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.

翻译：在2022年的研究中，Kuang等人提出了多元多项式公钥（MPPK）密码学，利用乘法与除法之间的逆运算关系构建量子安全公钥系统。他们将MPPK扩展为同态多项式公钥（HPPK），采用同态加密实现大规模隐藏环运算。最初为密钥封装（KEM）设计的HPPK，其安全性依赖于对公钥多项式的同态加密。本文将HPPK KEM扩展为数字签名方案，面临验证与解密本质差异带来的挑战。为将HPPK KEM适配至数字签名，作者引入Barrett约简算法的扩展版本，将素数域验证方程中的模乘法转化为除法运算。该扩展算法将签名非线性地嵌入公钥多项式系数中，解决了早期MPPK DS方案存在的安全缺陷。安全性分析表明，在环比特长度两倍于素数域尺寸的条件下，私钥恢复与伪造签名攻击均呈现指数级计算复杂度。