Inspired by the concept of fault tolerance quantum computation, this article proposes a framework dubbed Exact Homomorphic Encryption, EHE, enabling exact computations on encrypted data without the need for pre-decryption. The introduction of quantum gates is a critical step in constructing the message encryption and the computation encryption within the framework. Of significance is that both encryptions are respectively accomplished in a multivariate polynomial set generated by quantum gates. Two fundamental traits of quantum gates, the invertibility and the noncommutativity, establish the success of EHE. The employment of invertible gates allows exact decryptions for both an encrypted message and encrypted computation. The encrypted computation is exact as well because its encryption transformation is conducted with invertible gates. The second trait of noncommutativity among applied quantum gates brings forth the security for the two encryptions. In the message encryption, a plaintext is encoded into a ciphertext via a polynomial set generated by a product of noncommuting gates randomly chosen. Toward the computation encryption, a desired operation is encoded into an encrypted polynomial set generated by another product of noncommuting gates. The encrypted computation is then the evaluation of the encrypted polynomial set on the ciphertext and is referred to as the cryptovaluation. EHE is not only attainable on quantum computers, but also straightforwardly realizable on traditional computing environments. Surpassing the standard security 2^128 of quantum resilience, both the encryptions further reach a security greater than the suggested threshold 2^1024 and are characterized as hyper quantum-resilient. Thanks to the two essential traits of quantum gates, this framework can be regarded as the initial tangible manifestation of the concept noncommutative cryptography.

翻译：受容错量子计算概念的启发，本文提出了一种名为精确同态加密（EHE）的框架，该框架能够在无需预先解密的情况下对加密数据执行精确计算。量子门的引入是框架内消息加密与计算加密构建中的关键步骤。重要的是，这两种加密分别由量子门生成的多项式集合完成。量子门的两个基本特性——可逆性与非对易性——奠定了EHE的成功基础。可逆门的应用使得加密消息与加密计算均可实现精确解密。加密计算本身也是精确的，因为其加密变换通过可逆门执行。第二个特性，即所用量子门之间的非对易性，为两种加密提供了安全性保障。在消息加密中，明文通过随机选择的非对易门乘积生成的多项式集合编码为密文。在计算加密中，目标操作通过另一组非对易门乘积生成的加密多项式集合进行编码。加密计算即对密文执行加密多项式集合的评估，称为密码评估。EHE不仅可在量子计算机上实现，也能直接在传统计算环境中部署。超越量子抗性标准安全性2^128，两种加密的安全性进一步超过建议阈值2^1024，被定性为超量子抗性。得益于量子门的两个本质特性，该框架可被视为非对易密码学概念的首次具体实现。