Efficient Non-interactive Zero-Knowledge Proof for Graph 3-Coloring Problem

Zero-knowledge proof (ZKP) has a crucial role in the construction of cryptographic protocols and privacy protection. One of the core research components of zero-knowledge proof is the NP-complete (NPC) problem. This paper focus on a classic NPC problem graph 3-coloring problem (3-GCP). Firstly, we propose a non-interactive zero-knowledge (NIZK) proof scheme for the 3-GCP. In our scheme, the prover generates a proof p for each edge based on the graph and the coloring scheme. The verifier then chooses whether to trust the provers' proof based solely on p. This is the non-interaction between the prover and the verifier. Moreover, we optimize this scheme for efficiency based on the idea of homomorphic encryption. It allows each execution of the scheme to prove a vertex in the graph. Finally, we present the security analysis and computational cost of our solution, which again demonstrates that our solution is feasible.