Distinguishing a truncated random permutation from a random function

An oracle chooses a function f from the set of n bits strings to itself, which is either a randomly chosen permutation or a randomly chosen function. When queried by an n-bit string w, the oracle computes f(w), truncates the m last bits, and returns only the first n −m bits of f(w). How many queries does a querying adversary need to submit in order to distinguish the truncated permutation from a random function? In 1998, Hall et al. [2] showed an algorithm for determining (with high probability) whether or not f is a permutation, using O(2 m+n 2 ) queries. They also showed that if m < n/7, a smaller number of queries will not suffice. For m > n/7, their method gives a weaker bound. In this manuscript, we show how a modification of the method used by Hall et al. can solve the porblem completely. It extends the result to essentially every m, showing that Ω(2 m+n 2 ) queries are needed to get a non-negligible distinguishing advantage. We recently became aware that a better bound for the distinguishing advantage, for every m < n, follows from a result of Stam [3] published, in a different context, already in 1978.