# For discrete elliptic curves, can you find G, if you are given b and B?

I know you cannot find $$b$$ if you are given $$B$$ and $$G$$, where $$B = [b]G$$,

but can you find $$G$$ given $$b$$ and $$B$$?

• If you know the order of $G$, call it $q$, you can compute $b'=b^{-1}\bmod q$ and then $b'\cdot B$ should give you $G$ (if I'm doing my math right, else someone will correct me in another comment / answer, I'm too tired right now for a full answer / guaranteed info) Oct 23, 2019 at 22:38

$$G = [b^{-1} \bmod q]B$$ where $$q$$ is the order of the group generated by $$G$$, assuming $$\gcd(b, q) = 1$$.
• Use the extended Euclidean algorithm to compute the Bézout coefficients $s$ and $t$ so that $s b + t q = \gcd(b, q)$; then $s$ is your answer. Or, use Fermat's little theorem and compute $b^{q - 2} \bmod q$. Oct 24, 2019 at 15:48