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 for 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 encrypted computation is exact because its encryption transformation is conducted with invertible gates. In the same vein, decryptions for both an encrypted message and encrypted computation are exact. The second trait of noncommutativity among applied quantum gates brings forth the security for the two encryptions. Toward the message encryption, a plaintext is encoded into a ciphertext via a polynomial set generated by a product of noncommuting gates randomly chosen. In 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"（Exact Homomorphic Encryption, EHE）的框架，能够在无需预先解密的情况下对加密数据进行精确计算。在该框架中，量子门的引入是构建消息加密与计算加密的关键步骤。重要的是，这两种加密分别通过量子门生成的多项式集合实现。量子门的两个基本特性——可逆性与非交换性——奠定了EHE成功的基石。由于加密变换采用可逆门执行，加密计算具有精确性。同理，对加密消息与加密计算的解密过程亦为精确的。所用量子门的非交换性则为两种加密提供了安全保障。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，期望操作通过另一组非对易门乘积生成的加密多项式集合编码。加密计算即为对密文执行加密多项式集合的求值过程，称为密码求值（cryptovaluation）。EHE不仅可在量子计算机上实现，亦能直接部署于传统计算环境。两种加密的安全性超越了标准量子抗性2^128，进一步达到建议阈值2^1024以上，被定性为超量子抗性（hyper quantum-resilient）。得益于量子门的两个本质特性，该框架可被视为非交换密码学（noncommutative cryptography）概念的首个具体实现。