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Given a key $e = 3$ (the public exponent for RSA) and a group of numbers, say $Z_n^*$ (the multiplicative group of $n = pq$, where $p$ and $q$ are primes as requested by RSA).
I can find the decryption key: $d = e^{-1}$
Is it correct to say than any member $a$ in $Z_n^*$ has a source such that: $source = a ^ d$?
Edit: suppose we have gcd(3,phi(n)) = 1 as requested by RSA
If you're asking whether the power map $x\mapsto x^a \pmod n$ is injective, it suffices to take $a$ with $1 < a < \phi(n)$ and $\gcd(a,\phi(n)) = 1$.
Edit: I couldn't complete the answer earlier. As mentioned in the comments $gcd(a,n)$ is enough, but for computational efficiency one might as well take $a \pmod n$ as the exponent. In the RSA setup, $n$ is square free thus the function is bijective.
To see this, note that after all $e,d$ have similar roles and if one of them, say $e$, satisfies $\gcd(e,n)=1,$ then there is an integer $f$ such that $ed+fn=1$ (by the fact that they are inverses) thus demonstrating $gcd(d,n)=1.$)
