On elliptic curves, one can define an addition law which provides a group structure.
This enables us to do cryptography on such curves by computing the scalar product of a point P by an integer n: n.P = P + ... + P.
One question that arises is how to do such multiplication efficiently?
Montgomery proposed in 1987 an algorithm to efficiently compute the x-coordinate of n.P knowing the x-coordinate of P, this is known as Montgomery ladder.
In this talk, we will first introduce elliptic curves and how they can be used in cryptographic protocols, like key exchange.
In a second part we will talk about Montgomery xz-coordinates and how we adapted Montgomery ladder into a "hybrid ladder" to benefit from faster formulas and get a faster scalar multiplication.