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

2025-08-03

AI Summary

Constructing a graph function with provable one-wayness—where forward generation is efficient but inverse recovery of a hidden structure is computationally intractable—is a longstanding challenge in cryptography. Method: This paper proposes a Bloom-filter-based graph construction that implicitly embeds a clique of size $log n$, mapping clique vertices to Bloom filter bit positions via a black-box hash function. Edges are generated probabilistically to amplify the embedded clique while suppressing spurious ones, yielding a sparse graph containing a unique $log n$-clique. Crucially, the hash function remains unknown to the adversary. Contribution/Results: This work introduces the first application of Bloom filters to instantiate a strictly one-way graph function: forward construction runs in polynomial time, whereas recovering the hidden clique is proven to require at least EXPTIME under standard assumptions. It establishes a new paradigm and provides a rigorous theoretical lower bound for designing high-complexity one-way functions in cryptography.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Constructing graphs with Bloom filters and cliques
• Ensuring intractability of finding planted cliques
• Creating one-way functions for polynomial-time inversion

Innovation

• Bloom filter encodes edges via hash functions
• Planted clique ensures graph inversion intractability
• Polynomial-time graph construction with unique clique