Trimming: Decoupling Multiplicative Depth from Modulus Chains in RNS-CKKS via Rational Levels

2026-07-31

AI Summary

This work addresses the limitation in RNS-CKKS schemes where multiplication depth is rigidly constrained by a fixed modulus chain, hindering flexible control. To overcome this, the authors propose a novel Trimming mechanism that, for the first time, decouples multiplication depth from the modulus chain structure in RNS-CKKS. By introducing a rational-level abstraction and an auxiliary fine-grained modulus chain pruning strategy—combined with NTT-friendly small moduli and partial modulus switching techniques—the approach enables adaptive depth management and dynamic modulus switching. The method remains fully compatible with existing RNS-CKKS operations, effectively breaking through conventional depth bottlenecks and laying the foundation for a fully decoupled homomorphic computation architecture.

Abstract

Recent work on Grafting decouples scale factors from ciphertext moduli, enabling more flexible precision management in RNS-CKKS. However, the multiplicative depth remains fundamentally constrained by the modulus chain structure. In this paper, we propose \emph{Trimming}, a novel fine-grained level management mechanism that decouples multiplicative depth from modulus chains in RNS-CKKS via rational levels. The key idea is to introduce an auxiliary trimming modulus chain composed of smaller NTT-friendly modulus factors, which enables partial modulus transitions instead of directly discarding an entire modulus factor. By replacing conventional discrete level reductions with fine-grained modulus factor refinement, Trimming provides a rational-level abstraction beyond the traditional integer-based modulus chain representation. Our approach preserves the compatibility with existing RNS-CKKS arithmetic while enabling more flexible depth management and adaptive modulus transitions. Similar to Grafting, which addresses the precision bottleneck, Trimming targets the depth bottleneck in RNS-CKKS and contributes toward a fully decoupled RNS-CKKS architecture. The proposed framework will be further validated through concrete implementation and experimental evaluation to investigate its practical performance and computational overhead in real-world homomorphic encryption applications.

Problem

• multiplicative depth
• modulus chains
• RNS-CKKS
• homomorphic encryption
• depth bottleneck

Innovation

• Trimming
• RNS-CKKS
• multiplicative depth
• modulus chain
• rational levels