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Applying the scalar multiplication to points on elliptic curves over finite fields works the same way as over real numbers. The difference is that over a finite field, the operation can actually be - and it is - used for some real-world cryptography. If we look at the operation in detail, we also find some interesting features which may help or complicate things a bit when it comes to implementing such cryptography. This week we show how it works.

As we are heading towards the actual cryptography using elliptic curves, we need to define an operation that allows us to combine numbers and points. This operation is called scalar multiplication and allows us to - yes - multiply a point by a scalar value (that is - a number). Using this technique, we are getting closer to the ultimate goal of using elliptic curves for some cryptography to secure our online communication.

Last week, we have talked about certain situations where the operation performed with points of the elliptic curve does not produce a result which is a valid point of the elliptic curve over real numbers. In this article we present a different approach to explaining the point at infinity using visualizations of the projective plane. Once you can see it, you will find it rather easy to understand.

Setting aside the reasoning why we are building an algebraic system using points of elliptic curves (we will get into that later on), we omitted some bad things that can happen when applying the rules for point addition and doubling. We did this on purpose as it would be too confusing for some readers. But now the time has come and we can look at these problematic cases to find something new: the point at infinity.

After the introduction of the first two simple point operations on elliptic curves in simple Weierstrass form, we can now look at some more interesting operations available to us. Last of the three "primitive" operations specified for points of the elliptic curve is the point doubling operation. It should be the same as if we wanted to sum not two distinct but rather two equal points. As usually, I would like to encourage anyone to give me a feedback, especially about the visualizations used.