How to remove power in a modular exponent expression?

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 $$a\bmod N$$

• Based on the cryptgraphic scheme detailed here: link.springer.com/chapter/10.1007/0-387-34805-0_20 – Tabish Mir Jan 28 at 7:20
• Given y, it's quick to get y inverse mod p by the euclidean algorithm on p, y. So take x inverse, compute x, then you know the exponent used – Artimis Fowl Jan 28 at 7:25
• x is a confidential value only to be known by B. As such, A attempts to regenerate a mod N using his version of x, a^x mod N and the corresponding x inverse. If 'x' held by B is valid, he should be able to successfully generate a mod N – Tabish Mir Jan 28 at 7:32