Homomorphic Polynomial Public Key Cryptography for Quantum-secure Digital Signature

In their 2022 study, Kuang et al. introduced Multivariable Polynomial Public Key (MPPK) cryptography, leveraging the inversion relationship between multiplication and division for quantum-safe public key systems. They extended MPPK into Homomorphic Polynomial Public Key (HPPK), employing homomorphic encryption for large hidden ring operations. Originally designed for key encapsulation (KEM), HPPK's security relies on homomorphic encryption of public polynomials. This paper expands HPPK KEM to a digital signature scheme, facing challenges due to the distinct nature of verification compared to decryption. To adapt HPPK KEM to digital signatures, the authors introduce an extension of the Barrett reduction algorithm, transforming modular multiplications into divisions in the verification equation over a prime field. The extended algorithm non-linearly embeds the signature into public polynomial coefficients, addressing vulnerabilities in earlier MPPK DS schemes. Security analysis demonstrates exponential complexity for private key recovery and forged signature attacks, considering ring bit length twice that of the prime field size.