Author here, let me know if there are any questions or comments!
could you correct anything wrong in my understanding?
so we can create an isomorphism(?) between the field (Z_61, +, *) and points on a modular elliptic curve with a base point P using function g:= g(k) = k * P
g(k) is fast to compute with the doubling method, but the inverse requires brute force. Even if you know k_a * P and k_b * P, computing k_a * k_b * P is hard.
However, if you know k_a or k_b (either private key) you can easily find k_a * g(k_b) = k_b * g(k_a) = k_a * k_b * P.