2 = 1. These forms are the most promising candidates for efficient computations and thus considered in this talk. Nevertheless, the employed methods are capable of handling arbitrary elliptic curves.

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