International Association
for Cryptologic Research

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. 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

Omega ( 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