I am implementing a cryptography scheme which involves verifiying some data through the following process:
Suppose party A wants to verify data held by party B
Party A has: $a^x mod\ N$
Party B has: $x^{-1}$ i.e. modular inverse of $x$ with respect to some $p$ such that $xx^{-1} \equiv 1\ mod\ p$
To carry out the verification, B has to use $a^x \bmod N$ and $x^{-1}$ to attempt creating $a\bmod N$
If $x^{-1}$ is correctly calculated, then the data held by B is verified.
My question is, how can I, using $a^x \bmod N$ and modular inverse $x^{-1}$ attempt to generate $a\bmod N$