Consider graphs of n nodes and use a Bloom filter of length 2 log3 n bits. An edge between nodes i and j, with i < j, turns on a certain bit of the Bloom filter according to a hash function on i and j. Pick a set of log n nodes and turn on all the bits of the Bloom filter required for these log n nodes to form a clique. As a result, the Bloom filter implies the existence of certain other edges, those edges (x, y), with x < y, such that all the bits selected by applying the hash functions to x and y happen to have been turned on due to hashing the clique edges into the Bloom filter.
Constructing the graph consisting of the clique selected edges and those edges mapped to the turned on bits of the Bloom filter can be performed in polynomial time in n.
Choosing a large enough polylogarithmic in n Bloom filter yields that the graph has only one clique of size log n, the planted clique.
When the hash function is black-boxed, finding that clique is intractable and, therefore, inverting the function that maps log n nodes to a graph is not (likely to be) possible in polynomial time.