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6

It happens that a line usually (not always) cuts three points in a elliptic curve by the Bezout theorem. This is the case for the points and the curve you are asking for. So the sum of two points are defined like the inverse of the third point intercepted by the line that cut $P$ and $Q$ (let's name it $R$). So we need to find $-R$ because $P+Q=-R$ by ...


6

It is equivalent to the computational Diffie-Hellman problem; if you can one of the two problems, you can solve the other (with a polynomial number of queries to the oracle which solves the other). If you can solve the Diffie-Hellman problem, you can solve your problem: this can be seen by first noting that, with a Diffie-Hellman solver, given $g^b$, you ...


4

Not at all. It's very trivial: $$(g^{ab})^{b^{-1}}=g^a$$


3

The question as currently worded, and considering comments by its author, would boil down to: is it a hard problem finding $g^a\bmod p$, given large prime $p$ large integer $g$ less than $p$ that is a generator of $\mathbb Z_p$ [see note 1] prime $r$ less than $p$ knowledge that unknown $a$ is a positive integer less than $r$ positive integer $b$ less than ...



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