In Diffie-Hellman over $Z_p$, I know that if I choose a random $a$ in $Z_p$ and compute the modular inverse mod p-1, not mod p, then, with generator $g$ of my group, I can compute, for example, $(g^{ab})^{(a^{-1})} = g^{aa^{-1}b} = g^{b}$.

I am asking if a similar process is possible over elliptic curves? If I have a generator $P$ and I choose a random $a$ and compute $aP$. How do I do about calculating $a^{-1}$ such that $(aP)a^{-1} = P$?

  • $\begingroup$ Does this answer your question? elliptic curve multiplication with negative factor. AFAIK, both the question and answer will satisfy you. $\endgroup$
    – kelalaka
    May 3, 2020 at 1:59
  • $\begingroup$ You can use the extended euclidian algorithm to find k^-1. See bitcoin.stackexchange.com/questions/36382/… $\endgroup$
    – mti2935
    May 3, 2020 at 2:24
  • $\begingroup$ Yes, answered thank you. $\endgroup$
    – Joe
    May 3, 2020 at 6:32
  • $\begingroup$ @mti2935 actually, if you look at the link I've provided one doesn't need that. $n-k$ is enough. $\endgroup$
    – kelalaka
    May 3, 2020 at 13:00