Homomorphic polynomial public key with the Barrett transformation for digital signature

In their 2022 study, Kuang et al. introduced the multivariable polynomial public key (MPPK) cryptography, which is a quantum-safe public key cryptosystem that leverages the inversion relationship between multiplication and division. MPPK uses multiplication for key pair construction and division for decryption, generating public multivariate polynomials. Kuang and Perepechaenko expanded this into the homomorphic polynomial public key (HPPK) by transforming product polynomials over large hidden rings using homomorphic encryption. Initially designed for key encapsulation mechanism (KEM), HPPK ensures the security of public polynomials over concealed rings through homomorphic encryption. This article extends HPPK for KEM (HPPK KEM) to a digital signature (DS) scheme. To adapt HPPK KEM for DSs, we introduce an extension of the Barrett reduction algorithm which transforms modular multiplications over hidden rings into divisions in the verification equation. This extension nonlinearly embeds the signature into public polynomial coefficients, employing the floor function of large integer divisions. Our proposed scheme addresses forgery attacks observed in previous MPPK DS schemes by leveraging dual hidden rings and the Barrett reduction algorithm. This method provides nonlinear encryption for the HPPK public key, preventing shortcuts other than brute-force searches. Integrating signature elements into public polynomial coefficients adds complexity to forged signature attacks, with the nonlinear Barrett transformation significantly enhancing security. A toy example illustrates the functionality of the HPPK DS scheme, and security analysis indicates it achieves exponential complexity for both private key recovery and forged signature attacks. Future research will benchmark performance and compare it with National Institute of Standards and Technology (NIST)-standardized algorithms.