Eidolon: A Practical Post-Quantum Signature Scheme Based on k-Colorability in the Age of Graph Neural Networks

2026-02-02

AI Summary

This work proposes a post-quantum digital signature scheme based on the NP-complete graph k-coloring problem to address the threat posed by quantum computing to classical cryptography. By generalizing the Goldreich–Micali–Wigderson zero-knowledge protocol, integrating the Fiat–Shamir transform with Merkle tree compression, and introducing an innovative “silent” coloring mechanism to generate cryptographically hard graph instances, the scheme achieves both efficiency and practicality. Notably, this study is the first to employ graph neural networks (GNNs) for evaluating the security of combinatorial cryptographic constructions and presents a method for generating k-coloring hard instances resilient against both classical algorithms and machine learning attacks. Experimental results demonstrate that neither integer linear programming solvers nor custom-designed GNNs can recover the secret coloring on graphs with n ≥ 60, confirming the scheme’s robustness against contemporary cryptanalytic techniques and revitalizing combinatorial problems as viable foundations for post-quantum cryptography.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Economic aspects of blockchain and cryptocurrencies

Abstract

We propose Eidolon, a practical post-quantum signature scheme based on the NP-complete k-colorability problem. Our construction generalizes the Goldreich-Micali-Wigderson zero-knowledge protocol to arbitrary k>= 3, applies the Fiat-Shamir transform, and uses Merkle-tree commitments to compress signatures from O(tn) to O(t log n). Crucially, we generate hard instances via planted"quiet"colorings that preserve the statistical profile of random graphs. We present the first empirical security analysis of such a scheme against both classical solvers (ILP, DSatur) and a custom graph neural network (GNN) attacker. Experiments show that for n>= 60, neither approach recovers the secret coloring, demonstrating that well-engineered k-coloring instances can resist modern cryptanalysis, including machine learning. This revives combinatorial hardness as a credible foundation for post-quantum signatures.

Problem

• post-quantum signature
• k-colorability
• combinatorial hardness
• graph neural networks
• cryptanalysis

Innovation

• post-quantum signature
• k-colorability
• graph neural network
• zero-knowledge protocol
• Merkle-tree commitment