inverse problem about scalar multiplication on koblitz curves (or more exactly the secp256k1)

Question

My problem is given $Q=nP$ to find point $P$ given 257 bits long integer $n$ and point $Q$.
It’s something possible on other curves but Koblitz curves have extra characteristic and can’t be converted to Montgomery or Edwards curves.
P is a fixed point while n and thus Q can vary leading to several examples using the same unknown P.

So is such operation possible on the secp256k1 curve given a 237 bits integer n ? If yes, how to actually compute it in full ?

Answer 1

You can basically change $Q = n * P$ into: $P = Q / n$

which can be calculated as $P = Q * (1/n)$

where $1/n$ can be calculated as the modular inverse of n modulo the group order.

see here for an example.

in the example:

$1/123456789123456789123456789123456789$ (mod grouporder) == $0x9d8bc8ae304d195849e2173c96fe386c8f9ab51a6173af44b1e7d325c889db87$

calculate P:

$P=G*123456789123456789123456789123456789$

the inverse calculation:

$P/123456789123456789123456789123456789 == P*0x9d8bc8ae304d195849e2173c96fe386c8f9ab51a6173af44b1e7d325c889db87$