  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.
