Exact Homomorphic Encryption
Zheng-Yao Su, Ming-Chung Tsai
TL;DR
Problem: enabling exact computations on encrypted data without decryption under quantum adversaries. Approach: encode messages and computations into multivariate polynomials via invertible, noncommuting gates; IME for messages and cryptovaluation for computations, with sectional cryptovaluation enabling parallelism. Key contributions: exactness for both message and computation encryptions, security backed by NP-hardness of multivariate equations and noncommutativity (surpassing $2^{128}$ and $2^{1024}$), and hardware-backed experimental validation on CPU and GPU. Impact: provides a concrete, scalable, noncommutative cryptographic framework executable on classical hardware, with potential for secure multi-party computation and privacy-preserving data processing.
Abstract
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.