Montgomery curve

In mathematics the Montgomery curve is a form of elliptic curve, different from the usual Weierstrass form, introduced by Peter L. Montgomery in 1987. It is used for certain computations, and in particular in different cryptography applications.

A Montgomery curve over a field K is defined by the equation

for certain A, B ∈ K and with B(A² − 4) ≠ 0.

Generally this curve is considered over a finite field K (for example, over a finite field of q elements, K = F_q) with characteristic different from 2 and with A ∈ K ∖ {−2, 2}, B ∈ K ∖ {0}, but they are also considered over the rationals with the same restrictions for A and B.

It is possible to do some "operations" between the points of an elliptic curve: "adding" two points ${\displaystyle P,Q}$ consists of finding a third one ${\displaystyle R}$ such that ${\displaystyle R=P+Q}$; "doubling" a point consists of computing ${\displaystyle [2]P=P+P}$ (For more information about operations see The group law) and below.

...
Wikipedia