Thank you all very much in advance!
I will explain best with an example what is not clear to me.
Consider the elliptic curve
a = 1,
b = 2,
p = 11 of order
Also let's consider the "opposite" (negations of each other) two points of this elliptic curve
(2, 1) and
(2, 10) =
(2, -1) as starting points. These are the points for which we will do the multiplication.
For both of these two points, the order is 8 (we can only get 8 out of 16 points if we multiply them). I took this example arbitrarily just to make it as small as possible.
(2, 1) we get the following points (next to the points are numbers that indicate the number by which (2, 1) must be multiplied to get it):
(2, -1) we get the following points (next to the points are numbers that indicate the number by which (2, -1) must be multiplied to get it):
Until multiplying by 4 (in both cases) everything is as I expect. Multiplication just gives opposite points (the negation of each other) since we are working with the opposite points. For example,
2 * (2, 1) gives
(8, 4) and
2 * (2, -1) gives
(8, -4) =
(8, 7) since it is the opposite (negation) of
(8, 4). Same for multiplying by 3. This works literally the same as in case of working with an elliptic curve over real numbers. So everything is clear.
However, what confuses me is that after multiplying with 4 we're starting to sort of go backwards.
For example, multiplying
(2, 1) by 5, we get
(5, 2), which is also the 3rd point when multiplying
(2, -1) (that is,
3 * (2, -1) = (5, 2)). This also applies in the opposite way. Multiplying
(2, -1) by 5, we get
(5, -2), which is also 3rd point when multiplying by
It turns out that we are actually going backwards, which is different from the case when we work with the real numbers where we would just keep going by the principle before multiplying by 4 forever, so that there would be no such "repetition" of points
Also, it looks like multiplying by 4, i.e. the point
(10, 0) actually represents that TURNING POINT. Better said, up to that point everything goes like with real numbers, however, from there, we start going backwards.
So it goes as expected until some points. After that points, it starts to go backwards until we reach the beginning, that is,
(2, -1) point in our example. In other words, until we make the whole circle, when everything starts from the beginning.
- Why is this happening? Why do we have this circular behavior?
- What must be specific about a point (in this example
(10, 0)) for it to be a turning point? I thought that, in this example,
4 * (2, 1)and
4 * (2, -1)must give the same lambda for the point to be a turning point (since that's the only difference between working on a finite field and on real numbers), but that's not the case , that is, the same lambda is not obtained.