# Clarify EC point addition and multiplication

$P$-Generator Point; $a$ and $b$ are integers; $X$ and $Y$ are EC points, defined as follows:

1. $X = (a*P) + (b*P)$
2. $Y = (a+b)*P$

Questions:

1. Are the points $X$ and $Y$ equal?
2. Does computing $X$ take 1 scalar point multiplication (i.e. 0.5 + 0.5),
whereas computing $Y$ takes 0.5 scalar point multiplikations?
3. Does $(a*b)*P$ take 1 scalar point multiplication?
4. How can an EC-point computation be (equal to?) 1 or 0.5 scalar point multiplications?
• 1) Yes, since "scalar multiplication" is exponentiation in a group, hence the rule $aP+bP=(a+b)P$ holds for all points $P$ and integers $a,b$. – yyyyyyy Jul 9 '15 at 15:59
• I guess you are talking about computational complexity? In that case, I suspect there may be something wrong with the information you give — "half" operations do not make too much sense. Where did you get those questions? – yyyyyyy Jul 9 '15 at 16:12
• I quickly edited your question to include mathmatical symbols, to use SE's enumerated lists and I slightly reformulated your text. If I changed the meaning of your questions you can either edit again (using the "edit" button) or you can roll-back my edit (by clicking on the "edited ... ago" and selecting the previous version). – SEJPM Jul 9 '15 at 18:35
• 3) $(ab)P$ is equal to $cP$ (ignoring the costs for the computation of $c=ab$, meaning it takes (roughly) as much time as computing $Y$. – SEJPM Jul 9 '15 at 18:56