Complexity of inverting the Euler function

We present an algorithm to invert the Euler function φ(m). The algorithm, for a given integer n ≥ 1, in polynomial time “on average”, finds the set Ψ(n) of all solutions m to the equation φ(m) = n. In fact, in the worst case the set Ψ(n) is exponentially large and cannot be constructed by a polynomial time algorithm. In the opposite direction, we show, under some widely accepted number theoretic conjecture, that the problem of deciding whether φ(m) = n for some m is NP-complete. Finally, we establish close links between the problem of inverting the Euler function and the integer factorisation problem.