# "Add" points that are not on the same elliptic curve?

Assume elliptic curve in Weierstrass form.

$$y^2 = x^3 + a x + b$$ where $$x,y,a,b \in F$$

I noticed the point addition formula does not involve parameters $$a,b$$. Furthermore, one can always solve for $$a,b$$ given two distinct points.

Thus one can "add" 3 or more distinct points, as long as their coordinates are on $$F$$, without having them on the same curve. More formally, let $$O$$ be point of infinity, the set $$F \times F \cup O$$ is almost closed under point addition operation for short Weierstrass curve. (Almost closed since one has to "add" distinct points)

I did a quick check that associative law for groups no longer holds. What would be some interesting properties for the above setup?

• For $b$ yes, for $a$ no ($a$ needed for doubling). Could you support your claims? given two points to find $a$ and $b$ is here Commented Mar 21, 2022 at 14:41
• What if in $P+Q+R$, $Q+R = P$? now you turned to double. You may say, I can calculate $P+Q$ then add $R$. In this case, we can say, who is going to check these?. Instead, we prefer complete formulas if possible. Commented Mar 21, 2022 at 14:51
• You are mixin points of the curve and points of its quadratic twist. twist attack is based on this. Commented Mar 21, 2022 at 15:25
• Lack of associativity is a show-stopper in many cryptographic applications. Thus the question is about a solution looking for problem (and I don't know one).@kelalaka: performing point doubling the way you suggest requires associativity, which as pointed in the question does not hold.
– fgrieu
Commented Mar 21, 2022 at 15:55