assume $a$, $b$ are two integer numbers, and $G$ is a basepoint in an elliptic curve.
Is $(a+b)G$ equal to $aG+bG$ or not?
2026-04-06 08:41:07.1775464867
Addition on elliptic curves
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Yes. Point addition on an elliptic curve is an associative operation, and this rule is a consequence of that. If $a,b>0$ then $$ aG=G+G+\cdots+G, $$ where the sum has $a$ copies of $G$. The claimed equation means that $$ (G+G+\cdot+G)_a+(G+G+\cdots+G)_b=(G+G+\cdots+G)_{a+b}, $$ where the subscript indicates the number of summands. The identity then follows from our ability to move the parens around, i.e. associativity. If one or both of $a,b$ are negative, then you use $-G$ instead.
Associativity is needed for things like $aG$ to make sense in the first place. After all, we have only the addition of two points available as a primitive operation. Luckily associativity gives us everything we need. For example $3G$ could be either $$ (G+G)+G\qquad\text{or}\qquad G+(G+G), $$ and associtivity saves the day by declaring that the outcomes of those two calculations are equal.