# Is that possible to calculate modular inverse of a point on elliptic curves?

Imagine that you are given a point $$P$$ so that $$P=a\times G$$. If you have no knowledge of $$a$$ is that possible to calculate point $$I$$ so that $$I$$ is the modular inverse of $$P$$?

We know that over prime fields each member has a modular inverse, which means for $$a\in \Bbb F_n$$ there exists $$x$$ so that $$ax\equiv 1\pmod n$$. Now, My question is, is that possible to calculate point $$I$$ on the same curve as $$P$$ for which the equation $$I=x\times G$$ holds. This is to happen with no knowledge of either $$a$$ or $$x$$.

If such algorithm exists or become discovered in future will that be considered a weakness to the security of elliptic curves?

• Elliptic curve points do not have modular inverses. We can compute the additive inverse, $-P$, by negating the $Y$ coordinate of $P$ so that $-P + P$ is the identity element. It does not reveal anything about $a$. – user13741 Aug 2 '19 at 17:44

No, this is a known variation of the computational Diffie-Hellman problem, called Inverse computational Diffie-Hellman (InvCDH), according to which on input $$g$$, $$g^{x}$$, it is difficult to find $$g^{x^{-1}}$$.
• Will that reveal $a$ or $x$ if one can compute $I$ from $P$? – PouJa Aug 2 '19 at 19:30
• How would you compute $I$ without the knowledge of $a$ in the first place? It's even difficult to decide if $P$ and $I$ are made from a $x$ and it's inverse $x^{-1}$, respectively, in prime-order elliptic curves with large embedding degree (like secp256r1/k1), due to the Inverse decisional Diffie-Hellman assumption. – Kostas Chalkias Aug 2 '19 at 21:35