Towards EXPTIME One Way Functions: Bloom Filters, Succinct Graphs, Cliques, & Self Masking

TL;DR

This paper explores a construction of graphs using Bloom filters that encode large cliques, demonstrating a potential approach to creating one-way functions based on graph properties and hash functions.

Contribution

It introduces a novel method of constructing graphs with hidden large cliques using Bloom filters, linking graph intractability to hash function properties and one-way functions.

Findings

Constructs graphs with a unique large clique using Bloom filters.

Shows intractability of finding the clique when hash functions are black-boxed.

Proposes a connection between graph properties and cryptographic one-way functions.

Abstract

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.