Proof of a Parallel Mixed-Radix Conversion Algorithm for Accelerating Fully Homomorphic Encryption

The widespread adoption of Fully Homomorphic Encryption (FHE) has been hindered by a critical computational bottleneck: the sequential Mixed-Radix Conversion (MRC). This paper resolves that inefficiency by introducing a novel parallel MRC algorithm. Our methodology involves a fundamental re-derivation of the MRC equations to algebraically eliminate all sequential dependencies, which transforms the problem into an independent summation structure. We formally prove the algorithm's mathematical soundness via induction and demonstrate that its time complexity is $O(\log k)$, marking a fundamental improvement over the traditional $O(k^{2})$ complexity. By providing a clear and accelerated path to convert encrypted data, this work removes a critical performance bottleneck, enabling the real-time commercial deployment of FHE solutions.