I'm confused as to how he got (62, 44) as an answer within this article: https://www.coindesk.com/math-behind-bitcoin

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    $\begingroup$ What is your missing point? Is it point addition on the Elliptic Curves or the double-and-add algorithm work? $\endgroup$ – kelalaka Jan 13 at 7:38
  • $\begingroup$ Hi new user and welcome. Please make your title as descriptive as possible and try and tag your question the best way you can. ECDSA is not encryption. $\endgroup$ – Maarten Bodewes Jan 13 at 15:08

To obtain (62, 4) you just add points $\textbf{but on elliptic curves}$.

This is different from a "regular" addition, since the result must be a point of the curve (or a point said to be at infinity, I'm not explaining I try to keep things simple) .

Addition is defined and to do so either you use the heavy addition formulas (If you have seen groups theory), either you do as explain in the coindesk article :

let $P$ and $Q$ be two points on the curve to do $(P) + (Q)$ you trace the line passing by both $(P)$ and $(Q)$.

This line will intersect the curve on a third point, call it $(R')$, then $(P) + (Q) = (-R')$ and you have your result,

i.e. if $(R') = (x, y)$ then $(-R') = (x, -y)$. (In the coindesk article $(-R') = (R)$)

This is an $\textbf{incomplete}$ explanation but you should catch the thing, I didn't discuss the special points (when the last coordinate's member is $0$), inflexion points (when $P = Q = R$), or why there is another point on both the line and the curve.

If you want to double a point $(P)$ you trace the tangent passing by $(P)$, and this tangent will pass by a second point on the curve and you take the opposite point.

Personally I never use the explicite formulas (I didn't write them here) , which are too heavy in my opinion, I just do it with the last method.

For the first part you just break $76G$ in additions and doubles of $G$. Then you replace $G$ by its coordinates $(2, 22)$ and you just double or add.

For example: $2.(2, 22) = (2, 22) + (2, 22) = (52, 7)$ using your curve.

To check your results, or to get it easily, you can use : http://christelbach.com/ECCalculator.aspx an online calculator for elliptical curves.

You just have to set the parameters of your curve ($mod p = 67$, $A = 0$, $B = 7$) and you will get the correct answer (I checked).


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