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    $\begingroup$ Because of crypto.stackexchange.com/questions/112006 points are sampled independently of any other points/scalars. $\endgroup$ Commented Jul 27, 2024 at 20:43
  • $\begingroup$ @user2284570: that the $P_i$ "are sampled independently of any other points/scalars" is fully compatible with my hypothesis that the order of the points is actually $\ell$. Yes it's desirable in the context. The second subparagraph after "because" explains a method to generate the points. $\endgroup$ Commented Jul 27, 2024 at 20:49
  • $\begingroup$ So you are proposing a solution different from using the discrete logarithm, but would it be easier to compute than using the discrete logarithm? $\endgroup$ Commented Jul 27, 2024 at 21:41
  • $\begingroup$ @user2284570: Yes I'm considering solutions different from solving the DLP, including one that could be very easy if "only the first time and not at each computation" of the problem statement allows reuse of $P_1$, $P_2$, $P_3$ across several instances of the problem. This is a summary of the newly added summary. In the future, please strive to make questions accurate (e.g. "randomly sampled points on a curve" is missing of large prime order) and unambiguous (I misunderstood what's reused). $\endgroup$ Commented Jul 27, 2024 at 22:44
  • $\begingroup$ Points are fixed constants generated only 1 time. Only the scalars vary. $\endgroup$ Commented Jul 27, 2024 at 22:50