# Calculating the point on the curve during ECDSA signature verification

I'm confused as to how he got (62, 44) as an answer within this article: https://www.coindesk.com/math-behind-bitcoin • What is your missing point? Is it point addition on the Elliptic Curves or the double-and-add algorithm work? – kelalaka Jan 13 at 7:38
• 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. – Maarten Bodewes Jan 13 at 15:08

## 1 Answer

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).