How can I solve congruence modulo N?
$J_A \cdot a^e \: \equiv \: 1 \:\: \pmod{n} \;\;\;\;\; \iff \;\;\;\;\; a^e \: \equiv \;\; $$\operatorname{modinv}$$(J_A,\hspace{-0.02 in}n) \:\: \pmod{n}$

Since that is the RSA problem, the fastest known way to solve it is to factor $n$ which reveals $\lambda$$(n)$,

and then try $\;\;\; a \: = \: \operatorname{mod}\left(\hspace{-0.03 in}(\operatorname{modinv}(J_A,\hspace{-0.02 in}n))^{\operatorname{modinv}(e,\hspace{.02 in}\lambda(n))},n\hspace{-0.03 in}\right) \:\:\:\:$.