Correctness proof for a Ring-Learning-with-Errors multi-authority Ciphertext-Policy Attribute-Based Encryption scheme

The advent of quantum computing poses a significant threat to traditional cryptographic algorithms, including RSA, Diffie-Hellman, and Elliptic Curve Cryptography, due to the capabilities of quantum algorithms like Shor’s algorithm. Post-quantum cryptography (PQC) has emerged to address these challenges, with lattice-based cryptography (LBC) being a prominent candidate. LBC, underpinned by hard mathematical problems like Learning with Errors (LWE) and Ring-LWE (RLWE), offers robust security against quantum and classical adversaries. In parallel, Ciphertext-Policy Attribute-Based Encryption (CPABE) has become a critical tool for enabling fine-grained access control in data-sharing scenarios, such as secure cloud storage and IoT. While existing CP-ABE implementations rely on bilinear pairings vulnerable to quantum attacks, lattice-based CPABE schemes provide a quantum-resistant alternative. Despite their potential, these schemes face challenges in computational efficiency, collusion resistance, and implementation correctness. Our contributions include a detailed mathematical breakdown of one of the state-of-the-art (SOTA) lattice-based CP-ABE schemes and a novel correctness proof for the same scheme.